Walker control file keywords

Keyword 'C0'

Set Langevin SDE parameter C0

This keyword is used to specify a real number used to parameterize the Langevin model for the fluctuating velocity in homogeneous variable-density turbulence. Example: "C0 2.1".

Expected type: real

Keyword 'C3'

Set gamma (dissipation) SDE parameter C3

This keyword is used to specify a real number used to parameterize the gamma distribution dissipation (turbulence frequency) model for particles Example: "C3 1.0".

Expected type: real

Lower bound: 0.000000

Keyword 'C4'

Set gamma (dissipation) SDE parameter C4

This keyword is used to specify a real number used to parameterize the gamma distribution dissipation (turbulence frequency) model for particles Example: "C4 0.25".

Expected type: real

Lower bound: 0.000000

Keyword 'COM1'

Set gamma (dissipation) SDE parameter COM1

This keyword is used to specify a real number used to parameterize the gamma distribution dissipation (turbulence frequency) model for particles Example: "COM1 0.44".

Expected type: real

Lower bound: 0.000000

Keyword 'COM2'

Set gamma (dissipation) SDE parameter COM2

This keyword is used to specify a real number used to parameterize the gamma distribution dissipation (turbulence frequency) model for particles Example: "COM2 0.9".

Expected type: real

Lower bound: 0.000000

Keyword 'S'

Set SDE parameter(s) S

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "S 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'T'

Set SDE parameter(s) T

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "T 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'accurate'

Select the accurate algorithm for uniform RNG

This keyword is used to select the accurate method used to generate uniform random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'standard' and 'accurate'.

Keyword 'b'

Set SDE parameter(s) b

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "b 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'beta'

Introduce the beta SDE input block

This keyword is used to introduce a beta ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), with linear drift and quadratic diagonal diffusion, whose invariant is the joint beta distribution. For more details on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843 and src/DiffEq/Beta/Beta.hpp. Keywords allowed in a beta ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'kappa'. For an example beta ... end block, see doc/html/walker_example_beta.html.

Keyword 'beta_method'

Select an Intel MKL beta RNG method

This keyword is used to specify the method used to generate beta random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'cja' and 'cja_accurate'.

Expected type: string

Expected valid choices: 'cja' | 'cja_accurate'

Keyword 'betapdf'

Configure a beta distribution

This keyword is used to specify the configuration of beta distributions for the beta initialization policy. The configuration is given by four real numbers inside a betapdf...end block. Example: "betapdf 0.2 0.3 0.0 1.0 end", which specifies a univariate beta distribution with shape parameters 0.2 and 0.3, displacement 0.0, and scale 1.0. See also the help on keyword icbeta.

Expected type: 4 reals

Keyword 'boxmuller'

Select the Box-Muller algorithm for sampling a Gaussian

This keyword is used to select the Box-Muller method used to generate Gaussian random numbers using the Intel Math Kernel Library (MKL) random random number generators. Valid options are 'boxmuller', 'boxmuller2', and 'icdf'.

Keyword 'boxmuller2'

Select the Box-Muller 2 algorithm for sampling a Gaussian

This keyword is used to specify the Box-Muller 2 method used to generate Gaussian random numbers using the Intel Math Kernel Library (MKL) random number generators.

Keyword 'bprime'

Set SDE parameter(s) bprime

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "bprime 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'c'

Set SDE parameter(s) c

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "c 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'centering'

Specify data-centering for PDF output

This keyword is used to select the data centering of the probability value output on the sample space grid for file output of probability density functions (PDFs). Example: "centering elem", which selects element-centered values. Valid options are 'elem' and 'node', denoting cell-centered and point-centered output, respectively.

Expected type: string

Expected valid choices: 'elem' | 'node'

Keyword 'cja'

Select the Cheng, Johnk, Atkinson algorithm for sampling a beta

This keyword is used to select the Cheng-Johnk-Atkinson method used to generate beta random numbers using the Intel Math Kernel Library (MKL) random number generators.

Keyword 'cja_accurate'

Select the accurate Cheng, Johnk, Atkinson algorithm for sampling a beta

This keyword is used to select the accurate version of the Cheng-Johnk-Atkinson method used to generate beta random numbers using the Intel Math Kernel Library (MKL) random number generators.

Keyword 'coeff'

Select the coefficients policy

This keyword is used to select a coefficients policy. This is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Expected type: string

Expected valid choices: 'const_coeff' | 'decay' | 'homogeneous' | 'homdecay' | 'montecarlo_homdecay' | 'hydrotimescale' | 'const_shear' | 'stationary' | 'inst_velocity'

Keyword 'const_coeff'

Select constant coefficients policy

This keyword is used to select the 'constant coefficients' coefficients policy. A coefficients policy is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const_coeff", which selects 'constant coefficients' coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'const_shear'

Select constant shear coefficients policy

This keyword is used to select the prescribed constant shear coefficients policy, used to compute a homogeneous free shear flow using the Langevin model. This policy (or model) prescribes a constant mean shear in the y direction and computes the dissipation of turbulent kinetic energy specifically for this flow. The flow is a fully developed homogeneous turbulent shear flow with a uniform mean velocity gradient in one direction (y) and the mean flow is in predominantly in the x direction. The flow is considered to be far from solid boundaries. See Pope, S.B. (2000). Turbulent flows (Cambridge: Cambridge University Press).

Keyword 'cov'

Set SDE parameter(s) cov

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "cov 4.0 2.5 1.1 32.0 5.6 23.0 end" The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'decay'

Select decay coefficients policy

This keyword is used to select the decay coefficients policy. This policy (or model) is used to constrain a beta stochastic differential equation so that its variance, <y^2>, always decays. A coefficients policy, in general, is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'default'

Select the default ASCII floating-point output

This keyword is used to select the 'default' floating-point output format for ASCII floating-point real number output. Example: "format default", which selects the default floating-point output. Valid options are 'default', 'fixed', and 'scientific'. For more info on these various formats, see http://en.cppreference.com/w/cpp/io/manip/fixed.

Keyword 'depvar'

Select dependent variable (in a relevant block)

Dependent variable, e.g, in differential equations.

Expected type: character

Keyword 'diag_ou'

Introduce the diagonal Ornstein-Uhlenbeck SDE input block

This keyword is used to introduce a diag_ou ... end block, where 'diag_ou' stands for diagonal Ornstein-Uhlenbeck' and is used to specify the configuration of a system of stochastic differential equations (SDEs), with linear drift and constant diagonal diffusion, whose invariant is the joint normal distribution. Keywords allowed in a diagou ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'sigmasq', 'theta', 'mu'. For an example diagou ... end block, see doc/html/walker_example_diagou.html.

Keyword 'dirichlet'

Start configuration block for the Dirichlet SDE

This keyword is used to introduce a dirichlet ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), whose invariant is the Dirichlet distribution. For more details on the Dirichlet SDE, see https://doi.org/10.1155/2013/842981. Keywords allowed in a dirichlet ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'kappa'. For an example dirichlet ... end block, see doc/html/walker_example_dirichlet.html.

Keyword 'dirichletpdf'

Configure a Dirichlet distribution

This keyword is used to specify the configuration of a Dirichlet distribution for the Dirichlet initialization policy. The configuration is given by a vector of positive real numbers inside a dirichletpdf...end block. Example: "dirichletpdf 0.1 0.3 0.2 end" - prescribe a Dirichlet distribution with shape parameters 0.1, 0.3, and 0.2. See also the help on keyword icdirichlet.

Expected type: reals

Keyword 'dissipation'

Introduce the (particle) dissipation equation input block or coupling

This keyword is used to introduce a dissipation ... end block, used to specify the configuration of a system of deterministic or stochastic differential equations, governing a particle quantity that models the dissipation rate of turbulent kinetic energy, used to coupled to particle velocity model, e.g, the Langevin, model. Note that the random number generator r123_philox is automatically put on the list as a selected RNG if no RNG is selected. Keywords allowed in a dissipation ... end block: 'depvar', 'rng', 'init', 'coeff', 'velocity', 'For an example dissipation ... end block, see doc/html/walker_example_dissipation.html. (2) To specify a dependent variable (by a character) used to couple a differential equation system, in which the 'dissipation' keyword appears) to another labeled by a 'depvar'.

Keyword 'dt'

Select constant time step size

This keyword is used to specify the time step size that used as a constant during simulation. Setting 'cfl' and 'dt' are mutually exclusive. If both 'cfl' and 'dt' are set, 'dt' wins.

Expected type: real

Lower bound: 0.000000

Keyword 'elem'

Specify elem-centering for output

This keyword is used to select elem-centering for variable output. In walker for example, this is used to configure probability values on the sample space grid for file output of probability density functions (PDFs). Example: "centering elem", which selects element-centered values. Valid options are 'elem' and 'node', denoting cell-centered and point-centered output, respectively. In inciter this keyword is used in output variable specification blocks, prefixing variable names by either 'node' or 'elem', to specify their centering for output to file.

Keyword 'end'

End of an input block

The end of a block is given by the 'end' keyword in the input file. Example: "rngs ... end".

Keyword 'eq_A005H'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.05, IC:light<<heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.05, IC: light << heavy.

Keyword 'eq_A005L'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.05, IC:light>>heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.05, IC: light >> heavy.

Keyword 'eq_A005S'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.05, IC:light=heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.05, IC: light = heavy.

Keyword 'eq_A05H'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.5, IC:light<<heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.5, IC: light << heavy.

Keyword 'eq_A05L'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.5, IC:light>>heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.5, IC: light >> heavy.

Keyword 'eq_A05S'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.5, IC:light=heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.5, IC: light = heavy.

Keyword 'eq_A075H'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.75, IC:light<<heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.75, IC: light << heavy.

Keyword 'eq_A075L'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.75, IC:light>>heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.75, IC: light >> heavy.

Keyword 'eq_A075S'

Select inverse equilibrium hydro time scale from DNS of HRT, A=0.75, IC:light=heavy

Inverse equilibrium hydrodynamics time scale from DNS of homogeneous Rayleigh-Taylor instability, tau_eq, A = 0.75, IC: light = heavy.

Keyword 'evolution'

Select PDF output policy evolution

This keyword is used to select the 'evolution' output file policy for requested probability density functions (PDFs) within a pdfs ... end block. Example: "policy evolution", which selects the evolution output file policy. The evolution policy output appends new time step to the same output file for each time instant, yielding a time evolution of data in a single file. Valid PDF policy options are 'overwrite', 'multiple', and 'evolution'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Keyword 'exodusii'

Select ExodusII output

This keyword is used to select the ExodusII output file type readable by, e.g., ParaView of either a requested probability density function (PDF) within a pdfs ... end block or for mesh-based field output in a field_output ... end block. Example: "filetype exodusii", which selects ExodusII file output. For more info on ExodusII, see http://sourceforge.net/projects/exodusii.

Keyword 'filetype'

Select output file type

This keyword is used to specify the output file type of a requested probability density function (PDF) within a pdfs ... end block or for mesh-based field output in a field_output ... end block. Example: "filetype exodusii", which selects ExodusII output. Valid options depend on which block the keyword is used: in a pdfs ... end the valid choices are 'txt', 'gmshtxt', 'gmshbin', and 'exodusii', in a field_output ... end block the valid choices are 'exodusii' and 'root'.

Expected type: string

Expected valid choices: 'txt' | 'gmshtxt' | 'gmshbin' | 'root' | 'exodusii'

Keyword 'fixed'

Select the fixed ASCII floating-point output

This keyword is used to select the 'fixed' floating-point output format for ASCII floating-point real number output. Example: "format fixed", which selects the fixed floating-point output. Valid options are 'default', 'fixed', and 'scientific'. For more info on these various formats, see http://en.cppreference.com/w/cpp/io/manip/fixed.

Keyword 'fluctuation'

Select fluctuation (as the dependent variable) to solve for

This keyword is used to select the fluctuation of a random variable as what quantity to solve for, i.e., use as the dependent variable, e.g., in a position or velocity model for a stochastic particle. This configures how statistics must be interpreted.

Expected type: string

Keyword 'format'

Specify the ASCII floating-point output format

This keyword is used to select the floating-point output format for ASCII floating-point number output. Example: "format scientific", which selects the scientific floating-point output. Valid options are 'default', 'fixed', and 'scientific'. For more info on these various formats, see http://en.cppreference.com/w/cpp/io/manip/fixed.

Expected type: string

Expected valid choices: 'default' | 'scientific' | 'fixed'

Keyword 'fullvar'

Select full variable (as the dependent variable) to solve for

This keyword is used to select the full random (instantaneous) variable as what quantity to solve for, i.e., use as the dependent variable, in, e.g., a position or velocity model for a stochastic particle. This configures how statistics must be interpreted.

Expected type: string

Keyword 'gamma'

Introduce the gamma SDE input block

This keyword is used to introduce the gamma ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), with linear drift and linear diagonal diffusion, whose invariant is the joint gamma distribution.

Keyword 'gamma_method'

Select an Intel MKL gamma RNG method

This keyword is used to specify the method used to generate gamma random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'gnorm' and 'gnorm_accurate'.

Expected type: string

Expected valid choices: 'gnorm' | 'gnorm_accurate'

Keyword 'gammapdf'

Configure a gamma distribution

This keyword is used to specify the configuration of gamma distributions for the gamma initialization policy. The configuration is given by two real numbers inside a gammapdf...end block. Example: "gammapdf 0.2 0.3 end", which specifies a univariate gamma distribution with shape and scale parameters 0.2 and 0.3, respectively. See also the help on keyword icgamma.

Expected type: 2 reals

Keyword 'gaussian'

Configure a Gaussian distribution

This keyword is used to specify the configuration of Gaussian distributions for the jointgaussian initialization policy. The configuration is given by two real numbers inside a gaussian...end block. Example: "gaussian 0.2 0.3 end", which specifies a Gaussian distribution with 0.2 mean and 0.3 variance. See also the help on keyword icgaussian.

Expected type: 2 reals

Keyword 'gaussian_method'

Select an Intel MKL Gaussian RNG method

This keyword is used to specify the method used to generate Gaussian random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'boxmuller', 'boxmuller2', and 'icdf'.

Expected type: string

Expected valid choices: 'boxmuller' | 'boxmuller2' | 'icdf'

Keyword 'gaussianmv_method'

Select an Intel MKL multi-variate Gaussian RNG method

This keyword is used to specify the method used to generate multi-variate Gaussian random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'boxmuller', 'boxmuller2', and 'icdf'.

Expected type: string

Expected valid choices: 'boxmuller' | 'boxmuller2' | 'icdf'

Keyword 'gendir'

Start configuration block for the generalized Dirichlet SDE

This keyword is used to introduce a gendir ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), whose invariant is Lochner's generalized Dirichlet distribution. For more details on the generalized Dirichlet SDE, see https://doi.org/10.1063/1.4822416. Keywords allowed in a gendir ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'c', 'kappa'. For an example gendir... end block, see doc/html/walker_example_gendir.html.

Keyword 'glm'

Select the generalized Langevin model for the velocity PDF model

This keyword is used to select the generalized Langevin model for the Lagrangian velocity in turbulent flows.

Expected type: string

Keyword 'gmshbin'

Select Gmsh binary output for outputing PDFs

This keyword is used to select the binary output file type readable by Gmsh of a requested probability density function (PDF) within a pdfs ... end block. Example: "filetype gmshbin", which selects Gmsh binary file output. Valid options are 'txt', 'gmshtxt', 'gmshbin', and 'exodusii'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox. For more info on Gmsh, see http://www.geuz.org/gmsh.

Keyword 'gmshtxt'

Select Gmsh ASCII output for outputing PDFs

This keyword is used to select the ASCII (text) output file type readable by Gmsh of a requested probability density function (PDF) within a pdfs ... end block. Example: "filetype gmshtxt", which selects Gmsh ASCII file output. Valid options are 'txt', 'gmshtxt', 'gmshbin', and 'exodusii'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox. For more info on Gmsh, see http://www.geuz.org/gmsh.

Keyword 'gnorm'

Select the GNORM (see MKL doc) algorithm for sampling a gamma

This keyword is used to select the GNORM method used to generate gamma random numbers using the Intel Math Kernel Library (MKL) random number generators.

Keyword 'gnorm_accurate'

Select the accurate GNORM (see MKL doc) algorithm for sampling a gamma

This keyword is used to select the accurate version of the GNORM method used to generate gamma random numbers using the Intel Math Kernel Library (MKL) random number generator.

Keyword 'gravity'

Set Langevin SDE parameter gravity

This keyword is used to specify a vector of 3 real numbers used to parameterize the Langevin model for the fluctuating velocity in homogeneous variable-density turbulence, prescribing a gravy body force in the three coordinate directions, x, y, z. Example: "gravity 0.0 0.2 1.0 end".

Expected type: 3 reals

Keyword 'heavy'

Select the heavy-fluid normalization for the mixture Dirichlet SDE

This keyword is used to select the heavy-fluid normalization for the mixture Dirichlet PDF/SDE model for multi-material mixing in turbulent flows.

Expected type: string

Keyword 'homdecay'

Select homogeneous decay coefficients policy

This keyword is used to select the homogeneous decay coefficients policy. This policy (or model) is used to constrain a beta stochastic differential equation so that its variance, <y^2>, always decays and its mean, <R> = rho2/(1+r<RY>/<R>), where Y = <Y> + y, does not change in time. Note that R = rho2/(1+rY). This policy is similar to 'montecarlo_homdecay', but computes the SDE coefficient S in a different but statistically equivalent way. While 'homdecay' only requires the estimation of statistics, <R>, <r^2>, and <r^3>, 'montecarlo_homdecay' requires <R^2>, <YR^2>, and <Y(1-Y)R^3>. A coefficients policy, in general, is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'homogeneous'

Select homogeneous coefficients policy

This keyword is used to select the homogeneous coefficients policy. This policy (or model) is used to constrain a Dirichlet stochastic differential equation so that its mean density stays constant. A coefficients policy, in general, is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'hydroproductions'

Set MixMassFractionBeta SDE parameter(s) productions

This keyword is used to specify a vector of strings used to parameterize the system of stochastic differential equations, configured in a mixmassfracbeta ... end block. Within the mixmassfracbeta ... end block the coefficients policy must be set to 'hydrotimescale' in order for the hydroproductions ... end block to be in effect. The 'hydroproductions' keyword is then used to specify a list of strings, each specifying which turbulent kinetic energy production dividied by the dissipation rate (P/eps) data (from direct numerical simulations) should be used for the particular component integrated. Available P/eps data are defined in src/DiffEq/HydroProductions.hpp. Example: "productions prod_A05S prod_A05H prod_A05L prod_A05S prod_A05S end", which configures five P/eps data sets associated to 5 components, i.e., 5 scalar stochastic differential equations, integrated, specified and configured within the given mixmassfracbeta ... end block. The length of the hydroproductions vector depends on the number of scalar components and is controlled by the preceding keyword 'ncomp'. For mixmassfracbeta, ncomp is the actual number of scalar components * 4, since mixmassfractionbeta always computes 4 additional derived stochastic variables (in a diagnostic) fashion. See also MixMassFractionBeta::derived() in src/DiffEq/MixMassFractionBeta.hpp. Keywords allowed in a hydroproductions ... end block: 'prod_A005H', 'prod_A005S', 'prod_A005L', 'prod_A05H', 'prod_A05S', 'prod_A05L', 'prod_A075H', 'prod_A075S', 'prod_A075L'. For an example hydroproductions ... end block, see doc/html/walker_example_mixmassfracbeta.html.

Expected type: string(s)

Keyword 'hydrotimescale'

Select hydro-timescale coefficients policy

This keyword is used to select the hydrodynamics-timescale coefficients policy. This policy (or model) is used to constrain a beta stochastic differential equation (SDE) so that its variance, <y^2>, always decays and its mean, <R> = rho2/(1+r<RY>/<R>), where Y = <Y> + y, does not change in time. Note that R = rho2/(1+rY). This policy is similar to 'homdecay' as well as 'montecarlo_homdecay', but instead of simply constraining b' and kappa' to ensure decay in the evolution of <y^2>, b' and kappa' are specified as functions of an externally-specified hydrodynamics time scale, as a function of time. This policy is more similar to 'homdecay' than to 'montecarlo_homdecay' in that only requires the estimation of statistics, <R>, <r^2>, and <r^3>. A coefficients policy, in general, is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'hydrotimescales'

Set MixMassFractionBeta SDE parameter(s) hydrotimescales

This keyword is used to specify a vector of strings used to parameterize the system of stochastic differential equations, configured in a mixmassfracbeta ... end block. Within the mixmassfracbeta ... end block the coefficients policy must be set to 'hydrotimescale' in order for the hydrotimescales ... end block to be in effect. The 'hydrotimescales' keyword is then used to specify a list of strings, each specifying which inverse time scale should be used for the particular component integrated. Available time scales are defined in src/DiffEq/HydroTimescales.hpp. Example: "hydrotimescales eq_A05S eq_A05H eq_A05L eq_A05S eq_A05S end", which configures five inverse hydrodynamics time scales associated to 5 components, i.e., 5 scalar stochastic differential equations, integrated, specified and configured within the given mixmassfracbeta ... end block. The length of the hydrotimescales vector depends on the number of scalar components and is controlled by the preceding keyword 'ncomp'. For mixmassfracbeta, ncomp is the actual number of scalar components * 4, since mixmassfractionbeta always computes 4 additional derived stochastic variables (in a diagnostic) fashion. See also MixMassFractionBeta::derived() in src/DiffEq/Beta/MixMassFractionBeta.hpp. Keywords allowed in a hydrotimescales ... end block: 'eq_A005H', 'eq_A005S', 'eq_A005L', 'eq_A05H', 'eq_A05S', 'eq_A05L', 'eq_A075H', 'eq_A075S', 'eq_A075L'. For an example hydrotimescales ... end block, see doc/html/walker_example_mixmassfracbeta.html.

Expected type: string(s)

Keyword 'icbeta'

Introduce an icbeta...end block used to configure beta distributions

This keyword is used to introduce an icbeta...end block in which beta distributions are configured for the beta initialization policy. Example: "init jointbeta" - select beta init-policy,"icbeta betapdf 0.2 0.3 0.0 1.0 end end" - prescribe a univariate beta distribution with shape parameters 0.2 and 0.3, displacement 0.0, and scale 1.0. See also the help on keyword jointbeta and betapdf.

Keyword 'icdelta'

Introduce a icdelta...end block used to configure delta spikes

This keyword is used to introduce a icdelta...end block in which delta spikes are configured for the delta initialization policy. Example: "init jointdelta" - select joint delta init-policy,"icdelta spike 0.1 0.3 0.9 0.7 end end" - prescribe a univariate distribution that consists of two delta-spikes at sample space positions 0.1 and 0.9 with spike heights 0.3 and 0.7, respectively. Note that the sum of the heights must add up to unity. See also the help on keyword jointdelta and spike.

Keyword 'icdf'

Use inverse cumulative distribution function for sampling a Gaussian

This keyword is used to specify the inverse cumulative distribution function (ICDF) method used to generate Gaussian random numbers using the Intel Math Kernel Library (MKL) random number generators.

Keyword 'icdirichlet'

Configure a Dirichlet PDF as initial condition

This keyword is used to introduce an icdirichlet...end block in which a Dirichlet distribution is configured for the Dirichlet initialization policy

Keyword 'icgamma'

Configure a gamma distribution as initial condition

This keyword is used to introduce an icgamma...end block in which gamma distributions are configured for the gamma initialization policy. Example: "init jointgamma" - select gamma init-policy,"icgamma gammapdf 0.2 0.3 end end" - prescribe a univariate gamma distribution with shape and scale parameters 0.2 and 0.3, respectively. See also the help on keyword jointgamma and gammapdf.

Keyword 'icgaussian'

Configure a joint uncorrelated Gaussian as initial condition

This keyword is used to introduce an icgaussian...end block in which Gaussian distributions are configured for the jointgaussian initialization policy. Example: "init jointgaussian" - select jointgaussian init-policy,"icgaussian gaussian 0.2 0.3 end end" - prescribes a univariate Gaussian distribution with 0.2 mean and 0.3 variance. See also the help on keyword jointgaussian and gaussian.

Keyword 'icjointgaussian'

Configure an joint correlated Gaussian as initial condition

This keyword is used to introduce an icjointgaussian...end block in which a multi-variate joint Gaussian distribution is configured for the jointgaussian initialization policy. Example: "init jointgaussian" - select jointgaussian init-policy, " icjointgaussian mean 0.0 0.5 1.0 end cov 4.0 2.5 1.1 32.0 5.6 23.0 end end" - prescribes a tri-variate joint Gaussian distribution with means 0.0, 0.5 and 1.0, and a covariance matrix which must be symmetric positive definite. See also the help on keyword jointgaussian and gaussian.

Keyword 'init'

Select initialization policy

This keyword is used to select an initialization policy. This is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init raw", which selects raw initialization policy, which leaves the memory uninitialized. Note that this option may behave differently depending on the particular equation or physical model. See the init policies in DiffEq/InitPolicy.hpp for valid options.

Expected type: string

Expected valid choices: 'raw' | 'zero' | 'jointdelta' | 'jointbeta' | 'jointgaussian' | 'jointcorrgaussian' | 'jointgamma'

Keyword 'inst_velocity'

Select the instantaneous velocity coefficients policy

This keyword is used to select the instantaneous velocity coefficients policy. This is used to prescribe a coupling for instantaneous velocity to some other differential equation, e.g., to update Lagrangian particle position or to couple a mix model to velocity.

Keyword 'interval'

Set interval (in units of iteration count)

This keyword is used to specify an interval in units of iteration count (i.e., number of time steps). This must be used within a relevant block.

Expected type: uint

Lower bound: 0

Keyword 'jointbeta'

Select the joint beta initialization policy

This keyword is used to select the joint beta initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The joint beta initialization policy can be used to prescribe a multi-dimensional sample space where the samples are generated from a joint beta distribution with independent marginal univariate beta distributions.

Keyword 'jointcorrgaussian'

Select the joint correlated Gaussian initialization policy

This keyword is used to select the joint correlated Gaussian initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The joint correlated Gaussian initialization policy can be used to prescribe a joint correlated Gaussian on the sample space with a given covariance matrix. Example: "init jointcorrgaussian icjointgaussian mean 0.0 0.5 1.0 end cov 4.0 2.5 1.1 32.0 5.6 23.0 end end"

Keyword 'jointdelta'

Select the joint delta initialization policy

This keyword is used to select the joint delta initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The joint delta initialization policy can be used to prescribe delta-spikes on the sample space with given heights, i.e., probabilities. Example: "init jointdelta" - select delta init-policy, "delta spike 0.1 0.3 0.8 0.7 end end" - prescribe two delta-spikes at sample space positions 0.1 and 0.8 with spike heights 0.3 and 0.7, respectively. Note that the sum of the heights must add up to unity. See also the help on keyword spike.

Keyword 'jointdirichlet'

Select the Dirichlet initialization policy

This keyword is used to select the Dirichlet initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The Dirichlet initialization policy can be used to prescribe a Dirichlet distribution on the sample space with given shape parameters. Example: "init jointdirichlet" - select the Dirichlet init-policy, "dirichletpdf 0.1 0.3 0.2 end" - prescribe a Dirichlet distribution with shape parameters 0.1, 0.3, and 0.2. All shape parameters must be positive.

Keyword 'jointgamma'

Select the joint gamma initialization policy

This keyword is used to select the joint gamma initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The joint gamma initialization policy can be used to prescribe a joint gamma distribution on the sample space with given shape and scale parameters. Example: "init jointgamma" - select the (joint) gamma init-policy, "gammapdf 0.1 0.3 end" - prescribe a gamma distribution with shape 0.1 and scale 0.3 parameters, respectively. Note that both shape and scale must be positive. Multiple independent gamma PDFs can be specified and the they will be used for the different scalar components configured for the equation. No correlations between the gamma distributions (as the initial conditions) are supported.

Keyword 'jointgaussian'

Select the joint Gaussian initialization policy

This keyword is used to select the joint Gaussian initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. For an example, see tk::InitPolicies in DiffEq/InitPolicy.hpp for valid options.) The joint Gaussian initialization policy can be used to prescribe a joint Gaussian (joint Gaussian) on the sample space with given variances. Example: "init jointgaussian" - select (joint) Gaussian init-policy, "gaussian 0.1 0.3 0.8 0.7 end" - prescribe two Gaussians with mean 0.1 and variance 0.3, and with mean 0.8 and 0.7, respectively. Note that the means can be any real number while the variances must be positive. No correlations between the Gaussians (as the initial conditions) are supported.

Keyword 'kappa'

Set SDE parameter(s) kappa

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "kappa 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'kappaprime'

Set SDE parameter(s) kappaprime

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "kappaprime 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'lambda'

Set SDE parameter(s) lambda

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "lambda 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'light'

Select the light-fluid normalization for the mixture Dirichlet SDE

This keyword is used to select the light-fluid normalization for the mixture Dirichlet PDF/SDE model for multi-material mixing in turbulent flows.

Expected type: string

Keyword 'massfracbeta'

Introduce the massfracbeta SDE input block

This keyword is used to introduce a massfracbeta ... end block, used to specify the configuration of a system of number-fraction beta SDEs, a system of stochastic differential equations (SDEs), in which, in addition to the dependent variable, computed with linear drift and quadratic diagonal diffusion (whose invariant is joint beta), two additional variables are computed. In other words, this is a beta SDE but there are two additional stochastic variables computed based on the beta SDE. If Y is governed by the beta SDE, then the mass-fraction beta SDE additionally governs rho(Y) and V(Y), where both rho and V are random variables, computed by rho(Y) = rho2 / ( 1 + r Y ), and V(Y) = ( 1 + r Y ) / rho2. For more details on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843 and src/DiffEq/Beta/Beta.hpp. Keywords allowed in a massfracbeta ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'kappa', 'rho2', 'r'. For an example massfracbeta ... end block, see doc/html/walker_example_massfracbeta.html.

Keyword 'mean'

Set SDE parameter(s) mean

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "mean 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'mean_gradient'

Set prescribed mean gradient

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "mean_gradient 1.0 1.0 0.0 end". One use of a mean gradient vector is to specify a prescribed mean scalar gradient in 3 spatial directions for a scalar transprot equation.

Expected type: real(s)

Keyword 'mixdirichlet'

Start configuration block for the Mixture Dirichlet SDE

This keyword is used to introduce a mixdirichlet ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), whose invariant is the Dirichlet distribution constrained to model multi-material mixing in turbulent flows. For more details on the Dirichlet SDE, see https://doi.org/10.1155/2013/842981. Keywords allowed in a mixdirichlet ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'kappa', 'rho2', 'r'. For an example mixdirichlet ... end block, see doc/html/walker_example_mixdirichlet.html.

Keyword 'mixmassfracbeta'

Introduce the mixmassfracbeta SDE input block

This keyword is used in multiple ways: (1) To introduce a mixmassfracbeta ... end block, used to specify the configuration of a system of mix mass-fraction beta SDEs, a system of stochastic differential equations (SDEs), whose solution is the joint beta distribution and in which the usual beta SDE parameters b and kappa are specified via functions that constrain the beta SDE to be consistent with the turbulent mixing process. The mix mass-fraction beta SDE is similar to the mass-fraction beta SDE, only the process is made consistent with the no-mix and fully mixed limits via the specification of the SDE coefficients b and kappa. As in the mass-fraction beta SDE, Y is governed by the beta SDE and two additional stochastic variables are computed. However, in the mix mass-fraction beta SDE the parameters b and kappa are given by b = Theta * b' and kappa = kappa' * <y^2>, where Theta = 1

• <y^2> / [ <Y> ( 1 - <Y> ], the fluctuation about the mean, <Y>, is defined as usual: y = Y - <Y>, and b' and kappa' are user-specified constants. Similar to the mass-fraction beta SDE, there two additional random variables computed besides, Y, and they are rho(Y) and V(Y). For more detail on the mass-fraction beta SDE, see the help on keyword 'massfracbeta'. For more details on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843 and src/DiffEq/Beta/Beta.hpp. Keywords allowed in a mixmassfracbeta ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'bprime', 'S', 'kappaprime', 'rho2', 'hydrotimescales', 'hydroproductions', 'velocity', 'dissipation', 'r'. For an example mixmassfracbeta ... end block, see doc/html/walker_example_mixmassfracbeta.html. (2) To specify a dependent variable (by a character) used to couple a differential equation system, in which the 'mixmassfracbeta' keyword appears) to another labeled by a 'depvar'.

Keyword 'mixnumfracbeta'

Introduce the mixnumfracbeta SDE input block

This keyword is used to introduce a mixnumfracbeta ... end block, used to specify the configuration of a system of mix number-fraction beta SDEs, a system of stochastic differential equations (SDEs), whose solution is the joint beta distribution and in which the usual beta SDE parameters b and kappa are specified via functions that constrain the beta SDE to be consistent with the turbulent mixing process. The mix number-fraction beta SDE is similar to the number-fraction beta SDE, only the process is made consistent with the no-mix and fully mixed limits via the specification of the SDE coefficients b and kappa. As in the number-fraction beta SDE, X is governed by the beta SDE and two additional stochastic variables are computed. However, in the mix number-fraction beta SDE the parameters b and kappa are given by b = Theta * b' and kappa = kappa' * <x^2>, where Theta = 1

• <x^2> / [ <X> ( 1 - <X> ], the fluctuation about the mean, <X>, is defined as usual: x = X - <X>, and b' and kappa' are user-specified constants. Similar to the number-fraction beta SDE, there two additional random variables computed besides, X, and they are rho(X) and V(X). For more detail on the number-fraction beta SDE, see the help on keyword 'numfracbeta'. For more details on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843 and src/DiffEq/Beta/Beta.h. Keywords allowed in a mixnumfracbeta ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'bprime', 'S', 'kappaprime', 'rho2', 'rcomma'. For an example mixnumfracbeta ... end block, see doc/html/walker_example_mixnumfracbeta.html.

Keyword 'mkl_mcg31'

Select Intel MKL MCG31 RNG

This keyword is used to select 'VSL_BRNG_MCG31', a 31-bit multiplicative congruential random number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_mcg59'

Select Intel MKL MCG59 RNG

This keyword is used to select 'VSL_BRNG_MCG59', a 59-bit multiplicative congruential random number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_mrg32k3a'

Select Intel MKL MRG32K3A RNG

This keyword is used to select 'VSL_BRNG_MRG32K3A', a combined multiple recursive random number generator with two components of order 3, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_mt19937'

Select Intel MKL MT19937 RNG

This keyword is used to select 'VSL_BRNG_MT19937', a Mersenne Twister pseudorandom number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_mt2203'

Select Intel MKL MT2203 RNG

This keyword is used to select 'VSL_BRNG_MT2203', a set of 6024 Mersenne Twister pseudorandom number generators, available in Intel's Math Kernel Library (MKL).

Keyword 'mkl_niederr'

Select Intel MKL NIEDERR RNG

This keyword is used to select 'VSL_BRNG_NIEDERR', a 32-bit Gray code-based random number generator, producing low-discrepancy sequences for dimensions 1 .le. s .le. 318 with available user-defined dimensions, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_nondeterm'

Select Intel MKL NONDETERM RNG

This keyword is used to select 'VSL_BRNG_NONDETERM', a non-deterministic random number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_r250'

Select Intel MKL R250 RNG

This keyword is used to select 'VSL_BRNG_R250', a generalized feedback shift register random number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_sfmt19937'

Select Intel MKL SFMT19937 RNG

This keyword is used to select 'VSL_BRNG_SFMT19937', a SIMD-oriented Fast Mersenne Twister pseudorandom number generator, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_sobol'

Select Intel MKL SOBOL RNG

This keyword is used to select 'VSL_BRNG_SOBOL', a 32-bit Gray code-based random number generator, producing low-discrepancy sequences for dimensions 1 .le. s .le. 40 with available user-defined dimensions, provided by Intel's Math Kernel Library (MKL).

Keyword 'mkl_wh'

Select Intel MKL WH RNG

This keyword is used to select 'VSL_BRNG_WH', a set of 273 Wichmann-Hill combined multiplicative congruential random number generators, provided by Intel's Math Kernel Library (MKL).

Keyword 'montecarlo_homdecay'

Select Monte Carlo homogeneous decay coefficients policy

This keyword is used to select the Monte Carlo homogeneous decay coefficients policy. This policy (or model) is used to constrain a beta stochastic differential equation (SDE) so that its variance, <y^2>, always decays and its mean, <R> = rho2/(1+r<RY>/<R>), where Y = <Y> + y, does not change in time. Note that R = rho2/(1+rY). This policy is similar to 'homdecay', but computes the the SDE coefficient S in a different but statistically equivalent way. While 'homdecay' only requires the estimation of statistics, <R>, <r^2>, and <r^3>, 'montecarlo_homdecay' requires <R^2>, <YR^2>, and <Y(1-Y)R^3>. A coefficients policy, in general, is used to specify how the coefficients are set at each time step during time-integration. Example: "coeff const", which selects constant coefficients policy, which sets constant coefficients before t = 0 and leaves the coefficients unchanged during time integration. Note that this option may behave differently depending on the particular equation or physical model.

Keyword 'mu'

Set SDE parameter(s) mu

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "mu 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'multiple'

Select PDF output policy multiple

This keyword is used to select the 'multiple' output file policy for requested probability density functions (PDFs) within a pdfs ... end block. Example: "policy multiple", which selects the multiple output file policy. The multiple policy output creates a new file for each time step. Valid PDF policy options are 'overwrite', 'multiple', and 'evolution'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Keyword 'ncomp'

Set number of scalar components for a system of differential equations

This keyword is used to specify the number of scalar components of a vector. 'ncomp' means "number of components". It is also used for specifying the number of scalar components of a transporter scalar (see also the keywords 'transport').

Expected type: uint

Lower bound: 1

Keyword 'node'

Specify node-centering for output

This keyword is used to select node-centering for variable output. In walker for example, this is used to configure probability values on the sample space grid for file output of probability density functions (PDFs). Example: "centering elem", which selects element-centered values. Valid options are 'elem' and 'node', denoting cell-centered and point-centered output, respectively. In inciter this keyword is used in output variable specification blocks, prefixing variable names by either 'node' or 'elem', to specify their centering for output to file.

Keyword 'normalization'

Select mixture Dirichlet PDF model normalization type

This keyword is used to select the mixture Dirichlet PDF model normalization type.

Expected type: string

Expected valid choices: 'light' | 'heavy'

Keyword 'npar'

Set total number of particles

This keyword is used to specify the total number of particles in a simulation.

Expected type: uint

Lower bound: 1

Keyword 'nstep'

Set number of time steps to take

This keyword is used to specify the number of time steps to take in a simulation. The number of time steps are used in conjunction with the maximmum time specified by keyword 'term': the simulation stops whichever is reached first. Both 'nstep' and 'term' can be left unspecified, in which case their default values are used. See also 'term'.

Expected type: uint

Lower bound: 1

Keyword 'numfracbeta'

Introduce the numfracbeta SDE input block

This keyword is used to introduce a numfracbeta ... end block, used to specify the configuration of a system of number-fraction beta SDEs, a system of stochastic differential equations (SDEs), in which, in addition to the dependent variable, computed with linear drift and quadratic diagonal diffusion (whose invariant is joint beta), two additional variables are computed. In other words, this is a beta SDE but there are two additional stochastic variables computed based on the beta SDE. If X is governed by the beta SDE, then the number-fraction beta SDE additionally governs rho(X) and V(X), where both rho and V are random variables, computed by rho(X) = rho2 ( 1 - r' X ), and V(X) = 1 / [ rho2 ( 1 - r'X ) ]. For more details on the beta SDE, see https://doi.org/10.1080/14685248.2010.510843 and src/DiffEq/Beta/Beta.hpp. Keywords allowed in a numfracbeta ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'b', 'S', 'kappa', 'rho2', 'rcomma'. For an example numfracbeta ... end block, see doc/html/walker_example_numfracbeta.html.

Keyword 'omega'

Set SDE parameter(s) omega

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "omega 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'ornstein-uhlenbeck'

Introduce the Ornstein-Uhlenbeck SDE input block

This keyword is used to introduce an ornstein-uhlenbeck ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), with linear drift and constant diffusion, whose invariant is the joint normal distribution. Keywords allowed in an ornstein-uhlenbeck ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'sigmasq', 'theta', 'mu'. For an example ornstein-uhlenbeck ... end block, see doc/html/walker_example_ou.html.

Keyword 'overwrite'

Select PDF output policy overwrite

This keyword is used to select the the 'overwrite' output file policy for requested probability density functions (PDFs) within a pdfs ... end block. Example: "policy overwrite", which selects the overwrite output file policy. The overwrite policy overwrites the same output file containing a single time step. Valid PDF policy options are 'overwrite', 'multiple', and 'evolution'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Keyword 'pari'

Set particles output interval

This keyword is used to specify the interval in time steps for particles output during a simulation.

Expected type: uint

Lower bound: 1

Keyword 'pdfs'

Start of probability density function (PDF) input block

This keyword is used to start a block in the input file containing the descriptions and settings of requested output for probability density functions (PDFs). Example: "pdfs mypdf( y1 : 1.0e-2 ) end", which requests a single-variate PDF to be output to file, whose sample space variable is y1, using automatic determination of the bounds of the sample space, using 1.0e-2 as the sample space bin size, and call the PDF "mypdf". For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Keyword 'policy'

Select PDF output file policy

This keyword is used to select the output file policy for requested probability density functions (PDFs) within a pdfs ... end block. Example: "policy overwrite", which selects the overwrite output file policy. Valid options are 'overwrite', 'multiple', and 'evolution'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Expected type: string

Expected valid choices: 'overwrite' | 'multiple' | 'evolution'

Keyword 'position'

Introduce the (particle) position equation input block or coupling

This keyword is used to introduce a position ... end block, used to specify the configuration of a system of deterministic or stochastic differential equations, governing particle positions usually in conjunction with velocity model, e.g, the Langevin, model. Note that the random number generator r123_philox is automatically put on the list as a selected RNG if no RNG is selected. Keywords allowed in a position ... end block: 'depvar', 'rng', 'init', 'coeff', 'velocity', 'For an example position ... end block, see doc/html/walker_example_position.html. (2) To specify a dependent variable (by a character) used to couple a differential equation system, in which the 'position' keyword appears) to another labeled by a 'depvar'. Note that this keyword can also be used as a keyword that selects position in some other context-specific way, e.g., 'position' as opposed to 'velocity'.

Keyword 'precision'

Precision in digits for ASCII floating-point output

This keyword is used to select the precision in digits for ASCII floating-point real number output. Example: "precision 10", which selects ten digits for floating-point output, e.g., 3.141592654. The number of digits must be larger than zero and lower than the maximum representable digits for the given floating-point type. For more info on setting the precision in C++, see http://en.cppreference.com/w/cpp/io/manip/setprecision, and http://en.cppreference.com/w/cpp/types/numeric_limits/digits10

Expected type: int

Lower bound: 1

Upper bound: 16

Expected valid choices: integer between [1...16] (both inclusive)

Keyword 'prod_A005H'

Select production divided by dissipation rate from DNS of HRT, A=0.05, IC:light<<heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.05, IC: light << heavy.

Keyword 'prod_A005L'

Select production divided by dissipation rate from DNS of HRT, A=0.05, IC:light>>heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.05, IC: light >> heavy.

Keyword 'prod_A005S'

Select production divided by dissipation rate from DNS of HRT, A=0.05, IC:light=heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.05, IC: light = heavy.

Keyword 'prod_A05H'

Select production divided by dissipation rate from DNS of HRT, A=0.5, IC:light<<heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.5, IC: light << heavy.

Keyword 'prod_A05L'

Select production divided by dissipation rate from DNS of HRT, A=0.5, IC:light>>heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.5, IC: light >> heavy.

Keyword 'prod_A05S'

Select production divided by dissipation rate from DNS of HRT, A=0.5, IC:light=heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.5, IC: light = heavy.

Keyword 'prod_A075H'

Select production divided by dissipation rate from DNS of HRT, A=0.75, IC:light<<heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.75, IC: light << heavy.

Keyword 'prod_A075L'

Select production divided by dissipation rate from DNS of HRT, A=0.75, IC:light>>heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.75, IC: light >> heavy.

Keyword 'prod_A075S'

Select production divided by dissipation rate from DNS of HRT, A=0.75, IC:light=heavy

Production divided by dissipation rate from DNS of homogeneous Rayleigh-Taylor instability, P/e, A = 0.75, IC: light = heavy.

Keyword 'product'

Select product (as the dependent variable) to solve for

This keyword is used to select the product of multiple random variables as what quantity to solve for, i.e., use as the dependent variable, e.g., in a velocity model, solve for the product of the full density and the full velocity, i.e., the full momentum, for a stochastic particle. This configures how statistics must be interpreted.

Expected type: string

Keyword 'r'

Set SDE parameter(s) r

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "r 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'r123_philox'

Select Random123 Philox RNG

This keyword is used to select the Philox generator, based on Feistel network and integer multiplication, provided by the Random123 random number generator library. For more info on Random123 see http://dl.acm.org/citation.cfm?doid=2063405.

Keyword 'r123_threefry'

Select Random123 ThreeFry RNG

This keyword is used to select the ThreeFry generator, related to the Threefish block cipher from Skein Hash Function, provided by the Random123 random number generator library. For more info on Random123 see http://dl.acm.org/citation.cfm?doid=2063405.

Keyword 'raw'

Select the raw initialization policy

This keyword is used to select the raw initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init raw", which selects raw initialization policy, which leaves the memory uninitialized. Note that this option may behave differently depending on the particular equation or physical model. See the the init policies in DiffEq/InitPolicy.hpp for valid options.

Keyword 'rcomma'

Set SDE parameter(s) rcomma

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "rcomma 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'rho'

Set SDE parameter(s) rho

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "rho 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'rho2'

Set SDE parameter(s) rho2

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "rho2 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'rng'

Select random number generator (RNG) from pool of enabled RNGs

This keyword is used to select a particular random number generator (RNG) from a pre-selected set of (enabled and configured) pool of RNGs. The pool is specified by the 'rngs ... end' block and it must precede the selection of an RNG.

Expected type: string

Expected valid choices: 'r123_threefry' | 'r123_philox' | 'rngsse_gm19' | 'rngsse_gm29' | 'rngsse_gm31' | 'rngsse_gm55' | 'rngsse_gm61' | 'rngsse_gq58.1' | 'rngsse_gq58.3' | 'rngsse_gq58.4' | 'rngsse_mt19937' | 'rngsse_lfsr113' | 'rngsse_mrg32k3a' | 'mkl_mcg31' | 'mkl_r250' | 'mkl_mrg32k3a' | 'mkl_mcg59' | 'mkl_wh' | 'mkl_mt19937' | 'mkl_mt2203' | 'mkl_sfmt19937' | 'mkl_sobol' | 'mkl_niederr' | 'mkl_iabstract' | 'mkl_dabstract' | 'mkl_sabstract' | 'mkl_nondeterm' Remember: the RNG must be listed in the pool before it can be selected via this keyword!

Keyword 'rngs'

Start of a random number generators description input block

This keyword is used to start a block in the input file containing the descriptions and settings of requested random number generators. Example: "rngs mkl_mcg59 seed 2134 uniform_method accurate end end" which enables the MCG59 generator from MKL using the seed 2134. For more info on the structure of the rngs ... end block, see doc/pages/rngs_input.dox.

Keyword 'rngsse_gm19'

Select RNGSSE GM19 RNG

This keyword is used to select the GM19 random number generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gm29'

Select RNGSSE GM29 RNG

This keyword is used to select the GM29 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gm31'

Select RNGSSE GM31 RNG

This keyword is used to select the GM31 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gm55'

Select RNGSSE GM55 RNG

This keyword is used to select the GM55 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gm61'

Select RNGSSE GM61 RNG

This keyword is used to select the GM61 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gq58.1'

Select RNGSSE GQ58.1 RNG

This keyword is used to select the GQ58.1 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gq58.3'

Select RNGSSE GQ58.3 RNG

This keyword is used to select the GQ58.3 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSS2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_gq58.4'

Select RNGSSE GQ58.4 RNG

This keyword is used to select the GQ58.4 generator, using a method based on parallel evolution of an ensemble of transformations of a two-dimensional torus, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_lfsr113'

Select RNGSSE LFSR113 RNG

This keyword is used to select the LFSR113 generator, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_mrg32k3a'

Select RNGSSE MRG32K3A RNG

This keyword is used to select the MRG32K3A generator, a combined multiple recursive random number generator with two components of order 3, provided by the RNGSS2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'rngsse_mt19937'

Select RNGSSE MT19937 RNG

This keyword is used to select the MT19937 generator, a Mersenne Twister generator, provided by the RNGSSE2 random number generator library. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Keyword 'scientific'

Select the scientific ASCII floating-point output

This keyword is used to select the 'scientific' floating-point output format for ASCII floating-point real number output. Example: "format scientific", which selects the scientific floating-point output. Valid options are 'default', 'fixed', and 'scientific'. For more info on these various formats, see http://en.cppreference.com/w/cpp/io/manip/fixed.

Keyword 'seed'

Set random number generator seed

This keyword is used to specify a seed for a random number generator Example: rngmkl_mcg31 seed 1234 end

Expected type: uint

Keyword 'seqlen'

Specify the RNGSSE RNG sequence length

This keyword is used to select a random number generator sequence length, used by the RNGSSE2 random number generator library. Valid options are 'short', 'medium', and 'long'. For more info on RNGSSE see https://doi.org/10.1016/j.cpc.2011.03.022.

Expected type: string

Expected valid choices: 'short' | 'medium' | 'long'

Keyword 'sigmasq'

Set SDE parameter(s) sigmasq

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "sigmasq 4.0 2.5 1.1 32.0 5.6 23.0 end" The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'skew-normal'

Start configuration block for the Skew-normal SDE

This keyword is used to introduce a skew-normal ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), whose invariant is the joint skew-normal distribution. For more details on the skew-normal distribution, see http://www.jstor.org/stable/2337278. Keywords allowed in an skew-normal ... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'sigmasq', 'T', 'lambda'. For an example skew-normal... end block, see doc/html/walker_example_skewnormal.html.

Keyword 'slm'

Select the simplified Langevin model (SLM) for the velocity PDF model

This keyword is used to select the simplified Langevin model (SLM) for the Lagrangian velocity in turbulent flows.

Expected type: string

Keyword 'solve'

Select dependent variable to solve for

This keyword is used to select an the quantity (the dependent variable) to solve for in walker's position and/or velocity model. This configures how statistics must be interpreted.

Expected type: string

Expected valid choices: 'fullvar' | 'fluctuation' | 'product' | 'fluctuating_momentum'

Keyword 'spike'

Configure a delta spike

This keyword is used to specify the configuration of delta spikes for, the delta initialization policy. The configuration is given by an even set of real numbers inside a spike...end block. Example: "spike 0.1 1.0 end", which specifies a delta spike at sample space position 0.1 with relative height 1.0. The height must be between [0.0...1.0] inclusive and specifies a relative probability. See also the help on keyword icdelta.

Expected type: even reals

Keyword 'standard'

Select the standard algorithm for uniform RNG

This keyword is used to select the standard method used to generate uniform random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'standard' and 'accurate'.

Keyword 'stationary'

Select the stationary coefficients policy

This keyword is used to select the stationary coefficients policy. This policy will keep a stochastic differential equation at a constant statistically stationary state.

Keyword 'statistics'

Start of statistics input block

This keyword is used to start a block in the input file containing the descriptions and settings of requested output for statistical moments. Example: "statistics \<Y\> \<yy\> end", which requests the first two moments of the flutcutating variable 'Y'. For more info on the structure of the statistics ... end block, see doc/pages/statistics_output.dox.

Keyword 'term'

Set maximum non-dimensional time to simulate

This keyword is used to specify the termination time in a simulation. The termination time and number of time steps, specified by 'nstep', are used in conjunction to determine when to stop a simulation: whichever is reached first. Both 'nstep' and 'term' can be left unspecified, in which case their default values are used. See also 'nstep'.

Expected type: real

Lower bound: 0.000000

Keyword 'theta'

Set SDE parameter(s) theta

This keyword is used to specify a vector of real numbers used to parameterize a system of stochastic differential equations. Example: "theta 5.0 2.0 3.0 end". The length of the vector depends on the particular type of SDE system and is controlled by the preceding keyword 'ncomp'.

Expected type: real(s)

Keyword 'title'

Set analysis title

The analysis title may be specified in the input file using the 'title' keyword. The 'title' keyword must be followed by a double-quoted string specifying the analysis title. Example: title "Example problem". Specifying a title is optional.

Expected type: string

Keyword 'ttyi'

Set screen output interval

This keyword is used to specify the interval in time steps for screen output during a simulation.

Expected type: uint

Lower bound: 0

Keyword 'txt'

Select ASCII output for outputing PDFs

This keyword is used to select the text output file type of a requested probability density function (PDF) within a pdfs ... end block. Example: "filetype txt", which selects text-file output. Valid options are 'txt', 'gmshtxt', 'gmshbin', and 'exodusii'. For more info on the structure of the pdfs ... end block, see doc/pages/statistics_output.dox.

Keyword 'uniform_method'

Select an Intel MKL uniform RNG method

This keyword is used to specify the method used to generate uniform random numbers using the Intel Math Kernel Library (MKL) random number generators. Valid options are 'standard' and 'accurate'.

Expected type: string

Expected valid choices: 'standard' | 'accurate'

Keyword 'variant'

Select velocity PDF model variant

This keyword is used to select the velocity PDF model variant.

Expected type: string

Expected valid choices: 'slm' | 'glm'

Keyword 'velocity'

Introduce the velocity equation input block or coupling

This keyword is used to introduce a velocity ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), governed by the Langevin model for the fluctuating velocity in homogeneous variable-density turbulence. For more details on this Langevin model, see https://doi.org/10.1080/14685248.2011.554419 and src/DiffEq/Velocity/Velocity.hpp. Keywords allowed in a velocity ... end block: 'depvar', 'rng', 'init', 'coeff', 'hydrotimescales', 'hydroproductions', 'C0'. position', 'dissipation', 'mixmassfracbeta', 'For an example velocity ... end block, see doc/html/walker_example_velocity.html. (2) To specify a dependent variable (by a character) used to couple a differential equation system, in which the 'velocity' keyword appears) to another labeled by a 'depvar'.

Keyword 'walker'

Start configuration block of the random walker

This keyword is used to select the walker. Walker, is a random walker, that allows temporal integration of a system of ordinary or stochastic differential equations (SDEs) of various types and the collection of arbitrary coupled statistics and probability density functions. Walker is intended as a general mathematical tool to analyze the behavior of SDEs and its statistics.

Keyword 'wright-fisher'

Start configuration block for the Wright-Fisher SDE

This keyword is used to introduce a wright_fisher ... end block, used to specify the configuration of a system of stochastic differential equations (SDEs), whose invariant is the Dirichlet distribution. For more details on the Wright-Fisher SDE, see http://www.sciencedirect.com/science/article/pii/S0040580912001013. Keywords allowed in a wright-fisher... end block: 'depvar', 'ncomp', 'rng', 'init', 'coeff', 'omega'. For an example wright-fisher ... end block, see doc/html/walker_example_wf.html.

Keyword 'zero'

Select the zero initialization policy

This keyword is used to select the zero initialization policy. The initialization policy is used to specify how the initial conditions are set at t = 0 before time-integration. Example: "init zero", which selects zero initialization policy, which puts zeros in memory. Note that this option may behave differently depending on the particular equation or physical model. See the init policies in DiffEq/InitPolicy.hpp for valid options.