src/DiffEq/MassFractionBeta.h file

System of mass-fraction beta SDEs.


This file implements the time integration of a system of stochastic differential equations (SDEs) with linear drift and quadratic diagonal diffusion, whose invariant is the joint beta distribution. The main difference compared to the plain beta SDE (see DiffEq/Beta.h), is that in the mass-fraction beta SDE the dependent variable, there are two additional stochastic variables computed from the beta variables.

In a nutshell, the equation integrated governs a set of scalars, $0\!\le\!Y_\alpha$ , $\alpha\!=\!1,\dots,N$ , as

\[ \mathrm{d}Y_\alpha(t) = \frac{b_\alpha}{2}\left(S_\alpha - Y_\alpha\right) \mathrm{d}t + \sqrt{\kappa_\alpha Y_\alpha(1-Y_\alpha)} \mathrm{d}W_\alpha(t), \qquad \alpha=1,\dots,N \]
\[ \begin{split} \mathrm{d}Y_\alpha(t) = \frac{b_\alpha}{2}\left(S_\alpha - Y_\alpha\right) \mathrm{d}t + \sqrt{\kappa_\alpha Y_\alpha(1-Y_\alpha)} \mathrm{d}W_\alpha(t), \\ \alpha=1,\dots,N \end{split} \]

with parameter vectors $b_\alpha > 0$ , $\kappa_\alpha > 0$ , and $0 < S_\alpha < 1$ . This is the same as in DiffEq/Beta.h. Here $\mathrm{d}W_\alpha(t)$ is an isotropic vector-valued Wiener process with independent increments. The invariant distribution is the joint beta distribution. This system of SDEs consists of N independent equations. For more on the beta SDE, see

In addition to integrating the above SDE, there are two additional functions of $ Y_\alpha $ are computed as

\[ \begin{aligned} \rho(Y_\alpha) & = \frac{ \rho_{2\alpha} }{ 1 + r_\alpha Y_\alpha } \\ V(Y_\alpha) & = \frac{1}{ \rho(Y_\alpha) } \end{aligned} \]

These equations compute the instantaneous mixture density, $ \rho $ , and instantaneous specific volume, $ V_\alpha $ , for equation $ \alpha $ in the system. These quantities are used in binary mixing of variable-density turbulence between two fluids with constant densities, $ \rho_1, $ and $ \rho_2 $ . The additional parameters, $ \rho_2 $ and $ r' $ are user input parameters and kept constant during integration. Since we compute the above variables, $\rho,$ and $V$ , and call them mixture density and specific volume, respectively, $Y$ , governed by the beta SDE is a mass fraction, hence the name mass-fraction beta.

All of this is unpublished, but will be linked in here once published.


namespace walker
Walker declarations and definitions.


template<class Init, class Coefficients>
class walker::MassFractionBeta
MassFractionBeta SDE used polymorphically with DiffEq.