# Deterministic and stochastic differential equations in walker

This page collects some notes on the requirements and software design of the main ingredient of walker, the differential equations classes.

## General requirements

Numerical time integration of ordinary and stochastic differential equations (ODEs, SDEs) is probably the single most important ingredient of a continuum-realm particle-solver. This must be:

• High-performance,
• Easy to maintain,
• The design must scale well with adding new functionality, i.e., adding new equations and/or new models for already implemented equations should require as little code as possible,
• Should easily accommodate various advancement algorithms on different hardware end/or using various parallelization strategies.

There should be a possibility to quickly prototype a new equation, e.g., in a test-bed class. This would be used to:

• Verify its invariant probability density function (PDF),
• Explore the behavior of its statistics,
• Integrate multiple variables (coupled or non-coupled).

Questions:

• Should a base class hold a single random number generation (RNG) used by all specific (derived) SDEs or different SDEs should be able to instantiate and use their own (possibly different) RNGs?
Currently, each derived SDE may access its own RNG, but the user must configure it so. See also a more detailed page on Random number generators input.

## Requirements on a generic differential equation base class

ODEs and SDEs should inherit from a base class (if a multiple-policy design is adopted) that should have generic data and member functions, which facilitates code-reuse.

The base class should work for both N = 1 or N > 1, i.e., single-variate or multi-variate equation classes.

The differential equation base class should have pure virtual interfaces for:

• Setting initial conditions on the particles at t = 0, e.g., initialize()
• Advancing the particles in time, e.g., advance()

## Possible policies of a differential equation base class

Specific equation types (e.g., Ornstein-Uhlenbeck, Dirichlet, skew-normal, etc.), should derive from a base class, forwarding base class policies, i.e., a specific SDE class should not hard-code any base class policy.

Specific SDE classes may have their own policies (specific to the given SDE).

### Initialization policy

Specifies how the initialization of the particles happen at t = 0. Possible initialization policies:

• Do nothing: leave memory associated to particle data uninitialized
• Zero: zero particle properties
• Fill with one given constant: single-delta-spike PDF
• Fill with different constants given per variable
• Sample from given PDF, N = 1
• Sample from different PDF given per variable, N > 1 (independent)
• Sample from given JPDF, N > 1 (possibly non-independent)
• Pre-cycle properties using a given equation and its constant coefficients for:
• a given time period
• a given number of time steps
• until convergence is reached for given statistics and convergence criteria

### Coefficients policy

Specifies how the differential equation coefficients, e.g., b, S, and for the Dirichlet SDE, are used by the equation. Possible coefficients policies:

• Constant: initialized once, used for all t > 0
• Functional: advance() algorithm queries coefficients at every update via coefficients-policy functions, e.g., time, various statistics

### Time-integration policy

Specifies what time-integrator to use when advancing particles. Possible time-integration policies:

• Euler-Maruyama
• Milstein
• Runge-Kutta (with various orders)
• Various other explicit and implicit integrators, see Kloeden & Platen

Questions

• What new requirements and constraints does spatial inhomogeneity entail?
Most of the above is implemented under src/DiffEq/, with the forward Euler (Euler-Maruyama) time integration scheme.