Deterministic and stochastic differential equations in walker

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?

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).