Split-Step Methods for Fully Stiff Problems

When both drift and diffusion terms are stiff, split-step methods treat both parts implicitly. These methods can handle the most challenging stiff SDE problems.

Split-Step Implicit Methods

ISSEM - Implicit Split-Step Euler-Maruyama

ISSEulerHeun - Implicit Split-Step Euler-Heun

When to Use Split-Step Methods

Use ISSEM when:

• Both drift and diffusion are stiff
• Explicit methods fail even with small time steps
• Standard implicit methods (ImplicitEM) are insufficient
• Working with Itô interpretation

Use ISSEulerHeun when:

• Both drift and diffusion are stiff
• Working with Stratonovich interpretation
• Need implicit treatment of diffusion term

Understanding Full Stiffness

Drift Stiffness: Large negative eigenvalues in f(u,t) Diffusion Stiffness: Large coefficients in g(u,t) causing instability

Detection Signs:

• ImplicitEM still requires very small time steps
• Solutions become unstable despite implicit drift treatment
• Large diffusion coefficients cause numerical artifacts

Algorithm Structure

Split-step methods solve:

du = f(u,t)dt + g(u,t)dW

By treating both terms implicitly through operator splitting or fully implicit schemes.

Performance Considerations

• More expensive per step than drift-only implicit methods
• May require smaller time steps than expected
• Jacobian computations for both drift and diffusion
• Nonlinear solve complexity increases

Configuration

Same options as other implicit methods:

ISSEM(
    linsolve = KrylovJL_GMRES(),
    nlsolve = NLNewton(),
    theta = 1.0,
    autodiff = true
)

Alternative Approaches

If split-step methods are too expensive:

1. Try stabilized explicit methods (SROCK family)
2. Consider method of lines for PDE problems
3. Use shorter time intervals with restarts
4. Reformulate problem to reduce stiffness

References

• Implicit-explicit splitting schemes for SDEs
• Stochastic operator splitting methods