High Weak Order Methods
These methods are specifically designed for problems where weak convergence is more important than strong convergence. They are optimal for Monte Carlo simulations, computing expectations, moments, and other statistical properties of solutions.
Recommended High Weak Order Methods
DRI1 - Debrabant-Rößler Method (Weak Order 2)
DRI1NM - Debrabant-Rößler for Non-mixing Diagonal Problems
Other Weak Order 2 Methods
RI1, RI3, RI5, RI6 - Rößler Methods
RDI Methods - Alternative Weak Order 2
W2Ito1 - Efficient Weak Order 2
Fixed Step Methods
PL1WM, PL1WMA - Platen Methods
Stratonovich Methods
RS1, RS2 - Rößler Stratonovich Methods
NON, NON2 - Non-commutative Stratonovich
COM - Commutative Stratonovich
Specialized Methods
SIEA, SMEA, SIEB, SMEB - Tocino-Vigo-Aguiar Methods
Weak vs Strong Convergence
Strong Convergence: Measures pathwise error E[|X(T) - Xh(T)|^p] Weak Convergence: Measures error in expectations E[f(X(T))] - E[f(Xh(T))]
When to Use Weak Order Methods:
- Monte Carlo simulations
- Computing expectations and moments
- Statistical analysis of SDEs
- When pathwise accuracy is not critical
- Large ensemble simulations
Advantages:
- Often more efficient for statistical quantities
- Can use larger time steps while maintaining weak accuracy
- Optimized error constants for better practical performance
Method Selection Guide
- General purpose weak order 2: DRI1
- Non-mixing diagonal: DRI1NM
- Fixed step: PL1WM, RS1/RS2
- Stratonovich: RS1, RS2, NON, NON2
- Specialized applications: RI methods, RDI methods
References
- Debrabant, K. and Rößler A., "Families of efficient second order Runge–Kutta methods for the weak approximation of Itô stochastic differential equations"
- Rößler A., "Second Order Runge–Kutta Methods for Itô Stochastic Differential Equations"