Postprocessed integrators for the high order
integration of ergodic SDEs
G. Vilmart
Abstract. The concept of effective order is a popular methodology in the deterministic literature
for the construction of efficient and accurate integrators for differential equations over long times.
The idea is to enhance the accuracy of a numerical method by using an appropriate change
of variables called the processor.
We show that this technique can be extended to the stochastic context
for the construction of new high order integrators for the sampling
of the invariant measure of ergodic systems.
The approach is illustrated with modifications of the stochastic theta-method applied to Brownian dynamics,
where postprocessors achieving order two are introduced.
Numerical experiments, including stiff ergodic systems,
illustrate the efficiency and versatility of the approach.
Key Words. stochastic differential equations, effective order, postprocessor,
modified differential equations, invariant measure, ergodicity.