Kolloquium GRK 2339, Prof. Stefan Metzger (FAU Erlangen-Nürnberg)

An augmented SAV scheme for the stochastic Allen-Cahn equation

Abstract: The scalar auxiliary variable (SAV) method was originally
introduced in [Shen, Xu, Yang, J. Comput. Phys., 2018] for the
discretization of deterministic gradient flows. By introducing
an additional scalar auxiliary variable, they were able to
formulate linear numerical schemes that are still
unconditionally stable with respect to a modified energy.

This talk addresses the application of the SAV method to
nonlinear stochastic partial differential equations with
multiplicative noise. Using the stochastic Allen-Cahn equation
as a prototype problem, we motivate why a straightforward
application of the SAV method will not provide satisfactory
results and present an augmented SAV method that remedies the
shortcomings and allows for a rigorous convergence proof.

We conclude by presenting numerical simulations which underline
the practicality of the scheme and the importance of the
introduced augmentation terms.