Efficient positivity preserving schemes for stochastic complex systems

Abstract

We propose a Lagrange multiplier approach for constructing positive preserving scheme for stochastic complex systems. Adopting the approach for deterministic PDEs in Cheng and Shen (2022), we introduce the Karush–Kuhn–Tucker (KKT) condition to enforce positivity. For general stochastic partial complex systems with positive solution, we apply fully implicit schemes. But for stochastic Allen–Cahn equations, the discretization in time is an IMEX tamed Euler scheme and the discretization in space is a spectral Galerkin method with numerical integration, and for stochastic ordinary complex systems, the discretization is a tamed semi-implicit scheme. For all these cases, our schemes are proven to be unconditionally stable. Under regular assumptions for a broad class of SDEs and stochastic Allen–Cahn equation, we are able to provide an optimal strong error analysis. Numerical experiments are provided to validate our theoretical results.