Stochastic Discontinuous Galerkin Methods (SDGM) based on fluctuation-dissipation balance

Abstract

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms, handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows us to compensate for discrepancies in dissipative properties between the continuous operators and their discretized versions. This allows us to handle general heterogeneous discretizations, accurately capturing statistical relations. Our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.