Stochastic Discontinuous Galerkin Methods (SDGM) Based on Fluctuation-Dissipation Balance

TL;DR

This paper introduces a novel framework for approximating parabolic SPDEs using fluctuation-dissipation balance, leading to efficient stochastic discontinuous Galerkin methods that handle complex geometries and boundary conditions.

Contribution

The paper develops a general FDD framework for SDGM, enabling accurate, robust, and computationally efficient stochastic discretizations for complex domains and boundary conditions.

Findings

Methods with linear-time complexity for general geometries.

Effective handling of Dirichlet and Neumann boundary conditions.

Improved statistical accuracy in heterogeneous discretizations.

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 for compensating for differences in dissipative properties of discrete numerical operators relative to their continuum counter-parts. This allows us to handle general heterogeneous discretizations capturing accurately statistical relations. Our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.