Structure-preserving LDG methods for linear and nonlinear transport equations with gradient noise

TL;DR

This paper introduces structure-preserving LDG methods for stochastic conservation laws, ensuring stability and energy conservation/dissipation in complex linear and nonlinear transport equations with gradient noise.

Contribution

It develops semi-discrete LDG schemes that incorporate stochastic fluxes and Stratonovich corrections while maintaining hyperbolic stability and energy properties.

Findings

Numerical schemes are stable and high-order accurate.

Energy conservation or dissipation is achieved at the discrete level.

Methods are validated through numerical experiments.

Abstract

We develop local discontinuous Galerkin (LDG) methods for conservation laws with heterogeneous stochastic fluxes, where the Stratonovich-driven transport terms may be linear or nonlinear. Such equations arise, for example, in simplified turbulence models, mean field games, and fluctuating hydrodynamics. Starting from the It\^{o} formulation, we construct semi-discretizations that build the cancellation mechanism of transport noise into the numerical method. At the discrete energy level, the second-order Stratonovich-It\^{o} correction is balanced by the quadratic variation, up to numerical flux terms, so that the hyperbolic stability structure is retained. Suitable numerical fluxes yield discrete energy conservation or energy dissipation, valid either pathwise or in expectation. The resulting high-order schemes are proved well posed through stability estimates combined with a Khasminskii-type argument, without imposing linear growth assumptions. Numerical experiments confirm stability and high-order accuracy.