Energy Conserving Galerkin Approximation of Two Dimensional Wave Equations with Random Coefficients

TL;DR

This paper develops an energy-preserving stochastic Galerkin method combined with local discontinuous Galerkin discretization for 2D wave equations with random, possibly discontinuous coefficients, demonstrating optimal convergence and linear error growth.

Contribution

It introduces an energy-conserving Galerkin approximation for stochastic 2D wave equations with discontinuous coefficients, ensuring stability and optimal convergence.

Findings

Method preserves energy in semi- and fully discrete forms.

Convergence rate is proven to be optimal.

Error grows linearly over time.

Abstract

Wave propagation problems for heterogeneous media are known to have many applications in physics and engineering. Recently, there has been an increasing interest in stochastic effects due to the uncertainty, which may arise from impurities of the media. This work considers a two-dimensional wave equation with random coefficients which may be discontinuous in space. Generalized polynomial chaos method is used in conjunction with stochastic Galerkin approximation, and local discontinuous Galerkin method is used for spatial discretization. Our method is shown to be energy preserving in semi-discrete form as well as in fully discrete form, when leap-frog time discretization is used. Its convergence rate is proved to be optimal and the error grows linearly in time. The theoretical properties of the proposed scheme are validated by numerical tests.