A stochastic Galerkin method for general system of quasilinear hyperbolic conservation laws with uncertainty

TL;DR

This paper develops a stochastic Galerkin method using generalized polynomial chaos for solving uncertain quasilinear hyperbolic conservation laws, ensuring symmetric hyperbolicity and enabling high-order numerical schemes.

Contribution

It introduces a symmetrized stochastic Galerkin approach for hyperbolic systems with uncertainty, extending to 2D and demonstrating high accuracy and effectiveness.

Findings

The method maintains symmetric hyperbolicity of the gPC system.

Numerical experiments confirm high accuracy and stability.

Extension to 2D systems is successfully achieved.

Abstract

This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic Galerkin method is then applied to derive the equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is proved to be symmetrically hyperbolic. This important property then allows one to use a variety of numerical schemes for spatial and temporal discretization. Here a higher-order and path-conservative finite volume WENO scheme is adopted in space, along with a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional system is carried over via the operator splitting technique. Several 1D and 2D numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.