Weighted Essentially Non-Oscillatory stochastic Galerkin approximation for hyperbolic conservation laws

TL;DR

This paper introduces a weighted essentially non-oscillatory stochastic Galerkin scheme to improve the accuracy and stability of uncertainty quantification in hyperbolic conservation laws, reducing oscillations and Gibbs phenomena.

Contribution

The paper develops a novel WENO stochastic Galerkin method with slope limiting in stochastic space, enhancing high-order accuracy and stability over classical approaches.

Findings

Reduced oscillations and Gibbs phenomena in numerical tests

Improved total variation control compared to classical stochastic Galerkin

Effective high-order reconstruction in both physical and stochastic domains

Abstract

In this paper we extensively study the stochastic Galerkin scheme for uncertain systems of conservation laws, which appears to produce oscillations already for a simple example of the linear advection equation with Riemann initial data. Therefore, we introduce a modified scheme that we call the weighted essentially non-oscillatory (WENO) stochastic Galerkin scheme, which is constructed to prevent the propagation of Gibbs phenomenon into the stochastic domain by applying a slope limiter in the stochasticity. In order to achieve a high order method, we use a spatial WENO reconstruction and also compare the results to a scheme that uses WENO reconstruction in both the physical and the stochastic domain. We evaluate these methods by presenting various numerical test cases where we observe the reduction of the total variation compared to classical stochastic Galerkin.