A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods

TL;DR

This paper introduces a positivity-preserving limiting strategy for a locally-implicit Lax-Wendroff discontinuous Galerkin method, effectively handling singular solutions in hyperbolic conservation laws while maintaining high-order accuracy.

Contribution

It develops a novel LxW-DG scheme with a two-step predictor-corrector approach that ensures non-oscillatory, positivity-preserving solutions for complex hyperbolic PDEs.

Findings

Successfully applied to shallow water and Euler equations test cases.

Maintains high-order accuracy with positivity preservation.

Implemented in open-source Python code.

Abstract

Nonlinear hyperbolic conservation laws admit singular solutions such as shockwaves (discontinuities in conserved variables), rarefaction waves (discontinuities in derivatives), and vacuum states (loss of strong hyperbolicity). When ostensibly high-order numerical methods are applied in such solution regimes, unphysical oscillations present themselves that can lead to large errors and a breakdown of the numerical simulation. In this work we develop a new Lax-Wendroff discontinuous Galerkin (LxW-DG) method with a limiting strategy that keeps the solution non-oscillatory and positivity-preserving for relevant variables, such as height in the shallow water equations and density and pressure in the compressible Euler equations. The proposed LxW-DG scheme updates the solution over each time-step with a locally-implicit predictor followed by an explicit corrector. The locally-implicit prediction phase is formulated in terms of primitive variables, which greatly simplifies the solver. The resulting system of nonlinear algebraic equations are approximately solved via a Picard iteration, where the number of iterations is equal to the order of accuracy of the method. The correction phase is an explicit evaluation formulated in terms of conservative variables in order to guarantee numerical conservation. In order to achieve full positivity-preservation, limiting is required in both the prediction and correction steps. The resulting scheme is applied to several standard test cases for the shallow water and compressible Euler equations. All of the presented examples are written in a freely available open-source Python code.