A scalable exponential-DG approach for nonlinear conservation laws: with application to Burger and Euler equations

TL;DR

This paper introduces a scalable exponential DG method for nonlinear PDEs, combining linearization and exponential time integration to achieve stability, high-order accuracy, and computational efficiency, demonstrated on Burgers and Euler equations.

Contribution

The paper presents a novel exponential DG approach that decomposes PDE operators and employs exponential integrators, offering stability and efficiency advantages over existing methods.

Findings

Supports large Courant numbers (Cr > 1)

Achieves high-order accuracy in space and time

Demonstrates computational efficiency and scalability

Abstract

We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr > 1); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present detailed stability and convergence analyses for the exponential Euler DG scheme.