Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

TL;DR

This paper introduces a new high-order ALE one-step WENO finite volume scheme for solving nonlinear hyperbolic conservation laws on moving unstructured triangular meshes, combining advanced spatial and temporal discretizations.

Contribution

It develops a novel high-order ALE one-step WENO scheme with a local space-time Galerkin predictor on moving unstructured meshes, enhancing accuracy and simplicity.

Findings

Achieves up to sixth-order accuracy in space and time.

Demonstrates effectiveness on 2D Euler equations of gas dynamics.

Provides numerical convergence rates and classical test results.

Abstract

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor. For that purpose, a new element--local weak formulation of the governing PDE is adopted on moving space--time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.