Method of Relaxed Streamline Upwinding for Hyperbolic Conservation Laws

TL;DR

This paper introduces a novel finite element method called Relaxed Streamline Upwinding for hyperbolic conservation laws, utilizing relaxation systems to improve handling of nonlinearity and extendability to vector laws.

Contribution

The paper presents a new relaxation-based finite element scheme with symmetric velocity models, exact diffusion vectors, and demonstrated robustness for various hyperbolic PDEs.

Findings

Achieves optimal convergence rates in error analysis.

Demonstrates robustness across Burgers, Euler, and shallow water equations.

Provides exact solutions and stability analysis for complex test cases.

Abstract

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.