A robust first order meshfree method for time-dependent nonlinear conservation laws

TL;DR

This paper presents a new meshfree method for solving time-dependent nonlinear conservation laws, achieving first order accuracy and demonstrating promising convergence properties on challenging PDEs.

Contribution

It introduces a robust meshfree construction of summation by parts operators for nonlinear conservation laws, with efficient implementation and convergence analysis.

Findings

Operators achieve O(h) convergence in practice

Method effectively solves advection and Euler equations

Operators are only O(h^{1/2}) accurate in L^2 norm

Abstract

We introduce a robust first order accurate meshfree method to numerically solve time-dependent nonlinear conservation laws. The main contribution of this work is the meshfree construction of first order consistent summation by parts differentiations. We describe how to efficiently construct such operators on a point cloud. We then study the performance of such differentiations, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws, with focus on the advection equation and the compressible Euler equations. We observe numerically that, while the resulting mesh-free differentiation operators are only $O(h^\frac{1}{2})$ accurate in the $L^2$ norm, they achieve $O(h)$ rates of convergence when applied to the numerical solution of PDEs.