A kernel-based discretisation method for first order partial differential equations of evolution type

TL;DR

This paper introduces a novel kernel-based discretisation method for first order PDEs of evolution type, employing an Eulerian approach with proven stability and convergence, demonstrated through a Burgers equation example.

Contribution

It presents a new meshfree, kernel-based discretisation technique for first order PDEs that differs from SPH by using an Eulerian framework and provides theoretical stability and convergence results.

Findings

Method achieves stability and convergence under certain conditions.

Approximation order depends on kernel smoothness and solution regularity.

Numerical tests validate the method on a 1D Burgers equation.

Abstract

We derive a new discretisation method for first order PDEs of arbitrary spatial dimension, which is based upon a meshfree spatial approximation. This spatial approximation is similar to the SPH (smoothed particle hydrodynamics) technique and is a typical kernel-based method. It differs, however, significantly from the SPH method since it employs an Eulerian and not a Lagrangian approach. We prove stability and convergence for the resulting semi-discrete scheme under certain smoothness assumptions on the defining function of the PDE. The approximation order depends on the underlying kernel and the smoothness of the solution. Hence, we also review an easy way of constructing smooth kernels yielding arbitrary convergence orders. Finally, we give a numerical example by testing our method in the case of a one-dimensional Burgers equation.