A positive meshless finite difference scheme for scalar conservation laws with adaptive artificial viscosity driven by fault detection

TL;DR

This paper introduces a meshless finite difference scheme for scalar conservation laws that adaptively applies artificial viscosity near shocks, ensuring positivity, stability, and high accuracy on irregular nodes without fixed time step restrictions.

Contribution

It develops a novel meshless method with adaptive artificial viscosity driven by fault detection, enabling stable shock capturing on irregular nodes with no a priori time step constraints.

Findings

Robust performance on irregular nodes

Comparable accuracy to standard methods on Cartesian grids

Effective shock capturing with adaptive viscosity

Abstract

We present a meshless finite difference method for multivariate scalar conservation laws that generates positive schemes satisfying a local maximum principle on irregular nodes and relies on artificial viscosity for shock capturing. Coupling two different numerical differentiation formulas and the adaptive selection of the sets of influence allows to meet a local CFL condition without any {\it a priori}\ time step restriction. The artificial viscosity term is chosen in an adaptive way by applying it only in the vicinity of the sharp features of the solution identified by an algorithm for fault detection on scattered data. Numerical tests demonstrate a robust performance of the method on irregular nodes and advantages of adaptive artificial viscosity. The accuracy of the obtained solutions is comparable to that for standard monotone methods available only on Cartesian grids.