Blended numerical schemes for the advection equation and conservation laws

TL;DR

This paper introduces a novel approach to combine multiple explicit numerical schemes for advection equations and conservation laws, creating multiscale methods that leverage the strengths of each scheme.

Contribution

It proposes a coupling method for explicit schemes, including macroscopic and microscopic approaches, to develop new multiscale numerical schemes for PDEs.

Findings

Successfully couples Eulerian and Lagrangian schemes

Creates multiscale schemes with improved properties

Applicable to advection and nonlinear conservation laws

Abstract

In this paper we propose a method to couple two or more explicit numerical schemes approximating the same time-dependent PDE, aiming at creating new schemes which inherit advantages of the original ones. We consider both advection equations and nonlinear conservation laws. By coupling a macroscopic (Eulerian) scheme with a microscopic (Lagrangian) scheme, we get a new kind of multiscale numerical method.