On the use of Glimm‐like schemes for transport equations on multidimensional domain

Abstract

AbstractThis article is dedicated to a numerical method based on Glimm's idea and applied to the specific problem of advection of an indicator function. The numerical scheme relies on a fractional step approach for which: the first step is performed using any classical finite‐volume scheme, and the second step sharpens the approximated solution issued from the first step. This second step is based on a random choice as proposed in the original idea of Glimm. In order to assess the capabilities of this approach, several test cases have been investigated including convergence studies with respect to the mesh‐size. The method proposed in this article is very easy to implement, even for multiprocessor computations, and it naturally extends to multidimensional domains. Several numerical test cases, including both one‐ and two‐dimensional domains, have been considered in order to examine its behavior in terms of robustness, accuracy, and convergence rate.