On discretizations of axisymmetric systems

TL;DR

This paper examines the stability of different discretization methods for axisymmetric systems, highlighting potential instabilities in the cartoon method unless singular terms are carefully handled, with examples on the wave equation.

Contribution

It provides an analysis of stability issues in discretizations of axisymmetric systems, especially the cartoon method, emphasizing the importance of proper treatment of singular terms.

Findings

Discretizations can be unstable if singular terms are not properly treated.

The cartoon method's stability depends on how singularities are handled.

Examples demonstrate the impact on the linear axisymmetric wave equation.

Abstract

In this paper we discuss stability properties of various discretizations for axisymmetric systems including the so called cartoon method which was proposed by Alcubierre, Brandt et.al. for the simulation of such systems on Cartesian grids. We show that within the context of the method of lines such discretizations tend to be unstable unless one takes care in the way individual singular terms are treated. Examples are given for the linear axisymmetric wave equation in flat space.