Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration
The aim of the paper is the systematic calculation of stability limits for Runge-Kutta methods up to 7th order in combination with upwind or centered difference schemes up to 6th order, for the linear advection problem. The first section represents an introduction in nature. The second section gives a linear view to explicit Runge-Kutta schemes, showing that it is not necessary to make a distinction between different Runge-Kutta methods of a certain order. The author defines a very large class of Runge-Kutta methods, called Linear Case-RK (LC-RK), with exactly the same linear stability properties. In the third section the stability limits for LC-RK methods for different linear, spatial discretisation of the one-dimensional advection operator are calculated. The fourth section contains some numerical experiments carried out to highlight the practical meaning of the stability limits found in the previous section. In the fifth section the author presents an extension to the space splitted multi-dimensional case. Thus, he considers the two-dimensional advection equation. The main conclusions about the efficiency of these schemes are within the last section.
scientific article; zbMATH DE number 5303110