A Method for Approximate Analysis of Courant Stability of Central-Difference Schemes with Boundary Conditions

Abstract

We consider central-difference schemes for the transport equation that can be integrated over time by explicit multistage Runge–Kutta methods. Such algorithms are the basis for modeling in modern aeroacoustics problems. Their stability is determined by the hyperbolic Courant number. The effect of discrete boundary conditions on the stability of the schemes is investigated. A procedure is proposed for approximate analysis of high-frequency modes that are responsible for the value of the maximum time increment. A three-point central-difference scheme with various boundary conditions is studied in detail and quantitative results are obtained for the analytical method error. We conclude that the boundary conditions have but a marginal effect on the maximum Courant number.