Stability analysis for linear discretisations of the advection equation with Runge–Kutta time integration

Abstract

For the 1-dim. linear advection problem stability limits of Runge–Kutta (RK) methods from 1st to 7th order in combination with upwind or centered difference schemes from 1st to 6th order are presented. The analysis can be carried out in a rather general way by introduction of a broad class of Runge–Kutta methods, here called ‘Linear Case Runge–Kutta (LC-RK)’ methods, which behave completely similar for linear, time-independent and homogeneous ODE-systems and contain the ‘classical’ order = stage RK methods. The set of conditions for the coefficients of these LC-RK-schemes could be derived explicitly for arbitrary order N. From an efficiency viewpoint the LC-RK 3rd order methods in combination with upwind 3rd or 5th order or the LC-RK 4th order scheme with 4th order centered difference advection are a good choice. The analysis can be extended easily to multidimensional splited advection for which a necessary stability condition is presented.