Counterexamples to a stability barrier

Summary We consider multistep difference schemes for the linear, constant coefficient advection equationu _t=cu_x. In the last section of Strang 5], a barrier for the order of stable schemes has been given which was independent of the number of time levels. Here we give two types of counterexamples to this barrier. The first type consists of formulas of highest possible order to a given stencil, which are stable for small positive Courant numbers. Further a formula is given where one does not insist on having the highest possible order for the stencil and uses the gained freedom to ensure stability for small positive and negative Courant numbers. In addition, an explicit formula is derived for the schemes of highest possible order when stability is disregarded.This work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which is supported by the DFG