On the lack of convergence of unconditionally stable explicit rational Runge-Kutta schemes (Q1087319)

The authors investigate convergence properties of rational Runge-Kutta methods, when they are applied to ordinary differential equations which arise from the spatial discretization of \(u_ t-Cu_ x=0\) (hyperbolic equation) and \(u_{tt}-Du_{xx}=0\) (parabolic equation). Although the considered methods are consistent and unconditionally stable, it is shown that they are not convergent if the time steps are too large. This demonstrates the limited applicability of such time-integration schemes.