Linear stability of WENO schemes coupled with explicit Runge-Kutta schemes

This paper focuses on the results of the linear stability analysis of the finite-difference weighted essentially non-oscillatory (WENO) schemes with optimal weights. The standard WENO schemes between the third and 11th order, the order-optimised WENO schemes of the sixth and eighth order and the bandwidth-optimised WENO schemes of the third and fourth order are considered. Several explicit RungeKutta schemes including the recently published strong stability-preserving explicit RungeKutta schemes are considered for time discretisation. The stability limits as well as dissipation and dispersion properties dependent on the CourantFriedrichsLewy number are presented for a hyperbolic model equation. The different combinations of space and time discretisation schemes are compared in terms of their accuracy and efficiency. For a parabolic model equation, the viscous term is discretised with high-order central differences. The stability limits for the parabolic problem are presented as well. Numerical results of linear test cases are shown. Copyright (C) 2011 John Wiley & Sons, Ltd.