Linear Instability of the Fifth-Order WENO Method

Abstract

The weighted essentially nonoscillatory (WENO) methods are popular spatial discretization methods for hyperbolic partial differential equations. In this paper we show that the combination of the widely used fifth-order WENO spatial discretization (WENO5) and the forward Euler time integration method is linearly unstable when numerically integrating hyperbolic conservation laws. Consequently it is not convergent. Furthermore we show that all two-stage, second-order explicit Runge–Kutta (ERK) methods are linearly unstable (and hence do not converge) when coupled with WENO5. We also show that all optimal first- and second-order strong-stability-preserving (SSP) ERK methods are linearly unstable when coupled with WENO5. Moreover the popular three-stage, third-order SSP(3,3) ERK method offers no linear stability advantage over non-SSP ERK methods, including ones with negative coefficients, when coupled with WENO5. We give new linear stability criteria for combinations of WENO5 with general ERK methods of any order. We find that a sufficient condition for the combination of an ERK method and WENO5 to be linearly stable is that the linear stability region of the ERK method should include the part of the imaginary axis of the form $[-\iota \mu,\iota \mu]$ for some $\mu>0$. The linear stability analysis also provides insight into the behavior of ERK methods applied to nonlinear problems and problems with discontinuous solutions. We confirm the assertions of our analysis by means of numerical tests.