Do we know WENO?

The weighted essentially non-oscillatory (WENO) methods are popular spatial discretization methods for hyperbolic partial differential equations. In this talk I show that the combination of the widely used fifth-order WENO spatial discretization (WENO5) and several of the most popular time integration methods are in fact linearly unstable (and hence not convergent) when numerically integrating hyperbolic conservation laws. We find that a sufficient condition for the combination of an explicit Runge-Kutta (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 [-mu,mu], for some mu>0. The linear stability analysis also provides insight (and busts some myths) about the behaviour of ERK methods applied to nonlinear problems and problems with discontinuous solutions. We confirm the predictions of our analysis by means of numerical tests.