A posteriori error estimation for the Lax-Wendroff finite difference scheme

In many application domains, the preferred approaches to the numerical solution of hyperbolic partial differential equations such as conservation laws are formulated as finite difference schemes. While finite difference schemes are amenable to physical interpretation, one disadvantage of finite difference formulations is that it is relatively difficult to derive the so-called goal oriented a posteriori error estimates. A posteriori error estimates provide a computational approach to numerically compute accurate estimates in the error in specified quantities computed from a numerical solution.Widely used for finite element approximations, a posteriori error estimates yield substantial benefits in terms of quantifying reliability of numerical simulations and efficient adaptive error control. The chief difficulties in formulating a posteriori error estimates for finite difference schemes is introducing a variational formulation – and the associated adjoint problem – and a systematic definition of residual errors. In this paper, we approach this problem by first deriving an equivalency between a finite element method and the Lax–Wendroff finite volume method. We then obtain an adjoint based error representation formula for solutions obtained with this method. Results from linear and nonlinear viscous conservation laws are given. © 2013 Elsevier B.V. All rights reserved.