Observations on the fifth-order WENO method with non-uniform meshes

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Typical treatments of WENO methods assume a uniform mesh. In this paper, we give explicit formulas for the finite-volume, fifth-order WENO (WENO5) method on non-uniform meshes in a way that is amenable to efficient implementation. We then compare the performance of the non-uniform mesh approach with the classical uniform mesh approach for the finite-volume formulation of the WENO5 method. We find that the numerical results significantly favor the non-uniform mesh approach both in terms of computational efficiency as well as memory usage. We expect this investigation to provide a basis for future work on adaptive mesh methods coupled with the finite-volume WENO methods.