An Eulerian–Lagrangian WENO finite volume scheme for advection problems

We develop a locally conservative Eulerian–Lagrangian finite volume scheme with the weighted essentially non-oscillatory property (EL–WENO) in one-space dimension. This method has the advantages of both WENO and Eulerian–Lagrangian schemes. It is formally high-order accurate in space (we present the fifth order version) and essentially non-oscillatory. Moreover, it is free of a CFL time step stability restriction and has small time truncation error. The scheme requires a new integral-based WENO reconstruction to handle trace-back integration. A Strang splitting algorithm is presented for higher-dimensional problems, using both the new integral-based and pointwise-based WENO reconstructions. We show formally that it maintains the fifth order accuracy. It is also locally mass conservative. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy.