An Eulerian–Lagrangian Weighted Essentially Nonoscillatory scheme for nonlinear conservation laws

We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL‐WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi‐Lagrangian) scheme using a fixed velocity v. A flux correction computation accounts for particles that cross the v‐traceline during the time step. If v = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when v is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps. The v‐traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re‐averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 651–680, 2017