An implicit Eulerian-Lagrangian WENO3 scheme for nonlinear conservation laws

We present a new, formally third order, implicit Weighted Essentially NonOscillatory (iWENO3) finite volume scheme for solving systems of nonlinear conservation laws. We then generalize it to define an implicit Eulerian-Lagrangian WENO (iEL-WENO) scheme. Implicitness comes from the use of an implicit Runge-Kutta (RK) time integrator. A specially chosen two-stage RK method allows us to drastically simplify the computation of the intermediate RK fluxes, leading to a computationally tractable scheme. The iEL-WENO3 scheme has two main steps. The first accounts for particles being transported within a grid element in a Lagrangian sense along the particle paths. Since this particle velocity is unknown (in a nonlinear problem), a fixed trace velocity v is used. The second step of the scheme accounts for the inaccuracy of the trace velocity v by computing the flux of particles crossing the incorrect tracelines. The CFL condition is relaxed when v is chosen to approximate the characteristic velocity. A new Roe solver for the Euler system is developed to account for the Lagrangian tracings, which could be useful even for explicit EL-WENO schemes. Numerical results show that iEL-WENO3 is both less numerically diffusive and can take on the order of 2 to 5 times longer time steps than standard WENO3 for challenging nonlinear problems. An extension is made to the advection-diffusion equation. When advection dominates, the scheme retains its third order accuracy.