An Implicit Eulerian–Lagrangian WENO3 Scheme for Nonlinear Conservation Laws

Abstract

We present a new, formally third order, implicit Weighted Essentially Non-Oscillatory (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 about 2–3 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.