Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Abstract

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes. 1. Introduction. Nonlinear advection-diffusion equations often arise in the mod-eling of transport processes, especially in porous media. Typical examples are (en-hanced) oil recovery, CO 2 storage, and geothermal energy production. Since measurements of such processes are usually impossible or at best very difficult and thus very rare, numerical simulations are essential for an adequate understanding. The precise formulation of the underlying nonlinear advection-diffusion equations can involve strong heterogeneities due to largely varying physical properties and parameters. In turn, this raises significant mathematical and computational problems, such that the development and analysis of robust discretization methods becomes a non-trivial challenge. To still reach reasonable performance, a typical approach is the parallelization by a domain decomposition method, which is an established technique for steady problems; see [11, 45, 50, 52] and references therein. Regardless of the chosen dis-cretization method and of the linearization scheme, these methods aim at reducing the