Description
We introduce a new class of second-order in time and space numerical schemes, which are uniformly asymptotic preserving schemes. The proposed Implicit-Explicit (ImEx) approach, does not follow the usual path relying on the method of lines, either with multi-step methods or Runge-Kutta methods, or semi-discretized in time equations, but is inspired from the Lax-Wendroff approach with the proper level of implicit treatment of the source term.
We are able to rigorously show that both the second-order accuracy and the stability conditions are independent of the fast scales in every asymptotic regime, including the study of boundary conditions. The prototype system for the linear case is the hyperbolic heat equation, whereas Euler equations of gas dynamics with friction are the one for the nonlinear case. The method is also able to yield very accurate steady solutions in the nonlinear case when the source term depends on space. A thorough numerical assessment of the proposed strategy is provided by investigating smooth solutions, solutions with shocks and solutions leading to a steady state with variable source term in space. Our aim also includes plasma discharges with sheaths, where we have two small parameters related to Debye length and mass ratio, and we present some numerical simulations that assess and illustrate the potential of a method similar to the one we have introduced but applied to the isotherrmal Euler-Poisson equations. In particular, this last case can be regarded as a nonlinear low- Mach configuration coupled with a stiff source term. Numerical simulations assess and illustrate the potential of the method we have introduced.
