We consider hyperbolic systems of conservation laws with relaxation source terms leading to a diffusive asymptotic limit under a parabolic scaling. We introduce a new class of secondorder 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. As a result, it yields a very compact stencil in space and time and we are able to rigorously show that both the second-order accuracy and the stability conditions are independent of the fast scales in the asymptotic regime, including the study of boundary conditions. We provide an original derivation of l 2 and l $\infty$ stability conditions of the scheme that do not deteriorate the second order accuracy without relying on a limiter of any type in the linear case, in particular for shock solutions, and extend such results to the nonlinear case, showing the novelty of the method. The prototype system for the linear case is the hyperbolic heat equation, whereas barotropic 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 relaxation coefficient in 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 space dependent relaxation coefficient.
翻译:我们考虑超曲式的保护法体系,其排放源条件宽松,导致在抛物线缩放下出现偏差性无症状限值。我们采用一个新的时间和空间数字办法中第二阶级新等级,这些办法都是无症状的保存办法。拟议中的二阶准确度和稳定性条件都不遵循依赖线条方法的通常路径,要么采用多步法,要么使用Runge-Kutta方法,或者在时间方程中半分解,但受Lax-Wendroff方法的启发,该方法含有对源词进行适当程度的隐含处理。因此,在时间和空间数字和空间数字办法中,它产生一个非常紧凑的线条线条线次线次第二阶,因此,我们可以严格地表明,第二阶的准确性和稳定性条件都独立于线条线条法方法,包括边界条件研究。我们提供了该办法最初的l 2 和 l\\ inftyftytretrictredicretal ro condistrue条件, 在直线线性解决方案中不依赖任何类型的限制度解决方案,在空间平流中, 直径直线线线线线线路法则则则则则则是直径直线路法, 直线路路路路路法法,其结果是向一个直路路路路路路路路路路路路路路路路路路路,在直路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路路, 。