Multigrid Solution of Monotone Second-Order Discretizations of Hyperbolic Conservation Laws

Abstract

Abstract. This paper is concerned with two subjects: the construction of second-order accurate monotone upwind schemes for hyperbolic conservation laws and the multigrid solution of the resulting discrete steady-state equations. By the use of an appropriate definition of monotonicity, it is shown that there is no conflict between second-order accuracy and monotonicity (neither in one nor in more dimensions).It is shown that a symmetric block Gauss-Seidel underrelaxation (each block is associated with 4 cells) has satisfactory smoothing rates. The success of this relaxation is due to the fact that, by coupling the unknowns in such blocks, the nine-point stencil of a second-order 2D upwind discretization changes into a five-point block stencil.1. Introduction. To obtain solutions of first-order finite-volume upwind schemes for the 2D steady Euler equations, nested nonlinear multigrid (FMG-FAS) iteration has proved to be a very efficient solution process.[6], [7]. Encouraged by this successful application of nonlinear multigrid, it is natural to ask whether it is possible to use nonlinear multigrid for the efficient solution of second-order finite-volume monotone upwind schemes as well.To answer this question, we have to discuss the following subjects: how to construct a second-order montone upwind scheme and how to choose the nonlinear multigrid components such as the relaxation method, the restriction and prolongation operators, and the coarse grid operators.Because of the complexity of the Euler equations (a hyperbolic system of conservation laws), we start analyzing these subjects for the less complicated scalar hyperbolic conservation laws. Scalar hyperbolic conservation laws are interesting by themselves and, without the complexity of hyperbolic systems, the analysis is more complete and more transparent. The results of the scalar analysis can be generalized, in a straightforward manner, to systems of hyperbolic conservation laws such as the Euler equations. We will report on this in a separate paper.In Section 2 we describe the construction of second-order monotone upwind schemes. By using a definition of monotonicity based on positivity of coefficients, it is shown that there is no contradiction between monotonicity and second-order accuracy (neither in one nor in more dimensions). We emphasize that the concept of monotone schemes used in this paper is not equivalent with the definition of

On the link between weighted least-squares and limiters used in higher-order reconstructions for finite volume computations of hyperbolic equations

“…8 were computed using a uniform 64 × 32 grid. Both the first and second order (using SD-WLS scheme) results compare well with the results of Spekreijse [18]. The results obtained using the SDWLS scheme (in Fig.…”

“…The complexity of the test examples follows a logical course from the simplest hyperbolic equation-the one dimensional linear advection equation, to its non-linear counterpart, the inviscid Burger's equation in both 1D and 2D cases. To demonstrate multi-dimensional feasibility, two 2D examples are illustrated, one being a linear convection equation and the other being a non-linear convection equation [18]. These examples are computed on a structured grid for simplicity.…”

“…Here, the problem is a linear one with a square domain [0, 1] × [0, 1], as given by Spekreijse [18].…”

“…This is also a test problem solved by Spekreijse [18]. The problem is a linear convection problem on a rectangular…”

“…This test problem also solved by Spekreijse [18], is a non-linear one with a shock at the center of the square domain The governing equation is:…”