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

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An iterative method is developed for the .solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Osher's discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.

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

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