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.
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