Factorization in block-triangularly implicit methods for shallow water applications Original Research Article

The systems of first-order ordinary differential equations obtained by spatial discretization of the initial-boundary value problems modeling phenomena in shallow water in three spatial dimensions have right-hand sides of the form View the MathML source, where View the MathML source, View the MathML source and View the MathML source contain the spatial derivative terms with respect to the View the MathML source, View the MathML source and View the MathML source directions, respectively, and View the MathML source represents the forcing terms and/or reaction terms. It is typical for shallow water applications that the function View the MathML source is nonstiff and that the function View the MathML source corresponding with the vertical spatial direction is much more stiff than the functions View the MathML source and View the MathML source corresponding with the horizontal spatial directions. In order to solve the initial value problem for the system of ordinary differential equations numerically, we need a stiff solver. In a few earlier papers, we considered fully implicit Runge–Kutta methods and block-diagonally implicit methods. In the present paper, we analyze Rosenbrock type methods and the related DIRK methods (diagonally implicit Runge–Kutta methods) leading to block-triangularly implicit relations. In particular, we shall present a convergence analysis of various iterative methods based on approximate factorization for solving the triangularly implicit relations. Finally, the theoretical results are illustrated by a numerical experiment using a 3-dimensional shallow water transport model.