Factorization in block-triangularly implicit methods for shallow water applications

Abstract

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 f(t, y):= f 1(t, y)+ f 2(t, y)+ f 3(t, y)+ f 4(t, y) , where f 1 , f 2 and f 3 contain the spatial derivative terms with respect to the x 1 , x 2 and x 3 directions, respectively, and f 4 represents the forcing terms and/or reaction terms. It is typical for shallow water applications that the function f 4 is nonstiff and that the function f 3 corresponding with the vertical spatial direction is much more stiff than the functions f 1 and f 2 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.