Factorization in block-triangularly implicit methods for shallow water applications

The system of first-order ordinary differential equations obtained by spatial discretization of the initial-boundary value problems modelling phenomena in shallow water in 3 spatial dimensions have righthand sides of the form f(t,y) := f1(t,y) + f2(t,y) + f3(t,y) + f4(t,y), where f1, f2 and f3 contain the spatial derivative terms with respect to the x, y and z directions, respectively, and f4 represents the forcing terms and/or reaction terms. The number N of components of f is usually extremely large. It is typical for shallow water applications that the function f4 is nonstiff and that the function f3 corresponding with the vertical spatial direction is much more stiff than the functions f1 and f2 corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem for the system of ordinary differential equations numerically, we need a stiff solver. Stiff IVP solvers are necessarily implicit, requiring the solution of large systems of implicit relations. In a few earlier papers, we considered implicit Runge-Kutta methods leading to fully coupled, implicit systems whose dimension is a multiple of N, and block-diagonally implicit methods in which the implicit relations can be decoupled into subsystems of dimension N. In the present paper, we analyse 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.