Approximate factorization for time-dependent partial differential equations

AbstractThe first application of approximate factorization in the numerical solution of time-dependent partial differential equations (PDEs) can be traced back to the celebrated papers of Peaceman and Rachford and of Douglas of 1955. For linear problems, the Peaceman–Rachford–Douglas method can be derived from the Crank–Nicolson method by the approximate factorization of the system matrix in the linear system to be solved. This factorization is based on a splitting of the system matrix. In the numerical solution of time-dependent PDEs we often encounter linear systems whose system matrix has a complicated structure, but can be split into a sum of matrices with a simple structure. In such cases, it is attractive to replace the system matrix by an approximate factorization based on this splitting. This contribution surveys various possibilities for applying approximate factorization to PDEs and presents a number of new stability results for the resulting integration methods

Approximate factorization in shallow water applications

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the righthand side of the resulting system of ordinary differential equations can be split into terms f1, f2, f3 and f4, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term 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 (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported