Convergence and order reduction of diagonally implicit Runge-Kutta schemes in the method of lines. (English) Zbl 0642.65066
Numerical analysis, Proc. 11th Conf., Dundee/Scotl. 1985, Pitman Res. Notes Math. Ser. 140, 220-237 (1986).
[For the entire collection see Zbl 0637.00009.]
The method of lines (MOL) idea is simple in concept: for a given time dependent partial differential equation PDE) discretize the space variables so that the equation is converted into a continuous time system of ordinary differential equations (ODEs). This ODE system is then numerically integrated by an integration scheme, often one which can handle stiffness. Various known numerical schemes for PDEs can be viewed in this way. This contribution is devoted to an analysis for the full error of implicit Runge-Kutta MOL schemes. We will particularly concern ourselves with a class consisting of four known diagonally implicit methods although much of this paper will apply to other schemes as well. However, within the class of general implicit methods there is a significant computational advantage in diagonally implicit RK methods, especially for PDEs. With the exception of special circumstances, other types of implicit RK methods are in fact of rather limited practical value here.
The method of lines (MOL) idea is simple in concept: for a given time dependent partial differential equation PDE) discretize the space variables so that the equation is converted into a continuous time system of ordinary differential equations (ODEs). This ODE system is then numerically integrated by an integration scheme, often one which can handle stiffness. Various known numerical schemes for PDEs can be viewed in this way. This contribution is devoted to an analysis for the full error of implicit Runge-Kutta MOL schemes. We will particularly concern ourselves with a class consisting of four known diagonally implicit methods although much of this paper will apply to other schemes as well. However, within the class of general implicit methods there is a significant computational advantage in diagonally implicit RK methods, especially for PDEs. With the exception of special circumstances, other types of implicit RK methods are in fact of rather limited practical value here.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
