GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
VODE: a variable-coefficient ODE solver
SIAM Journal on Scientific and Statistical Computing
Iterative schemes for three-stage implicit Runge-Kutta methods
Applied Numerical Mathematics
Triangularly Implicit Iteration Methods for ODE-IVP Solvers
SIAM Journal on Scientific Computing
A note on stability of the Douglas splitting method
Mathematics of Computation
Physics-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion
Journal of Computational Physics
Approximate factorization for time-dependent partial differential equations
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
An Implicit-Explicit Runge--Kutta--Chebyshev Scheme for Diffusion-Reaction Equations
SIAM Journal on Scientific Computing
IRKC: an IMEX solver for stiff diffusion-reaction PDEs
Journal of Computational and Applied Mathematics
A variable time-step-size code for advection-diffusion-reaction PDEs
Applied Numerical Mathematics
A comparison of AMF- and Krylov-methods in Matlab for large stiff ODE systems
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
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This paper is concerned with the time integration of semi-discretized, multi-dimensional PDEs of advection-diffusion-reaction type. To cope with the stiffness of these ODEs, an implicit method has been selected, viz., the two-stage, third-order Radau IIA method. The main topic of this paper is the efficient solution of the resulting implicit relations. First, a modified Newton process has been transformed into an iteration process in which the 2 stages are decoupled and, moreover, can exploit the same LU-factorization of the iteration matrix. Next, we apply a so-called Approximate Matrix Factorization (AMF) technique to solve the linear systems in each Newton iteration. This AMF approach is very efficient since it reduces the 'multi-dimensional' system to a series of 'one-dimensional' systems. The total amount of linear algebra work involved is reduced enormously by this approach. The idea of applying AMF to two-dimensional problems is quite old and goes back to Peaceman and Rachford in the early fifties. The situation in three space dimensions is less favourable and will be analyzed here in more detail, both theoretically and experimentally. Furthermore, we analyze a variant in which the AMF-technique has been used to really solve ('until convergence') the underlying Radau IIA method so that we can rely on its excellent stability and accuracy characteristics. Finally, the method has been tested on several examples. Also, a comparison has been made with the existing codes VODPK and IMEXRKC, and the efficiency (CPU time versus accuracy) is shown to be at least competitive with the efficiency of these solvers.