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__Refinements in the Approximate Matrix Factorization for the time integration of advection-diffusion-reaction PDEs__

S.G. Pinto, D.H. Abreu and S.P. Rodriguez

**Abstract**

The numerical integration of PDEs of Advection Diffusion Reaction
type in several spatial variables in the MoL framework is
considered. The spatial discretization is based on Finite
Differences and the time integration is carried out by using
splitting techniques applied to some Rosenbrock-type methods. The focus
is to provide a way of making some refinements to the usual
Approximate Matrix Factorization (AMF), here considered as the splitting technique to solve the large linear systems of equations. The AMF-refinements
allow to recover the convergence order of the underlying method and in some cases to enlarge the linear stability regions and the Courant numbers with regard to the standard AMF-scheme. Most of these methods
belong to the class of the W-methods. A few numerical
experiments on some important 2D and 3D non-linear PDE problems with
applications in Physics are presented. The development of some integration codes (in Fortran, Matlab, R) is in progress.