Speaker
Description
Results of convergence in $\ell_p$-norms for a MoL (Method of Lines) approach applied to $m$-dimensional ($m\ge 2$) linear parabolic problems of diffusion-reaction type on rectangular domains, are supplied. The space semi-discretization is based on central differences and the time integration is carried out with AMF-W-methods, which reduce the algebra computational costs to the level of one-dimensional problems on each spatial direction, in a similar way as methods of Alternating Direction Implicit-type (ADI) do.
Most of known results on convergence (PDE convergence) are restricted to the $\ell_2$-norm and currently depend on the number of spatial dimensions $m$. Here, we present results of convergence (PDE convergence) of order two in time (and in space) in both norms ($\ell_2$ and $\ell_\infty$), independently of the number of space dimensions $m$ and of the spatial grid, when time-independent boundary conditions are considered. In case of time-dependent boundary conditions, the PDE order of convergence is almost two in the $\ell_2$-norm when $m\ge 2$ and order one in the uniform norm for $m\ge 3$. Besides for $m=2$, a slight modification of the AMF-W method allows to get convergence of order almost two in the uniform norm. Some ideas about the proofs are presented and some numerical examples to illustrate the theory are also given.
References
[1] S. Gonzalez-Pinto, E. Hairer, D. Hernandez-Abreu,
Convergence in $\ell_2$ and $\ell_{\infty}$ norm of one-stage AMF-W methods for parabolic problems}, http://www.unige.ch/hairer/preprints.html (submitted for publication 2019).
[2] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer 2003, in Springer Series in Computational Mathematics.
[3] G. I. Marchuk, Splitting and alternating direction methods. In: Handbook of Numerical Analysis I, Eds. P.G. Ciarlet, J. L. Lions, North-Holland 1990, Amsterdam, pp. 197-462.
