\documentclass{simai-2025} \usepackage{pslatex} %\usepackage{amsmath} %\usepackage{amsfonts} %\usepackage{amssymb} \title{Fourth order discretization in space for Advection Diffusion Reaction equations under Dirichlet boundary conditions} \author{S. Gonz\'alez-Pinto $^{*}$, E. Hairer $^{\dag}$ and D. Hern\'andez-Abreu$^{*}$} \address{$^{*}$ Dpto An\'alisis Matem\'atico, Universidad de la Laguna La Laguna, Tenerife, Spain\\ e-mail: spinto@ull.edu.es, web page: https://portalciencia.ull.es/investigadores/81839/publicaciones/ \and $^{\dag}$ Université de Genève Section de mathématiques rue du Conseil-Général 7-9 CH-1205 Genève Switzerland, web page: https://www.unige.ch/~hairer/} \begin{document} %\maketitle \begin{center} \bf ABSTRACT \end{center} A fourth order discretization in space based on finite differences for time-dependent advection diffusion reaction PDEs under Dirichlet boundary conditions is considered. The discretization is quite simple to implement, since it makes use of directional splitting on each space variable and its stencil only takes five points for the interior points of the space domain (based on central differences of four order) and three points for the neighbouring points to the boundary (based on second order central differences). This space discretization has been successfully used in \cite{GHH-apnum2025}, where the global fourth order of approximation has been numerically confirmed. The point here, is the theoretical justification of this global fourth order approximation, which is somewhat surprising since only a second order approach is used in the neighbouring points to the boundary. The fourth order discretization is quite useful for parabolic problems in practice, since it allows a very low number of grid-points on each space dimension when it is compared with the usual second order stencil based on second order central differences. Some ideas to develop such a discretization and to prove the main results are based on the material in \cite[Chapt.I-IV]{HV-spinger2003} and references therein. \begin{thebibliography}{99} \bibitem{GHH-apnum2025} Gonz\'alez-Pinto and Hern\'andez-Abreu, D. Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems. Applied Numerical Mathematics (2025), 210:95-112. \bibitem{HV-spinger2003} Hundsdorfer, W. and Verwer J.G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computatio