This manuscript deals with the time integration of spatially discretized parabolic Partial Differential Equations (PDEs) subject to Dirichlet boundary conditions on a rectangular m-dimensional domain. A class of linearly implicit methods (TASE W-methods, [11]) in combination with Approximate Matrix Factorization (AMF) [5] based on an alternating direction implicit approach is considered. The so-called AMF-TASE W-methods are efficient methods for the numerical solution of large systems of Ordinary Differential Equations arising from the spatial discretization of s uch PDE problems since they allow for parallelism and their algebraic cost is reduced to the level of one-dimensional problems.
Optimal results on PDE-convergence of AMF-TASE W-methods are provided for linear problems, the Euclidean norm and arbitrary spatial dimensions \(m\ge 2\) . In case of time-independent Dirichlet boundary conditions, the nonstiff order conditions for order p, with \(p\le 3\) , are shown to be sufficient for PDE-convergence of order p (regardless of the spatial resolution) under mild stability assumptions. PDE-convergence of order \(p=3.25-\epsilon \) , for every \(\epsilon >0\) , is obtained assuming order conditions of order four. For time-dependent boundary conditions, the order of PDE-convergence is limited to \(p=2\) . Numerical experiments are presented to assess the sharpness of the PDE-convergence results.
