AMF-TASE W-Methods and PDE-Convergence in Euclidean Norm for Linear Parabolic Problems

Abstract 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 $$m\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . In case of time-independent Dirichlet boundary conditions, the nonstiff order conditions for order p , with $$p\le 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , are shown to be sufficient for PDE-convergence of order p (regardless of the spatial resolution) under mild stability assumptions. PDE-convergence of order $$p=3.25-\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3.25</mml:mn> <mml:mo>-</mml:mo> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:math> , for every $$\epsilon &gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ϵ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , is obtained assuming order conditions of order four. For time-dependent boundary conditions, the order of PDE-convergence is limited to $$p=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . Numerical experiments are presented to assess the sharpness of the PDE-convergence results.