Year: 2025
Type: article
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 $$m\ge 2$$ m ≥ 2 . In case of time-independent Dirichlet boundary conditions, the nonstiff order conditions for order p , with $$p\le 3$$ p ≤ 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 $$p=3.25-\epsilon $$ p = 3.25 - ϵ , for every $$\epsilon >0$$ ϵ > 0 , is obtained assuming order conditions of order four. For time-dependent boundary conditions, the order of PDE-convergence is limited to $$p=2$$ p = 2 . Numerical experiments are presented to assess the sharpness of the PDE-convergence results.
Source: Journal of Scientific Computing
Institution: Universidad de La Laguna
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
Award: G5828273025
OpenAlex ID: W4414620422