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Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems

S. Gonzalez-Pinto, D. Hernandez-Abreu

TL;DR

This paper tackles the classical order-reduction problem in time integration for multidimensional parabolic PDEs with time-dependent boundary conditions when using directional splitting methods. It combines a fourth-order spatial discretization (MoL framework) with high-order AMF-W ADI-type time integrators and introduces two boundary-correction strategies—an interpolant construction and an operator-extension approach—to recover the PDE-convergence order. Theoretical results establish bounds in weighted and maximum norms under Dirichlet BCs, showing potential restoration of temporal order to as high as $p=3$ (and up to $3.25^{*}$ in the weighted norm) for time-independent BCs, with reductions under time-dependent BCs explained and mitigated. Numerical experiments on 2D and 3D problems, including nonlinear cases, corroborate the improved temporal convergence and fourth-order spatial accuracy, validating the proposed techniques and outlining paths to extend to Neumann/Robin BCs and broader applications.

Abstract

This work considers two boundary correction techniques to mitigate the reduction in the temporal order of convergence in PDE sense (i.e., when both the space and time resolutions tend to zero independently of each other) of $d$ dimension space-discretized parabolic problems on a rectangular domain subject to time dependent boundary conditions. We make use of the MoL approach (method of lines) where the space discretization is made with central differences of order four and the time integration is carried out with $s$-stage AMF-W-methods. The time integrators are of ADI-type (alternating direction implicit by using a directional splitting) and of higher order than the usual ones appearing in the literature which only reach order 2. Besides, the techniques here explained also work for most of splitting methods, when directional splitting is used. A remarkable fact is that with these techniques, the time integrators recover the temporal order of PDE-convergence at the level of time-independent boundary conditions.

Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems

TL;DR

This paper tackles the classical order-reduction problem in time integration for multidimensional parabolic PDEs with time-dependent boundary conditions when using directional splitting methods. It combines a fourth-order spatial discretization (MoL framework) with high-order AMF-W ADI-type time integrators and introduces two boundary-correction strategies—an interpolant construction and an operator-extension approach—to recover the PDE-convergence order. Theoretical results establish bounds in weighted and maximum norms under Dirichlet BCs, showing potential restoration of temporal order to as high as (and up to in the weighted norm) for time-independent BCs, with reductions under time-dependent BCs explained and mitigated. Numerical experiments on 2D and 3D problems, including nonlinear cases, corroborate the improved temporal convergence and fourth-order spatial accuracy, validating the proposed techniques and outlining paths to extend to Neumann/Robin BCs and broader applications.

Abstract

This work considers two boundary correction techniques to mitigate the reduction in the temporal order of convergence in PDE sense (i.e., when both the space and time resolutions tend to zero independently of each other) of dimension space-discretized parabolic problems on a rectangular domain subject to time dependent boundary conditions. We make use of the MoL approach (method of lines) where the space discretization is made with central differences of order four and the time integration is carried out with -stage AMF-W-methods. The time integrators are of ADI-type (alternating direction implicit by using a directional splitting) and of higher order than the usual ones appearing in the literature which only reach order 2. Besides, the techniques here explained also work for most of splitting methods, when directional splitting is used. A remarkable fact is that with these techniques, the time integrators recover the temporal order of PDE-convergence at the level of time-independent boundary conditions.
Paper Structure (18 sections, 3 theorems, 56 equations, 1 figure, 12 tables)

This paper contains 18 sections, 3 theorems, 56 equations, 1 figure, 12 tables.

Table of Contents

  1. Introduction
  2. Boundary Corrections in the time integration of spatially discretized parabolic PDEs
  3. The PDE problems considered
  4. The Spatial Semidiscretization
  5. $s$-stage AMF-W method $(A,L,b,\theta)$
  6. Illustration of the reduction in the temporal PDE-convergence order on a 3D parabolic academic example
  7. Some theoretical convergence results
  8. Convergence results in the $\ell_\infty$-norm
  9. Mitigating the order reduction through interpolants
  10. How can the interpolant $\varphi$ be built in the interior points of $\Omega$?
  11. Building the interpolant
  12. Boundary corrections based on extending the operator to the boundary
  13. Additional numerical experiments
  14. The PDE problems
  15. The AMF-W methods used
  16. ...and 3 more sections

Key Result

Theorem 1

(Weighted Euclidean norm, GHH-ESAIM2023). Consider linear systems of type (eq2-2) obtained from the space discretization of constant coefficient PDEs of type (eq1-0) with $a_j>0,\;b_j=0,\;j=1,\ldots,d$, by using second order central differences and Dirichlet boundary conditions. Additionally, assume Then, for the global errors $\epsilon_h(t_n):=u(\vec{x}_G,t_n)-V_h(\vec{x}_G,t_n)$, there exist two

Figures (1)

  • Figure 1: Stencils used on each direction for the diffusion case (operator $\partial^{(h)}_{x_j,x_j}V$). Observe that they are based on central second order differences for the adjacent points to the boundary and in central fourth order differences for the other interior points. For the advection terms (operator $\partial^{(h)}_{x_j}V$) the corresponding approaches of second and fourth order will be taken at the same points.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3