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On the convergence of split exponential integrators for semilinear parabolic problems

Marco Caliari, Fabio Cassini, Lukas Einkemmer, Alexander Ostermann

TL;DR

The paper addresses order reduction in exponential integrators caused by splitting φ-functions when solving semilinear parabolic problems with commuting operators. It develops an abstract semigroup framework to derive local error bounds for split φ1 and φℓ approximations and analyzes two split ETD2RK variants (ERK2L and ERK2) to establish their convergence properties, supported by numerical experiments in two and three spatial dimensions. The results show that split φ-function approximations can incur order reduction if boundary conditions are not satisfied, with ERK2L capable of reduced order while ERK2 remains robustly second-order under the stated assumptions. The work provides practical guidance for designing efficient, high-order split exponential integrators for stiff parabolic PDEs and points to future work on higher-order split φ-approximations.

Abstract

Splitting the exponential-like $\varphi$ functions, which typically appear in exponential integrators, is attractive in many situations since it can dramatically reduce the computational cost of the procedure. However, depending on the employed splitting, this can result in order reduction. The aim of this paper is to analyze different such split approximations. We perform the analysis for semilinear problems in the abstract framework of commuting semigroups and derive error bounds that depend, in particular, on whether the vector (to which the $\varphi$ functions are applied) satisfies appropriate boundary conditions. We then present the convergence analysis for two split versions of a second-order exponential Runge--Kutta integrator in the context of analytic semigroups, and show that one suffers from order reduction while the other does not. Numerical results for semidiscretized parabolic PDEs confirm the theoretical findings.

On the convergence of split exponential integrators for semilinear parabolic problems

TL;DR

The paper addresses order reduction in exponential integrators caused by splitting φ-functions when solving semilinear parabolic problems with commuting operators. It develops an abstract semigroup framework to derive local error bounds for split φ1 and φℓ approximations and analyzes two split ETD2RK variants (ERK2L and ERK2) to establish their convergence properties, supported by numerical experiments in two and three spatial dimensions. The results show that split φ-function approximations can incur order reduction if boundary conditions are not satisfied, with ERK2L capable of reduced order while ERK2 remains robustly second-order under the stated assumptions. The work provides practical guidance for designing efficient, high-order split exponential integrators for stiff parabolic PDEs and points to future work on higher-order split φ-approximations.

Abstract

Splitting the exponential-like functions, which typically appear in exponential integrators, is attractive in many situations since it can dramatically reduce the computational cost of the procedure. However, depending on the employed splitting, this can result in order reduction. The aim of this paper is to analyze different such split approximations. We perform the analysis for semilinear problems in the abstract framework of commuting semigroups and derive error bounds that depend, in particular, on whether the vector (to which the functions are applied) satisfies appropriate boundary conditions. We then present the convergence analysis for two split versions of a second-order exponential Runge--Kutta integrator in the context of analytic semigroups, and show that one suffers from order reduction while the other does not. Numerical results for semidiscretized parabolic PDEs confirm the theoretical findings.

Paper Structure

This paper contains 10 sections, 11 theorems, 79 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Splitting of phi functions
  3. Splitting of phi1
  4. Splitting of phiell
  5. Numerical experiments on the split approximations
  6. Convergence analysis of two split versions of ETD2RK
  7. Split exponential integrator ERK2L
  8. Split exponential integrator ERK2
  9. Numerical experiments
  10. Conclusions and future work

Key Result

Lemma 1

Under assum:scsassum:commut we have

Figures (4)

  • Figure 1: Observed decay rate of the local error $\lVert u_1-u(\tau)\rVert_\infty$ (see formula \ref{['eq:Strang1']}) for the function $v(x,y)=(16x(1-x)y(1-y))^p$, with $p=0,1,2,3$. The spatial discretization is performed with $250^2$ interior points and standard second-order finite differences. The slope of the dotted line is one, that of the dashed line is two and that of the solid line is three.
  • Figure 2: Observed decay rates of the local errors (in the infinity norm) of the split approximations to $\tau\varphi_1(\tau (A+B))v$ and $\tau\varphi_2(\tau(A+B))v$ for the different functions $v=v(x,y)$, see \ref{['tab:loc']}. The slope of the dotted line is one, that of the dashed line is two and that of the solid line is three.
  • Figure 3: Observed decay rates of the global errors (at $T=0.1$ in the infinity norm) of ERK2L \ref{['eq:ETD2RKdsg']}, ERK2 \ref{['eq:ETD2RKdsf']} and ETD2RK \ref{['eq:ETD2RKf']} for the two-dimensional example \ref{['eq:semilinear_2d']}. The slope of the dashed line is one and that of the solid line is two.
  • Figure 4: Observed decay rate of the global error (at $T=0.1$ in the infinity norm) of ERK2L \ref{['eq:ETD2RKdsg']} and ERK2 \ref{['eq:ETD2RKdsf']} for the three-dimensional example \ref{['eq:semilinear_3d']}. The slope of the dashed line is one and that of the solid line is two.

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • ...and 14 more