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Exponential integrators for parabolic problems with non-homogeneous boundary conditions

Carlos Arranz-Simón, Alexander Ostermann

TL;DR

This paper addresses order reduction in exponential Runge-Kutta methods for parabolic problems with non-homogeneous boundary conditions by introducing a smooth boundary extension to reformulate the problem as homogeneous with a modified source term. It proves that linear problems recover the expected order after correction, and that Gauss collocation can achieve order $2s$ with suitable corrections; for semilinear problems, convergence orders governed by stiff order conditions are preserved. The authors further show that additional boundary-compatible corrections can raise the linear order to $s+m+1$, with Gauss methods attaining classical orders under appropriate conditions, and validate these results through extensive numerical experiments in 1D and 2D. The approach incurs negligible overhead and is extendable to other exponential-type integrators, making high-accuracy integration feasible for parabolic systems with non-homogeneous boundaries.

Abstract

Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this paper, we extend classical convergence results to the case of non-homogeneous boundary conditions. Since non-homogeneous boundary conditions typically cause order reduction, we introduce a correction strategy based on smooth extensions of the boundary data. This results in a reformulation as a homogeneous problem with modified source term, to which standard exponential integrators can be applied. For linear problems, we prove that the corrected schemes recover the expected convergence order, and hat higher orders can be attained with suitable quadrature rules, reaching order $2s$ for s-stage Gauss collocation methods. For semilinear problems, our approach preserves the convergence orders guaranteed by exponential Runge-Kutta methods satisfying the corresponding stiff order conditions. Numerical experiments validate the theoretical findings.

Exponential integrators for parabolic problems with non-homogeneous boundary conditions

TL;DR

This paper addresses order reduction in exponential Runge-Kutta methods for parabolic problems with non-homogeneous boundary conditions by introducing a smooth boundary extension to reformulate the problem as homogeneous with a modified source term. It proves that linear problems recover the expected order after correction, and that Gauss collocation can achieve order with suitable corrections; for semilinear problems, convergence orders governed by stiff order conditions are preserved. The authors further show that additional boundary-compatible corrections can raise the linear order to , with Gauss methods attaining classical orders under appropriate conditions, and validate these results through extensive numerical experiments in 1D and 2D. The approach incurs negligible overhead and is extendable to other exponential-type integrators, making high-accuracy integration feasible for parabolic systems with non-homogeneous boundaries.

Abstract

Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this paper, we extend classical convergence results to the case of non-homogeneous boundary conditions. Since non-homogeneous boundary conditions typically cause order reduction, we introduce a correction strategy based on smooth extensions of the boundary data. This results in a reformulation as a homogeneous problem with modified source term, to which standard exponential integrators can be applied. For linear problems, we prove that the corrected schemes recover the expected convergence order, and hat higher orders can be attained with suitable quadrature rules, reaching order for s-stage Gauss collocation methods. For semilinear problems, our approach preserves the convergence orders guaranteed by exponential Runge-Kutta methods satisfying the corresponding stiff order conditions. Numerical experiments validate the theoretical findings.
Paper Structure (6 sections, 6 theorems, 69 equations, 9 tables)

This paper contains 6 sections, 6 theorems, 69 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Linear problems
  3. More corrections and some numerical illustrations
  4. Semilinear problems
  5. Conclusions
  6. Acknowledgments

Key Result

Theorem 1

Consider the exponential Runge--Kutta method (unumlinearscheme) for the numerical solution of (linearproblem). If $f^{(s)} + k^{(s)} \in L^1\left(0,T;X\right)$, then the method converges with order $s$ and the error satisfies for $0 \leq t_n \leq T$. The constant $C$ depends on $T$, but is independent of $n$ and $\left\lbrace \tau_n \right\rbrace_{n=0}^{N-1}$.

Theorems & Definitions (18)

  • Theorem 1
  • proof
  • Theorem 2
  • Remark
  • proof : Proof of Theorem \ref{['thm:order-s+1']}
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • ...and 8 more