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Derivation of sixth-order exponential Runge--Kutta methods for stiff systems

Vu Thai Luan, Trky Alhsmy

TL;DR

This work addresses high-order time integration for stiff parabolic systems $u'(t)=A u(t)+g(t,u(t))$ by developing ExpRK methods of order $p=6$ using exponential $B$-series. It derives the full set of $36$ stiff order conditions (with $20$ being new) and demonstrates that they can be satisfied either strongly or with a weakened No. 17, enabling two parallelizable sixth-order families, ExpRK6s15 and ExpRK6s16. Convergence is established with a bound $ orm{e_n}\u2264 C h^6$ under the strong or weakened conditions, and the methods are designed to exploit parallel internal stages to achieve costs comparable to a 6-stage method. Numerical experiments validate sixth-order accuracy and parallel efficiency, while also highlighting potential order reduction under certain time-dependent boundary conditions, a phenomenon mitigated in lower-order ExpRK schemes.

Abstract

This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK methods of arbitrary order based on an essential set of trees only. Then, we explicitly provide the 36 order conditions required for sixth-order methods and present convergence results. In addition, we are able to solve the 36 stiff order conditions in both their weak and strong forms, resulting in two families of sixth-order parallel stages ExpRK schemes. Interestingly, while these new schemes require a high number of stages, they can be implemented efficiently similar to the cost of a 6-stage method. Numerical experiments are given to confirm the accuracy and efficiency of the new schemes.

Derivation of sixth-order exponential Runge--Kutta methods for stiff systems

TL;DR

This work addresses high-order time integration for stiff parabolic systems by developing ExpRK methods of order using exponential -series. It derives the full set of stiff order conditions (with being new) and demonstrates that they can be satisfied either strongly or with a weakened No. 17, enabling two parallelizable sixth-order families, ExpRK6s15 and ExpRK6s16. Convergence is established with a bound under the strong or weakened conditions, and the methods are designed to exploit parallel internal stages to achieve costs comparable to a 6-stage method. Numerical experiments validate sixth-order accuracy and parallel efficiency, while also highlighting potential order reduction under certain time-dependent boundary conditions, a phenomenon mitigated in lower-order ExpRK schemes.

Abstract

This work constructs the first-ever sixth-order exponential Runge--Kutta (ExpRK) methods for the time integration of stiff parabolic PDEs. First, we leverage the exponential B-series theory to restate the stiff order conditions for ExpRK methods of arbitrary order based on an essential set of trees only. Then, we explicitly provide the 36 order conditions required for sixth-order methods and present convergence results. In addition, we are able to solve the 36 stiff order conditions in both their weak and strong forms, resulting in two families of sixth-order parallel stages ExpRK schemes. Interestingly, while these new schemes require a high number of stages, they can be implemented efficiently similar to the cost of a 6-stage method. Numerical experiments are given to confirm the accuracy and efficiency of the new schemes.
Paper Structure (7 sections, 2 theorems, 25 equations, 1 figure, 1 table)

This paper contains 7 sections, 2 theorems, 25 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Stiff order conditions for ExpRK methods of order 6
  3. Convergence results for sixth-order stiff ExpRK methods
  4. Derivation of sixth-order stiff ExpRK methods
  5. A family of sixth-order parallel-stage ExpRK schemes with $s=15$
  6. A family of sixth-order parallel-stage ExpRK schemes with $s=16$
  7. Numerical experiments

Key Result

Lemma 2.1

Assuming that $A$ is the infinitesimal generator of an analytical semigroup ${\rm e}\space^{tA}$ on a Banach space $X$ and the nonlinearity $g\colon X \to X$ is sufficiently often Fréchet differentiable in a strip along the exact solution, and that $u\colon [t_0, t_{\text{end}}]\to X$ is sufficientl where $\mathcal{S}_i(\tau_k)(u)$ are the elementary differentials given recursively as with

Figures (1)

  • Figure 1: Order plots (left) and total CPU times (right) of $\mathtt{ExpRK6s15}$ and $\mathtt{ExpRK6s16}$ when applied to Example \ref{['ex1']}.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Example 5.1