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Two new families of fourth-order explicit exponential Runge--Kutta methods with four stages for first-order differential systems

Xianfa Hu, Yonglei Fang, Bin Wang

TL;DR

The paper develops two fourth-order explicit exponential Runge–Kutta families (MVERK and SVERK) with four stages for stiff first-order systems $y'(t) + M y(t) = f(y(t))$. It shows the ERK order conditions match the classical RK conditions and that the schemes reduce to RK methods as $M \to \mathbf{0}$ while integrating the linear part exactly via $e^{-hM}$. A convergence theorem is established for the SVERK family under a local Lipschitz condition, and stability is analyzed using partitioned Dalquist tests, with results indicating slightly smaller stability regions than standard ERK4 methods. Numerical experiments across multiple challenging models demonstrate comparable accuracy to standard exponential integrators at lower computational cost, with MVERK generally outperforming SVERK due to simpler correction terms. Overall, the work provides practical, high-order ERK solvers that preserve linear dynamics exactly and are well-suited for a range of physics and engineering applications.

Abstract

In this paper, two new families of fourth-order explicit exponential Runge--Kutta (ERK) methods with four stages are studied for solving first-order differential systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor series of the exact solution, the order conditions of these ERK methods are derived, which are exactly identical to the order conditions of explicit Runge--Kutta methods, and these ERK methods reduce to classical Runge--Kutta methods once $M\rightarrow \mathbf{0}$. Moreover, we analyze the stability properties and the convergence of the new methods. Several numerical examples are implemented to illustrate the accuracy and efficiency of these ERK methods by comparison with standard exponential integrators.

Two new families of fourth-order explicit exponential Runge--Kutta methods with four stages for first-order differential systems

TL;DR

The paper develops two fourth-order explicit exponential Runge–Kutta families (MVERK and SVERK) with four stages for stiff first-order systems . It shows the ERK order conditions match the classical RK conditions and that the schemes reduce to RK methods as while integrating the linear part exactly via . A convergence theorem is established for the SVERK family under a local Lipschitz condition, and stability is analyzed using partitioned Dalquist tests, with results indicating slightly smaller stability regions than standard ERK4 methods. Numerical experiments across multiple challenging models demonstrate comparable accuracy to standard exponential integrators at lower computational cost, with MVERK generally outperforming SVERK due to simpler correction terms. Overall, the work provides practical, high-order ERK solvers that preserve linear dynamics exactly and are well-suited for a range of physics and engineering applications.

Abstract

In this paper, two new families of fourth-order explicit exponential Runge--Kutta (ERK) methods with four stages are studied for solving first-order differential systems . By comparing the Taylor series of the exact solution, the order conditions of these ERK methods are derived, which are exactly identical to the order conditions of explicit Runge--Kutta methods, and these ERK methods reduce to classical Runge--Kutta methods once . Moreover, we analyze the stability properties and the convergence of the new methods. Several numerical examples are implemented to illustrate the accuracy and efficiency of these ERK methods by comparison with standard exponential integrators.
Paper Structure (6 sections, 3 theorems, 64 equations, 6 figures)

This paper contains 6 sections, 3 theorems, 64 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A modified version of fourth-order explicit ERK methods
  3. A simplified version of fourth-order explicit ERK methods
  4. Convergence analysis
  5. Numerical Experiments
  6. Conclusions

Key Result

Theorem 2.2

If the coefficients of the four-stage explicit MVERK method with $w_4(z)$ where and $g(y_0)=-My_0+f(y_0)$, satisfy the order conditions order condition, then the explicit MVERK method has order four.

Figures (6)

  • Figure 1: (a): Stability region for the fourth-order explicit MVERK (EXMVERK41) method \ref{['MVERK41']} with four stages. (b): Stability region for the fourth-order explicit SVERK (EXSVERK41) method \ref{['SVERK41']} with four stages.
  • Figure 2: Results for Problem 1. (a): The $\log$-$\log$ plots of global errors (GE) against $h$. (b): The $\log$-$\log$ plots of global errors against the CPU time.
  • Figure 3: Results for Problem 2. (a): The $\log$-$\log$ plots of global errors (GE) against $h$. (b): The $\log$-$\log$ plots of global errors against the CPU time.
  • Figure 4: Results for Problem 3. (a): The $\log$-$\log$ plots of global errors (GE) against $h$. (b): The $\log$-$\log$ plots of global errors against the CPU time.
  • Figure 5: Results for Problem 4. (a): The $\log$-$\log$ plots of global errors (GE) against $h$. (b): The $\log$-$\log$ plots of global errors against the CPU time.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 4.2