A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

Jin Cui; Yayun Fu; Jin Cui; Yayun Fu

doi:10.3934/nhm.2023016

Networks and Heterogeneous Media

2023, Volume 18, Issue 1: 399-411. doi: 10.3934/nhm.2023016

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A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

Jin Cui ¹,
Yayun Fu ^{2
,
,}

1.
Department of Basic Sciences, Nanjing Vocational College of Information Technology, Nanjing 210023, China
2.
School of Science, Xuchang University, Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China

Received: 05 September 2022 Revised: 13 December 2022 Accepted: 14 December 2022 Published: 06 January 2023

In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.
- Dispersive equation,
- Camassa-Holm equation,
- energy conservation,
- high-order linearly implicit scheme,
- Partitioned Runge-Kutta method
Citation: Jin Cui, Yayun Fu. A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations[J]. Networks and Heterogeneous Media, 2023, 18(1): 399-411. doi: 10.3934/nhm.2023016

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Abstract

In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.

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