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Phase Error Analysis of Implicit Runge–Kutta Methods: New Classes of Minimal Dissipation Low Dispersion High Order Schemes

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Abstract

In the current research, we analyze dissipation and dispersion characteristics of the most accurate two and three-stage Gauss–Legendre implicit Runge–Kutta (R-K) methods. These methods, known for their A-stability and high stage order, are observed to carry minimum dissipation error along with the highest possible dispersive order in their respective classes. Investigation reveals that these schemes are inherently optimized to carry low phase error only at small wavenumber. It is noticed that a unique scheme, although usually sought, might not be best across diverse temporal step sizes. As larger temporal step size is imperative in conjunction with implicit R-K methods for physical problems, we thoroughly investigate to derive a class of minimum dissipation and optimally low dispersion implicit R-K schemes. Schemes are obtained by cutting down amplification error and maximum reduction of weighted phase error, suggest better accuracy for relatively bigger and varied CFL numbers. A potentially generalizable algorithm is used to design stable implicit R-K methods. As the work focuses on two and three-stage schemes, a comprehensive comparison using numerical test cases document only modest gain in accuracy or efficiency over Gauss methods.

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Data Availability

The data sets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are thankful to the anonymous reviewers for their fruitful suggestions. These suggestions have enhanced the quality of the manuscript. The authors are thankful to Dr. Anjan K. Chakrabarty, Department of Mathematics, Indian Institute of Technology Guwahati, India for some intense discussions. Authors acknowledge the use of facilities at the High-Performance Computing Centre, Tezpur University sponsored by DeitY, India in collaboration with C-DAC, India. The first author is supported by University Grant Commission, India under Rajiv Gandhi National Fellowship (F1\(-\)17.1/2015-16/RGNF-2015-17-SC-WES-12451). The second author is thankful to Science & Engineering Research Board, India for assistance under Mathematical Research Impact Centric Support (File Number: MTR/2017/000038).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tezpur University, Tezpur, Sonitpur, Assam, 784028, India

    Subhajit Giri & Shuvam Sen

  2. Department of Mathematics, Faculty of Science and Technology, The ICFAI University Tripura, West Tripura, 799210, India

    Subhajit Giri

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  1. Subhajit Giri
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Correspondence to Shuvam Sen.

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Giri, S., Sen, S. Phase Error Analysis of Implicit Runge–Kutta Methods: New Classes of Minimal Dissipation Low Dispersion High Order Schemes. J Sci Comput 96, 9 (2023). https://doi.org/10.1007/s10915-023-02220-7

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  • DOI: https://doi.org/10.1007/s10915-023-02220-7

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