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Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation

Author

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  • Yi Ji

    (MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China)

  • Yufeng Xing

    (Institute of Solid Mechanics, Beihang University, Beijing 100083, China)

Abstract

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2 m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10 −16 (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

Suggested Citation

  • Yi Ji & Yufeng Xing, 2023. "Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation," Mathematics, MDPI, vol. 11(3), pages 1-36, January.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:593-:d:1044687
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    References listed on IDEAS

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    1. Jinyue Zhang & Lei Shi & Tianhao Liu & De Zhou & Weibin Wen, 2021. "Performance of a Three-Substep Time Integration Method on Structural Nonlinear Seismic Analysis," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-20, December.
    2. Mahmoud Saleh & Endre Kovács & Imre Ferenc Barna & László Mátyás, 2022. "New Analytical Results and Comparison of 14 Numerical Schemes for the Diffusion Equation with Space-Dependent Diffusion Coefficient," Mathematics, MDPI, vol. 10(15), pages 1-26, August.
    3. Yi Ji & Huan Zhang & Yufeng Xing, 2022. "New Insights into a Three-Sub-Step Composite Method and Its Performance on Multibody Systems," Mathematics, MDPI, vol. 10(14), pages 1-28, July.
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    Cited by:

    1. Issa Omle & Ali Habeeb Askar & Endre Kovács & Betti Bolló, 2023. "Comparison of the Performance of New and Traditional Numerical Methods for Long-Term Simulations of Heat Transfer in Walls with Thermal Bridges," Energies, MDPI, vol. 16(12), pages 1-27, June.
    2. Sergei Sitnik, 2023. "Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”," Mathematics, MDPI, vol. 11(15), pages 1-7, August.

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