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A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

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  • Jiong Weng

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Xiaojing Liu

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Youhe Zhou

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

  • Jizeng Wang

    (Key Laboratory of Mechanics on Disaster and Environment in Western China, The Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China)

Abstract

A space-time fully decoupled wavelet integral collocation method (WICM) with high-order accuracy is proposed for the solution of a class of nonlinear wave equations. With this method, wave equations with various nonlinearities are first transformed into a system of ordinary differential equations (ODEs) with respect to the highest-order spatial derivative values at spatial nodes, in which all the matrices in the resulting nonlinear ODEs are constants over time. As a result, these matrices generated in the spatial discretization do not need to be updated in the time integration, such that a fully decoupling between spatial and temporal discretization can be achieved. A linear multi-step method based on the same wavelet approximation used in the spatial discretization is then employed to solve such a semi-discretization system. By numerically solving several widely considered benchmark problems, including the Klein/sine–Gordon equation and the generalized Benjamin–Bona–Mahony–Burgers equation, we demonstrate that the proposed wavelet algorithm possesses much better accuracy and a faster convergence rate than many existing numerical methods. Most interestingly, the space-associated convergence rate of the present WICM is always about order 6 for different equations with various nonlinearities, which is in the same order with direct approximation of a function in terms of the proposed wavelet approximation scheme. This fact implies that the accuracy of the proposed method is almost independent of the equation order and nonlinearity.

Suggested Citation

  • Jiong Weng & Xiaojing Liu & Youhe Zhou & Jizeng Wang, 2021. "A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations," Mathematics, MDPI, vol. 9(22), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:22:p:2957-:d:683154
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    References listed on IDEAS

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