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High order multiquadric trigonometric quasi-interpolation method for solving time-dependent partial differential equations

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Abstract

In this paper, we propose a high order multiquadric trigonometric quasi-interpolation method for function approximation and derivative approximation based on periodic sampling data. Moreover, we apply it to solve time-dependent nonlinear partial differential equations (PDEs) with periodic solutions. Learning from the construction of the trigonometric B-spline quasi-interpolation, we derive the new quasi-interpolation scheme by replacing the truncated trigonometric polynomial with a high degree, infinitely smooth multiquadric trigonometric kernel. By appropriately choosing the shape parameter in the kernel, we prove that the proposed method achieves higher order of convergence than the existing multiquadric trigonometric quasi-interpolation method. Finally, we conduct extensive numerical experiments to demonstrate the accuracy and efficiency of the proposed method for approximating unknown function and solving different types of PDEs including the one-dimensional KdV equation and two-dimensional Allen-Cahn equation.

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Data Availability

The datasets and algorithms generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors wish to thank the reviewers for their suggestions and for carefully reading the manuscript.

Funding

This work is supported by the National Natural Science Foundation of China (Grant No. 12101310), National Natural Science Foundation of China Key Project (Grant No. 11631015), Natural Science Foundation of Jiangsu Province (Grant No. BK20210315), and 2021 Jiangsu Shuangchuang Talent Program (JSSCBS 20210222).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, China

    Zhengjie Sun

  2. School of Public Finance and Taxation, Nanjing University of Finance & Economics, Nanjing, China

    Yuyan Gao

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  1. Zhengjie Sun
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Contributions

Zhengjie Sun: theoretical analysis, writing, review. Yuyan Gao: coding, methodology, validation, review. All authors reviewed the manuscript.

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Correspondence to Yuyan Gao.

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Appendix: A. The expression of coefficients α i,j

Appendix: A. The expression of coefficients α i,j

For convenience, we present the formula to compute the coefficients αi,j appeared in (2.4) which was given in [21]. Let 1 ≤ l ≤ k and

$$t_{i}\leq \tau_{i,1}<\tau_{i,2}<\cdots<\tau_{i,l}\leq t_{i+k}$$

lie in the support [ti,ti+k] of the B-spline \({T_{i}^{k}}\) for 1 ≤ i ≤ n. Then for each 1 ≤ j ≤ l, let

$$ \alpha_{i,j}=\frac{\sum\limits_{\mathbf{\iota}=1}^{k-1}~_{k-1}\prod\limits_{v=1}^{l-1}\sin(\frac{t_{i+i_{v}}-\theta_{v}}{2})\prod\limits_{v=1}^{(k-l)/2}\cos(\frac{t_{i+i_{l+2v-1}}-t_{i+i_{l+2v-2}}}{2})}{(k-1)!\prod\limits_{v=1,v\neq j}^{l} \sin(\frac{\tau_{i,v}-\tau_{i,j}}{2})}, $$
(6.1)

where {𝜃1,…,𝜃l− 1} := {τi,1,…,τi,j− 1,τi,j+ 1,…,τi,l}. The first sum in the numerator is defined for multiple sums (eq.(5.7), [21]. Suppose \(A_{\iota }:=A_{i_{1},\ldots ,i_{m}}\) are real numbers defined for 1 ≤ i1,…,im ≤ k. Then

$$ \sum\limits_{\iota=1}^{k}~_{m}A_{\iota}:=\sum\limits_{i_{1}=1}^{k}\sum\limits_{\underset{i_{2}\neq i_{1}}{i_{2}=1}}^{k}\cdots\sum\limits_{\underset{i_{m}\neq i_{1},i_{2},\ldots,i_{m-1}}{i_{m}=1}}^{k}A_{i_{1},i_{2},\ldots,i_{m}}, $$

where ι stands for the multi-index (i1,…,im).

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Sun, Z., Gao, Y. High order multiquadric trigonometric quasi-interpolation method for solving time-dependent partial differential equations. Numer Algor 93, 1719–1739 (2023). https://doi.org/10.1007/s11075-022-01486-6

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