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A Convergent Iterated Quasi-interpolation for Periodic Domain and Its Applications to Surface PDEs

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Abstract

The paper first provides an iterated quasi-interpolation scheme for function approximation over periodic domain and then attempts its applications to solve time-dependent surface partial differential equations (PDEs). The key feature of our scheme is that it gives an approximation directly just by taking a weighted average of the available discrete function values evaluated at sampling centers in the periodic domain. As such, it is simple, easy to compute, and the implementation process is stable. Moreover, if the sampling centers distribute uniformly over the periodic domain, it even preserves the same convergence order to all the derivatives. To employ the iterated quasi-interpolation scheme for solving time-dependent surface PDEs, we follow the framework of the method-of-lines. More precisely, we first reformulate the PDEs in terms of the parametric form of the surface. Then we use our quasi-interpolation scheme to approximate both the analytic solution and its spatial derivatives in the reformulated form (of PDEs) to get a semi-discrete ordinary differential equation (ODE) system. Finally, we adopt an appropriate time-integration technique to obtain the full-discrete scheme. As concrete examples, we take the torus for illustration and solve some benchmark reaction-diffusion equations imposed on the torus at the end of the paper. However, the proposed method is general and works on time-dependent PDEs defined on any smooth closed parameterized surfaces without coordinate singularity.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant Nos. 12101310, 11871074, 11501006, 61672032), National Natural Science Foundation of China Key Project (Grant No. 11631015), Natural Science Foundation of Jiangsu Province (Grant No. BK20210315), 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 Big Data and Statistics, Anhui University, Hefei, China

    Wenwu Gao

  3. Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, China

    Ran Yang

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Correspondence to Ran Yang.

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Appendix A: Fourier Coefficients of Multiquadrics

Appendix A: Fourier Coefficients of Multiquadrics

For \(\kappa (r)=(a^2+r^2)^{\lambda }\) with \(\lambda >0\), the Fourier coefficient of \(\phi \) is given by

$$\begin{aligned} {\widehat{\phi }}(\ell )=\mathcal {A}_{\lambda ,\ell ,d}\cdot F\left( \ell -\lambda ,\ell +\frac{d-1}{2};2l+d-1;\frac{4}{4+a^2}\right) , \end{aligned}$$
(A.1)

where

$$\begin{aligned} \mathcal {A}_{\lambda ,\ell ,d}=\frac{(-1)^{\ell }\pi ^{\frac{d-1}{2}}2^{2l+d-1}}{(4+c^2)^{\ell -\lambda }}\frac{\Gamma (\lambda +1)}{\Gamma (\lambda -\ell +1)}\frac{\Gamma (\ell +\frac{d-1}{2})}{\Gamma (2l+d-1)}, \end{aligned}$$

and F(a, b, c; z) is the Gauss hypergeometric series defined by

$$\begin{aligned} F(a,b,c;z)=\frac{\Gamma (c)}{\Gamma (a)\Gamma (b)}\sum _{n=0}^{\infty }\frac{\Gamma (a+n)\Gamma (b+n)}{\Gamma (c+n)}\frac{z^n}{n!}. \end{aligned}$$

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Sun, Z., Gao, W. & Yang, R. A Convergent Iterated Quasi-interpolation for Periodic Domain and Its Applications to Surface PDEs. J Sci Comput 93, 37 (2022). https://doi.org/10.1007/s10915-022-01998-2

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