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

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.

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

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