Radial basis function-based quasi-interpolation performs efficiently in high-dimensional approximation and its applications, which can attain the approximant and its derivatives directly without solving any large-scale linear system. In this paper, the bivariate multi-quadrics (MQ) quasi-interpolation is used to simulate two-dimensional (2-D) Burgers' equation. Specifically, the spatial derivatives are approximated by using the quasi-interpolation, and the time derivatives are approximated by forward finite difference method. One advantage of the proposed scheme is its simplicity and easy implementation. More importantly, the proposed scheme opens the gate to meshless adaptive moving knots methods for the high-dimensional partial differential equations (PDEs) with shock or soliton waves. The scheme is also applicable to other non-linear high-dimensional PDEs. Two numerical examples of Burgers' equation (shock wave equation) and one example of the Sine-Gordon equation (soliton wave equation) are presented to verify the high accuracy and efficiency of this method.
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High-dimensional PDE, Meshless method, Quasi-interpolation, Shockwave
Zhang, Shenggang, Chungang Zhu, and Qinjiao Gao. "Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation." Mathematics 7, no. 8 (2019): 734.