Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

Abstract

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.

A Computational Approach to Solve a System of Transcendental Equations with Multi-Functions and Multi-Variables

“…The advent of the computer has made explicit solution and visualization of transcendental equations (TE) easier [1]. Computing has, indeed, expanded the possibilities of modelling and simulation of complex phenomena, processes, and systems [2]. However, encountering non-zero TE of the form f(x) = g(x) in science and engineering poses challenges, particularly when the TE is included in a system of equations to create a system of transcendental equations (SoTE).…”

Abel–Goncharov Type Multiquadric Quasi-Interpolation Operators with Higher Approximation Order

“…e authors have given that the approximating capacity of the operator L Λ n is comparable with that of the operator L H 2m− 1 . Furthermore, many researchers applied multiquadric quasiinterpolants to solve differential equations [15][16][17][18][19][20][21][22][23][24][25][26]. Meanwhile, Ali et al [27] constructed the SDI using Timmer triangular patches, which are used to visualize the energy data, i.e., spatial interpolation in visualizing rainfall data.…”