Numerical Scheme Of Two-dimensional Camassa-Holm Equation By Quasi-interpolation Technique

Soliton plays a central role in the theory of nonlinear partial differential equation.Therefore,this thesis is devoted to the study of shallow water equations with soliton so-lutions by applying the Multiquadric（MQ）quasi-interpolation method.Firstly,the MQ quasi-interpolation method is used to solve the one-dimensional Camassa-Holm（CH）and Degasperis-Process（DP）equations numerically,and the con-vergence analysis of the methods are demonstrated with errors of O(Δth^2/3+h¹)and O（Δth¹+h²）,respectively.Secondly,the scheme for solving the two-dimensional Sine-Gordon equation is to ap-ply bivariate MQ quasi-interpolation to approximate spatial derivatives,and Euler forward formula to approximate time derivatives.The numerical results indicate that excellent er-ror accuracy can be achieved in a short period of time,the L₂and L_∞errors are obtained as10^-5and 10^-4,separately,which provides a reliable basis for adopting the bivariate MQ quasi-interpolation operator to the numerical solution of two-dimensional or even higher dimensional partial differential equations.Finally,the two-dimensional Camassa-Holm equation is approximated by bivariate MQ quasi-interpolation and third-order Runge-Kutta technique in space and time,respec-tively.The numerical results show the feasibility of this means.