Radial Basis Function Interpolation And Its Application In Shallow Water Wave Equation——Error Analysis And Algorithm Testing

The radial basis function method is a powerful tool for solving partial differential equations.We mainly analyze the error of Multi-Quadri quasi-interpolation?MQQI?in combination with the properties of shallow water wave equations in this thesis.Therefore,we focus on the interpolation error estimation formula and algorithm test.The main research content is divided into the following two parts.First,we display the MQQI and the Finite Difference Method?FDM?to solve the KdV equation separately.Afterwards,we analyze the error of the exact solution and numerical solution of the FDM and the error of the approximate solution of the FDM and the interpolation solution of the MQQI so that the error estimation formula of MQQI solving KdV equation is derived.According to the above process,we know that the error is O(?1+?t?h^min{2,l-1})when initial value condition u0 satisfies C^k?k?5?.Take the KdV equation as an example,numerical examples illustrate that the feasibility and effectiveness of the constructed error analysis method.Second,we introduce the MQQI and the Finite Difference Method to solve the Camassa-Holm?C-H?equation and the Degasperis-Procisi?D-P?equation respectively.Combining the error estimation of between the exact solution and the numerical solu-tion obtained by MQQI and FDM and between the approximate solution of the FDM and the interpolation solution of the MQQI,we obtain the interpolation error of the Camassa-Holm equation and the Degasperis-Procisi equation.The error of the Camassa-Holm equation is O(?th^min{2,l-1})+O(h^min{2,l-1})when initial value condition u0 satisfies C^k?k?4?.For a short time,the error of the Degasperis-Procisi equation is O??th²??O(h^min{2,l-1})when initial value condition u0 satisfies C^k?k?3?.