The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi-discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function-finite difference method. The convergence rate and stability of the fully-discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme.
{"title":"A radial basis function (RBF)-finite difference method for solving improved Boussinesq model with error estimation and description of solitary waves","authors":"Mostafa Abbaszadeh, AliReza Bagheri Salec, Taghreed Abdul-Kareem Hatim Aal-Ezirej","doi":"10.1002/num.23077","DOIUrl":"https://doi.org/10.1002/num.23077","url":null,"abstract":"The Boussinesq equation has some application in fluid dynamics, water sciences and so forth. In the current paper, we study an improved Boussinesq model. First, a finite difference approximation is employed to discrete the derivative of the temporal variable. Then, we study the existence and uniqueness of solution of the semi-discrete scheme according to the fixed point theorem. In addition, the unconditional stability and convergence of the semi-discrete scheme are presented. Then, we construct the fully discrete formulation based upon the radial basis function-finite difference method. The convergence rate and stability of the fully-discrete scheme are analyzed. In the end, some examples in 1D and 2D cases are studied to corroborate the capability of the proposed scheme.","PeriodicalId":19443,"journal":{"name":"Numerical Methods for Partial Differential Equations","volume":"53 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138532623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work develops two temporal second-order alternating direction implicit (ADI) numerical schemes for solving multidimensional parabolic-type integrodifferential equations with multi-term weakly singular Abel kernels. For the two-dimensional (2D) case, applying the Crank–Nicolson method and product integration rule to discretizations of temporal derivative and integral terms, respectively, and the spatial discretization is proposed using a compact difference formulation combined with the ADI algorithm; for the three-dimensional case, the method of temporal discretization is the same as the 2D case, and then we employ the standard finite difference in space to construct a fully discrete ADI finite difference scheme. The ADI technique is used to reduce the calculation cost of the high-dimensional problem. Besides, the stability and convergence of two ADI schemes are rigorously proved by the energy argument, in which the first scheme converges to the order