A shift‐adaptive meshfree method for solving a class of initial‐boundary value problems with moving boundaries in one‐dimensional domain

Abstract

A new shift‐adaptive meshfree method for solving a class of time‐dependent partial differential equations (PDEs) in a bounded domain (one‐dimensional domain) with moving boundaries and nonhomogeneous boundary conditions is introduced. The radial basis function (RBF) collocation method is combined with the finite difference scheme, because, unlike with Kansa's method, nonlinear PDEs can be converted to a system of linear equations. The grid‐free property of the RBF method is exploited, and a new adaptive algorithm is used to choose the location of the collocation points in the first time step only. In fact, instead of applying the adaptive algorithm on the entire domain of the problem (like with other existing adaptive algorithms), the new adaptive algorithm can be applied only on time steps. Furthermore, because of the radial property of the RBFs, the new adaptive strategy is applied only on the first time step; in the other time steps, the adaptive nodes (obtained in the first time step) are shifted. Thus, only one small system of linear equations must be solved (by LU decomposition method) rather than a large linear or nonlinear system of equations as in Kansa's method (adaptive strategy applied to entire domain), or a large number of small linear systems of equations in the adaptive strategy on each time step. This saves a lot in time and memory usage. Also, Stability analysis is obtained for our scheme, using Von Neumann stability analysis method. Results show that the new method is capable of reducing the number of nodes in the grid without compromising the accuracy of the solution, and the adaptive grading scheme is effective in localizing oscillations due to sharp gradients or discontinuities in the solution. The efficiency and effectiveness of the proposed procedure is examined by adaptively solving two difficult benchmark problems, including a regularized long‐wave equation and a Korteweg‐de Vries problem. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1622–1646, 2016

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

“…In Tables 2-4 we present an extensive and detailed analysis on the RBF-PUM-FD collocation method, which is also compared with the global RBF-FD method studied in [24]. More precisely, considering four sets of uniform and Halton points and IMQ as local RBF approximant, we report the MAE computed with "optimal" ǫ, the CPU time required to solve the sparse linear system (29) and the CN.…”

“…Then, in Tables 7-10 we compare our RBF-PUM-FD scheme with the RBF-FD method studied in [24]. Therefore, for some sets of uniform and Halton points we show the numerical results obtained by using IMQ and GA as local RBF approximants.…”

“…The target of our numerical analysis is thus two-fold: on the one hand, analyzing the efficiency expressed in terms of CPU times of the RBF-PUM-FD; on the other hand, verifying its accuracy and stability. In addition, we also compare our collocation method with two existing techniques: the RBF-FD method presented in [24] and the RBF-PUM proposed in [43].…”

“…The semi-implicit scheme is somewhere between the explicit and implicit schemes. Although the conclusion as to which type of scheme should be chosen is to consider case by case, usually the implicit scheme is preferred for stiff initial value problems [36,24].…”

“…In this work, we thus have a two-fold scope: first, exhibiting efficiency of our RBF-PUM-FD method in terms of CPU times; then, analyzing its numerical accuracy and stability. Finally, in order to show goodness of our approach, we compare our collocation method with two existing computational techniques: the global RBF-FD method proposed in [24] and the RBF-PUM studied in [43].…”