Numerical solutions of the perturbed Sine-Gordon equation in two space variables, arising from a Josephson junction are presented. The method proposed arises from a two-step, one parameter method for the numerical solution of second-order ordinary differential equations. Though implicit in nature, the method is applied explicitly. Global extrapolation in both space and time is used to improve the accuracy. The method is analysed with respect to stability criteria and numerical dispersion. Numerical results are obtained for various cases involving line and ring solitons.

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SUMMARYSimple, mesh=grid free, explicit and implicit numerical schemes for the solution of linear advection-di usion problems is developed and validated herein. Unlike the mesh or grid-based methods, these schemes use well distributed quasi-random points and approximate the solution using global radial basis functions. The schemes can be seen as generalized ÿnite di erences with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no need for complex mesh=grid structure and with no extra implementation di culties.

Please cite this article as: Zhe Sun , K. Djidjeli , Jing T. Xing , Fai cheng , Modified mps method for the 2d fluid structure interaction problem with free surface, Computers and Fluids (2015), doi: 10.1016/j.compfluid.2015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Meshless methods for solving fluid and fluid-structure problems have become a promising alternative to the finite volume and finite element methods. In this paper, a mesh-free computational method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier—Stokes (NS) equations in stream function vorticity form. This compact RBF-FD formulation generates sparse coefficient matrices, and hence advancing solutions will in time be of comparatively lower cost. The spatial discretization of the incompressible NS equations is done using the RBF-FD method and the temporal discretization is achieved by explicit Euler time-stepping and the Crank—Nicholson method. A novel ghost node strategy is used to incorporate the no-slip boundary conditions. The performance of the RBF-FD scheme with the ghost node strategy is validated against a variety of benchmark problems, including a model fluid—structure interaction problem, and is found to be in a good agreement with the existing results. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the NS equations.

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SUMMARYAn e cient numerical method is developed for the numerical solution of non-linear wave equations typiÿed by the regularized long wave equation (RLW) and its generalization (GRLW). The method developed uses a pseudo-spectral (Fourier transform) treatment of the space dependence together with a linearized implicit scheme in time.An important advantage to be gained from the use of this method, is the ability to vary the mesh length, thereby reducing the computational time. Using a linearized stability analysis, it is shown that the proposed method is unconditionally stable. The method is second order in time and all-order in space.The method presented here is for the RLW equation and its generalized form, but it can be implemented to a broad class of non-linear long wave equations (Equation (2)), with obvious changes in the various formulae.Test problems, including the simulation of a single soliton and interaction of solitary waves, are used to validate the method, which is found to be accurate and e cient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm.