scholarly journals A Direct Meshless Method for Solving Two-Dimensional Second-Order Hyperbolic Telegraph Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Fuzhang Wang ◽  
Enran Hou
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Meshless Method ◽  
Space Variable ◽  
Second Order ◽  
Time Variable ◽  
Time Dependent ◽  
Two Dimensional ◽  
Radial Basis ◽  
Telegraph Equations

In this paper, a direct meshless method (DMM), which is based on the radial basis function, is developed to the numerical solution of the two-dimensional second-order hyperbolic telegraph equations. Since these hyperbolic telegraph equations are time dependent, we present two schemes for the basis functions from radial and nonradial aspects. The first scheme is fulfilled by considering time variable as normal space variable to construct an “isotropic” space-time radial basis function. The other scheme considered a realistic relationship between space variable and time variable which is not radial. The time-dependent variable is treated regularly during the whole solution process and the hyperbolic telegraph equations can be solved in a direct way. Numerical experiments performed with the proposed numerical scheme for several two-dimensional second-order hyperbolic telegraph equations are presented with some discussions, which show that the DMM solutions are converging very fast in comparison with the various existing numerical methods.

scholarly journals Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Elmiloud Chaabelasri
Keyword(s):  
Differential Equation ◽  
Radial Basis Function ◽  
Basis Function ◽  
Meshless Method ◽  
Artificial Viscosity ◽  
Dam Break ◽  
Two Dimensional ◽  
Radial Basis ◽  
Temporal Derivative ◽  
Irregular Bed Topography

A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge–Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. Then, we apply our scheme for several theoretical two-dimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Ram Jiwari ◽  
Alf Gerisch
Keyword(s):  
Meshless Method ◽  
Order Of Convergence ◽  
Method Of Lines ◽  
Reaction Diffusion ◽  
Diffusion Models ◽  
Second Order ◽  
Differential Quadrature ◽  
Time Dependent ◽  
Content Type ◽  
Radial Basis

Purpose This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time. Design/methodology/approach In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight. Findings The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems. Originality/value The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

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