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.

scholarly journals An Efficient Local Formulation for Time-Dependent PDEs

2019 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Muhammad Ahsan ◽  
Masood Ahmad ◽  
Zaheer ud Din ◽  
Poom Kumam
Keyword(s):  
Meshless Method ◽  
Order Of Convergence ◽  
Shape Parameter ◽  
Computational Cost ◽  
Basis Functions ◽  
Time Dependent ◽  
Irregular Domains ◽  
Time Integrators ◽  
Radial Basis ◽  
Function Approximations

In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs, due to the flexibility with respect to geometry and high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focus towards localized radial basis function approximations, as the local meshless method is proposed here. The local meshless procedures is used for spatial discretization whereas for temporal discretization different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation using regular and irregular domains.

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.

The meshless method of radial basis functions for the numerical solution of time fractional telegraph equation

2014 ◽  
Vol 24 (8) ◽  
pp. 1636-1659 ◽  
Author(s):  
Akbar Mohebbi ◽  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan
Keyword(s):  
Radial Basis Functions ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Telegraph Equation ◽  
Basis Functions ◽  
Content Type ◽  
Fractional Telegraph Equation ◽  
Radial Basis ◽  
Collocation Approach ◽  
Derivatives Of

Purpose – The purpose of this paper is to show that the meshless method based on radial basis functions (RBFs) collocation method is powerful, suitable and simple for solving one and two dimensional time fractional telegraph equation. Design/methodology/approach – In this method the authors first approximate the time fractional derivatives of mentioned equation by two schemes of orders O(τ3−α) and O(τ2−α), 1/2<α<1, then the authors will use the Kansa approach to approximate the spatial derivatives. Findings – The results of numerical experiments are compared with analytical solution, revealing that the obtained numerical solutions have acceptance accuracy. Originality/value – The results show that the meshless method based on the RBFs and collocation approach is also suitable for the treatment of the time fractional telegraph equation.

scholarly journals Numerical Solution of Heat Equation in Polar Cylindrical Coordinates by the Meshless Method of Lines

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Arshad Hussain ◽  
Marjan Uddin ◽  
Sirajul Haq ◽  
Hameed Ullah Jan
Keyword(s):  
Heat Equation ◽  
Numerical Solution ◽  
Meshless Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Cylindrical Coordinates ◽  
Runge Kutta Method ◽  
Radial Basis ◽  
Rectangular Coordinates ◽  
Meshless Method Of Lines

We propose a numerical solution to the heat equation in polar cylindrical coordinates by using the meshless method of lines approach. The space variables are discretized by multiquadric radial basis function, and time integration is performed by using the Runge-Kutta method of order 4. In radial basis functions (RBFs), much of the research are devoted to the partial differential equations in rectangular coordinates. This work is an attempt to explore the versatility of RBFs in nonrectangular coordinates as well. The results show that application of RBFs is equally good in polar cylindrical coordinates. Comparison with other cited works confirms that the present approach is accurate as well as easy to implement to problems in higher dimensions.

A conservative, piecewise–analytical, transversal method of lines for reaction–diffusion equations

2019 ◽  
Vol 29 (11) ◽  
pp. 4093-4129 ◽  
Author(s):  
J.I. Ramos
Keyword(s):  
Finite Difference ◽  
Boundary Layers ◽  
Diffusion Coefficients ◽  
Method Of Lines ◽  
Reaction Diffusion ◽  
Reaction Rates ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Content Type ◽  
One Dimensional

Purpose The purpose of this paper is to develop a new transversal method of lines for one-dimensional reaction–diffusion equations that is conservative and provides piecewise–analytical solutions in space, analyze its truncation errors and linear stability, compare it with other finite-difference discretizations and assess the effects of the nonlinear diffusion coefficients, reaction rate terms and initial conditions on wave propagation and merging. Design/methodology/approach A conservative, transversal method of lines based on the discretization of time and piecewise analytical integration of the resulting two-point boundary-value problems subject to the continuity of the dependent variables and their fluxes at the control-volume boundaries, is presented. The method provides three-point finite difference expressions for the nodal values and continuous solutions in space, and its accuracy has been determined first analytically and then assessed in numerical experiments of reaction-diffusion problems, which exhibit interior and/or boundary layers. Findings The transversal method of lines presented here results in three-point finite difference equations for the nodal values, treats the diffusion terms implicitly and is unconditionally stable if the reaction terms are treated implicitly. The method is very accurate for problems with the interior and/or boundary layers. For a system of two nonlinearly-coupled, one-dimensional reaction–diffusion equations, the formation, propagation and merging of reactive fronts have been found to be strong function of the diffusion coefficients and reaction rates. For asymmetric ignition, it has been found that, after front merging, the temperature and concentration profiles are almost independent of the ignition conditions. Originality/value A new, conservative, transversal method of lines that treats the diffusion terms implicitly and provides piecewise exponential solutions in space without the need for interpolation is presented and applied to someone.

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