Two-level meshless local Petrov Galerkin method for multi-dimensional nonlinear convection–diffusion equation based on radial basis function

2018 ◽  
Vol 74 (4) ◽  
pp. 685-698 ◽  
Author(s):  
Jingwei Li ◽  
Jianping Zhao ◽  
Lingzhi Qian ◽  
Xinlong Feng
Keyword(s):  
Diffusion Equation ◽  
Galerkin Method ◽  
Radial Basis Function ◽  
Basis Function ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Radial Basis

scholarly journals Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1735
Author(s):  
Cheng-Yu Ku ◽  
Jing-En Xiao ◽  
Chih-Yu Liu
Keyword(s):  
Diffusion Equation ◽  
Radial Basis Function ◽  
Time Domain ◽  
Basis Function ◽  
Numerical Solutions ◽  
Space Time ◽  
Diffusion Equations ◽  
Numerical Examples ◽  
Convection Diffusion ◽  
Convection Diffusion Equation

This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of n–dimensional Euclidean space and time into an n + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation.

Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 69-76
Author(s):  
Fuzhang Wang ◽  
Kehong Zheng ◽  
Imtiaz Ahmad ◽  
Hijaz Ahmad
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Meshless Method ◽  
Absolute Error ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Radial Basis ◽  
Linear And Nonlinear ◽  
Physical Phenomena ◽  
Diffusion Problems

Abstract In this study, we propose a simple direct meshless scheme based on the Gaussian radial basis function for the one-dimensional linear and nonlinear convection–diffusion problems, which frequently occur in physical phenomena. This is fulfilled by constructing a simple ‘anisotropic’ space–time Gaussian radial basis function. According to the proposed scheme, there is no need to remove time-dependent variables during the whole solution process, which leads it to a really meshless method. The suggested meshless method is implemented to the challenging convection–diffusion problems in a direct way with ease. Numerical results show that the proposed meshless method is simple, accurate, stable, easy-to-program and efficient for both linear and nonlinear convection–diffusion equation with different values of Péclet number. To assess the accuracy absolute error, average absolute error and root-mean-square error are used.

scholarly journals Finite point method of nonlinear convection diffusion equation

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1517-1533
Author(s):  
Xinqiang Qin ◽  
Gang Hu ◽  
Gaosheng Peng
Keyword(s):  
Diffusion Equation ◽  
Domain Size ◽  
Nonlinear Flow ◽  
Step Size ◽  
Finite Point ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Numerical Oscillations ◽  
The One

Aiming at the nonlinear convection diffusion equation with the numerical oscillations, a numerical stability algorithm is constructed. The basic principle of the finite point algorithm is given and the computational scheme of the nonlinear convection diffusion equation is deduced. Then, the numerical simulation of the one-dimensional and two-dimensional nonlinear convection-dominated diffusion equation is carried out. The relationship between the calculation result and the support domain size, step size and time is discussed. The results show that the algorithm has the characteristics of simplicity, stability and efficiency. Compared with the traditional finite element method and finite difference method, the new algorithm can attain a higher calculation accuracy. Simultaneously, it proves that the method given in this paper is effective to solve the nonlinear flow diffusion equation and can eliminate the numerical oscillations.

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