Numerical simulation of nonlinear coupled Burgers’ equation through meshless radial point interpolation method
TL;DR: In this article, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of nonlinear coupled Burgers' equation in two dimensions, and the results of numerical experiments confirm the accuracy and efficiency of the presented scheme.
Abstract: In present paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the solution of nonlinear coupled Burgers’ equation in two dimensions. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The aim of this paper is to show that the SMRPI method is suitable for the treatment of nonlinear coupled Burgers’ equation. With regard to test problems that have not exact solutions, we consider two strategies for checking the stability of time difference scheme and for survey the convergence of the fully discrete scheme. The results of numerical experiments confirm the accuracy and efficiency of the presented scheme.
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TL;DR: To numerically simulate these models and in order to overcome their difficulties such as computational complexity, multi-dimensionality, non-linearity, having large spatial and temporal domains and also having a discrete initial data, a strongly meshless technique based on Wendland compactly supported radial basis function in conjunction with an operator splitting algorithm is proposed.
Abstract: Cognitive disorders especially epilepsy are closely linked with synchronization/desynchronization of neurons in the brain. In this paper, the dynamical modeling and behavior analysis of a FitzHugh–Nagumo neuron and also synchronization control of a network of FitzHugh–Nagumo neurons which promise the understanding of cognitive processing, are studied. To numerically simulate these models and in order to overcome their difficulties such as computational complexity, multi-dimensionality, non-linearity, having large spatial and temporal domains and also having a discrete initial data, we propose the use of a strongly meshless technique based on Wendland compactly supported radial basis function in conjunction with an operator splitting algorithm. The main advantage of the proposed algorithm is its computational complexity (for example O ( N x 1 3 + N x 2 3 ) of the inversion of all matrices) which is much lower than the complexity of pure radial basis functions method ( O ( N x 1 3 N x 2 3 ) of the inversion of matrices). Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.
20 citations
TL;DR: A new high order algorithm based on the coupling of Radial Point Interpolation Method and a high order continuation to solve the geometrically nonlinear elasticity problems under a strong formulation is proposed.
Abstract: In this work, we propose a new high order algorithm based on the coupling of Radial Point Interpolation Method (RPIM) and a high order continuation to solve the geometrically nonlinear elasticity problems under a strong formulation. The high order continuation has an adaptive step length which is very efficient and performed especially for solving the nonlinear problems. The specificity of RPIM is the exact implementation of boundary conditions because its shape functions have the Kronecker delta function property as in the conventional Finite Element Method (FEM). Therefore, it has proven that the RPIM shape functions have not only possess all advantages of the enforcing boundary conditions, but also can accurately reflect the properties of stresses distribution. This algorithm allows obtaining the solution with a less expensive CPU time to that of incremental iterative methods. A numerical comparison between the proposed algorithm and the others of literature is illustrated on some examples of geometrically nonlinear elasticity problems.
19 citations
TL;DR: Numerical experiments show that the proposed geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations is effective and accurate for the CNSG equations.
Abstract: In the paper, we derive a geometric meshless method for coupled nonlinear sine-Gordon (CNSG) equations. Approximate solutions of the CNSG equations are supposed to be expressed as the moving Kriging (MK) shape functions in the space direction. Global weak form of the CNSG equations is obtained, and then, a system of ODEs in time coordinate is extracted after imposing the MK meshless method. Then the geometric integrator, namely group preserving scheme, is offered to approximate the solution of obtained system of ODEs. Stability analysis of the method is numerically investigated. Numerical experiments show that the proposed method is effective and accurate for the CNSG equations.
18 citations
TL;DR: In this article, the improved backward substitution method was applied to solve nonlinear coupled Burgers' equations, where the temporal variable is discretized by the Crank-Nicolson finite difference scheme and the primary approximation is obtained from the boundary condition and its corresponding correcting solution.
Abstract: In this paper, we make the first attempt to extend the improved backward substitution method for solving unsteady nonlinear coupled Burgers’ equations. The temporal variable is discretized by the Crank-Nicolson finite difference scheme. Then the improved backward substitution method is applied to solve the corresponding system. The solution to the discretized system is approximated by the primary approximation which is obtained from the boundary condition and its corresponding correcting solution. A simple iteration scheme is used to eliminate the non-linearity of considered problems. To illustrate the accuracy and efficiency, we consider five examples and results are compared with existing results in literatures. Numerical experiments demonstrate that the present method has potential for real engineering problems.
10 citations
TL;DR: This paper proposes the use of a strongly meshless technique based on radial basis functions in conjunction with a (quasi)linearization algorithm for dynamical modeling and behavior analysis of the inverse boundary Stefan problem which promising the understanding of modeling brain tumor treatment.
Abstract: In this paper, the dynamical modeling and behavior analysis of the inverse boundary Stefan problem which promising the understanding of modeling brain tumor treatment, are studied. To umerical simulate these models and to overcome their difficulties such as, non-linearity, free boundary property and having a non-rectangular domain, we propose the use of a strongly meshless technique based on radial basis functions in conjunction with a (quasi)linearization algorithm. Numerical examples are given to show the good accuracy and stability of the presented method.
8 citations
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TL;DR: In this article, the application of statistical analysis and statistical mechanics to the problem of turbulent fluid motion has attracted much attention in recent years, and the authors investigated a complicated system of nonlinear equations, in order to find out enough about the properties of the solutions of these equations that insight can be obtained into the various patterns exhibited by the field and that data can be derived concerning the relative frequencies of these patterns in the hope that in this way a basis may be found for the calculation of important values.
Abstract: Publisher Summary This chapter discusses that the application of methods of statistical analysis and statistical mechanics to the problem of turbulent fluid motion has attracted much attention in recent years. It investigated a complicated system of nonlinear equations, in order to find out enough about the properties of the solutions of these equations that insight can be obtained into the various patterns exhibited by the field and that data can be derived concerning the relative frequencies of these patterns in the hope that in this way a basis may be found for the calculation of important values. The difficulties encounter are of a twofold nature: in part they are connected with the complicated geometrical character of the hydrodynamical equations; in part they are dependent upon the presence of nonlinear terms, containing derivatives of the first order of the velocity components, along with derivatives of the second order multiplied by the very small coefficient of viscosity. The latter feature is responsible for a number of important, characteristics of turbulence, among which are prominent those connected with the balance of energy and with the appearance of dissipation layers. These layer an important part in the energy exchange, as they represent the main regions where energy is dissipated.
2,202 citations
TL;DR: The meshless local radial point interpolation method (LRPIM) is adopted to simulate the two-dimensional nonlinear sine-Gordon (S-G) equation and a simple predictor–corrector scheme is performed to eliminate the nonlinearity.
184 citations
TL;DR: In this article, the two-dimensional Burgers' equation is used as a model equation for comparing the accuracy of different computational algorithms, and the authors present a more detailed review of the two dimensions of the Burgers's equation.
Abstract: Burgers’ equation is well suited to modelling fluid flows as it incorporates directly the interaction between the non-linear convection processes and the diffusive viscous processes. In one dimension the Cole-Hopf procedure transforms Burgers’ equation into the linear heat conduction equation. As a result many exact solutions of Burgers’ equation are available in the literature. Thus Burgers’ equation has often been used as a model equation for comparing the accuracy of different computational algorithms. This aspect of Burgers’ equation is reviewed by Fletcher.’ The two-dimensional Burgers’ equations
175 citations
TL;DR: Efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector are developed.
Abstract: Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen--Cahn reaction-diffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator.
170 citations
TL;DR: The present method performs well in solving the two-dimensional Burgers' equations in fully implicit finite-difference form and is examined by comparison with other analytical and numerical results.
158 citations