A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs (original) (raw)
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Engineering Analysis with Boundary Elements, 2012
A comparison of the performance of the global and the local radial basis function collocation meshless methods for three dimensional parabolic partial differential equations is performed in the present paper. The methods are structured with multiquadrics radial basis functions. The time-stepping is performed in a fully explicit, fully implicit and Crank-Nicolson ways. Uniform and non-uniform node arrangements have been used. A three-dimensional diffusion-reaction equation is used for testing with the Dirichlet and mixed Dirichlet-Neumann boundary conditions. The global methods result in discretization matrices with the number of unknowns equal to the number of the nodes. The local methods are in the present paper based on seven-noded influence domains, and reduce to discretization matrices with seven unknowns for each node in case of the explicit methods or a sparse matrix with the dimension of the number of the nodes and seven non-zero row entries in case of the implicit method. The performance of the methods is assessed in terms of accuracy and efficiency. The outcome of the comparison is as follows. The local methods show superior efficiency and accuracy, especially for the problems with Dirichlet boundary conditions. Global methods are efficient and accurate only in cases with small amount of nodes. For large amount of nodes, they become inefficient and run into ill-conditioning problems. Local explicit method is very accurate, however, sensitive to the node position distribution, and becomes sensitive to the shape parameter of the radial basis functions when the mixed boundary conditions are used. Performance of the local implicit method is comparatively better than the others when a larger number of nodes and mixed boundary conditions are used. The paper represents an extension of our recently made similar study in two dimensions.
A new meshless collocation method for partial differential equations
Communications in Numerical Methods in Engineering, 2008
A collocation meshless method is developed for the numerical solution of partial differential equations (PDEs) on the scattered point distribution. The meshless shape functions are constructed on a group of selected nodes (stencil) arbitrarily distributed in a local support domain by means of a polynomial interpolation. This shape function formulation possesses the Kronecker delta function property, and hence many numerical treatments are as simple as those of the finite element method (FEM). The nearest neighbor algorithm is used for the support domain nodes collection, and a search algorithm based on the Gauss–Jordan pivot method is applied to select a suitable stencil for the construction of the shape functions and their derivatives. This search technique is subject to a monitoring procedure which selects appropriate stencil in order to keep the condition number of the resulting linear systems small. Various meshless collocation schemes for the solution of elliptic, parabolic, and hyperbolic PDEs are investigated for the proposed method. The proposed method is pure meshless since it only uses scattered sets of collocation points and thus no nodes connectivity information is required. Different types of PDEs, i.e. the Poisson equation, the Wave equation, and Burgers' equation are studied as test cases and all of the computational results are examined. Copyright © 2007 John Wiley & Sons, Ltd.
Comparison of local weak and strong form meshless methods for 2-D diffusion equation
Engineering Analysis with Boundary Elements, 2012
A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.
Meshless solution of a diffusion equation with parameter optimization and error analysis
Engineering Analysis with Boundary Elements, 2008
Derivation and implementation of a numerical solution of a time-dependent diffusion equation is given in detail, based on the meshless local Petrov-Galerkin method (MLPG). A simple method is proposed that ensures a constant number of support nodes for each point. Numerical integrations are carried out over local square domains. The implicit Crank-Nicolson scheme is used for time discretization. A detailed convergence study was performed experimentally to optimize the number of support nodes, quadrature domain size and other parameters. The accuracy of the MLPG solution is compared with that of standard methods on a unit square and on an irregularly shaped test domain. As expected, the finite difference method on a regular mesh is incompetitive on irregularly shaped domains. MLPG is significantly more accurate when using moving least square shape functions of degree two than with degree one. It is comparable to the finite element method of degree two in the H 1 error norm and about two times less accurate in the L 2 error norm.
2015
In the recent years, meshless methods are applied for solution of partial differential equations in the various engineering fields. In these types of solution procedures, nodes are the main subject for the governing equations discretization and these nodes were disturbuted in the domain in both regular and irregular scheme. High flexibility and excellent accuracy of meshless methods are two major advantages of these methods in compare with mesh based Eulerian approaches. Recently, the mixed formulation technique is used in Discrete Least square meshless (DLSM) method for the solution of elliptic partial differential equations. This method so called MDLSM is based on minimizing the least squares functional which calculated at nodes in the study area and its boundaries. The least square functional is defined as the weighted summation of the squared residuals. To construct the meshless shape functions a Moving Least Squares (MLS) approximation is used. Theoretically, the accuracy of nu...
International Journal of Differential Equations, 2015
Meshless method of line is a powerful device to solve time-dependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause ill-conditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasi-uniform and locally bounded meshes. These avoid the ill-conditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arc-length monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes.
An Efficient Local Formulation for Time–Dependent PDEs
Mathematics
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. This local approach has flexibility with respect to geometry along with 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 focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure 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 on regular and irregular domains.
A moving least square reproducing polynomial meshless method
Interest in meshless methods has grown rapidly in recent years in solving boundary value problems (BVPs) arising in science and engineering. In this paper, we present the moving least square radial reproducing polynomial (MLSRRP) meshless method as a generalization of the moving least square reproducing kernel particle method (MLSRKPM). The proposed method is established upon the extension of the MLSRKPM basis by using the radial basis functions. Some important properties of the shape functions are discussed. An interpolation error estimate is given to assess the convergence rate of the approximation. Also, for some class of time-dependent partial differential equations, the error estimate is acquired. The efficiency of the present method is examined by several test problems. The studied method is applied to the parabolic two-dimensional transient heat conduction equation and the hyperbolic two-dimensional sine-Gordon equation which are discretized by the aid of the meshless local Petrov–Galerkin (MLPG) method.