Method Of Approximate Particular Solutions Research Articles

A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.

The MAPS based on trigonometric basis functions for solving elliptic partial differential equations with variable coefficients and Cauchy–Navier equations

In this paper, we extended the method of approximate particular solutions (MAPS) using trigonometric basis functions to solve two-dimensional elliptic partial differential equations (PDEs) with variable-coefficients and the Cauchy–Navier equations. The new approach is based on the closed-form particular solutions for second-order differential operators with constant coefficients. For the Cauchy–Navier equations, a reformulation of the equations is required so that the particular solutions for the new differential operators are available. Five numerical examples are provided to demonstrate the effectiveness of the proposed method.

Localized method of approximate particular solutions with polynomial basis functions

The method of particular solutions (MPS) using polynomials as the basis functions has been successfully developed for solving a large class of partial differential equations. However, when a large number of collocation points are required, the above mentioned approach is not feasible since the resultant matrix is dense and ill-conditioned. This restriction is common for global methods. One of the alternative approaches is to employ the localized scheme in which only a small number of neighboring points are being used in the solution process. As a result, the resultant matrix of the localized method is sparse and thus can be solved efficiently.In this paper, the localized method has been employed to extend the recently developed MPS using polynomial basis functions for solving large-scale science and engineering problems. Overall, the proposed approach is stable and highly accurate. Four numerical examples are presented to validate the proposed numerical method.

Influence of magnetic fields on simultaneous stationary solutions of two-dimensional sudden expansion channel flow at low Rem

The influence of uniform magnetic fields on stationary simultaneous solutions in a channel with sudden expansion is studied in this work. Reynolds number (Re) values between 250 ≤ Re ≤ 550 and magnetic fields with Hartmann numbers (Ha) between 0 ≤ Ha ≤ 100 are considered. The global meshless method of approximate particular solutions (MAPS) is implemented to obtain the simultaneous stationary solutions that are presented in Re values above an approximate critical value of Rec ≈ 219. The base functions used in the MAPS to approximate velocity and pressure fields solve an auxiliary Stokes system. The four-sided lid-driven cavity (4SLDC) and backward facing step flows are considered as validation problems in order to show the ability of the method to study bifurcations and solve magnetohydrodynamic problems, respectively. The influence of transverse magnetic fields on the sudden expansion flow for low Ha contrasts with the few effects that a stream-wise field produces. The asymmetric flow is eliminated if a transverse magnetic field with Ha = 5 is imposed over the flow and recirculation zones almost disappear at Ha = 20 for Re values considered in this work.

A high accurate simulation of thin plate problems by using the method of approximate particular solutions with high order polynomial basis

In this paper, a closed-form particular solution of polyharmonic splines has been obtained for high order partial differential operators. Instead of using complex derivation, the new particular solution is derived simply by adding or subtracting several available particular solutions. The proposed particular solution is further coupled with polynomial basis for numerically solving thin plate problems. The relationship between number of nodes and order of polynomials are fully studied. Numerical examples with irregular domains are presented to demonstrate the effectiveness of the proposed algorithm.

The extended method of approximate particular solutions to simulate two-dimensional electromagnetic scattering from arbitrary shaped anisotropic objects

The numerical simulation of electromagnetic scattering problem by cylinders of arbitrary cross-section made of homogeneous and anisotropic material is carried out. The anisotropic materials are characterized by symmetric and positive definite magnetic permeability tensors. An efficient and powerful meshfree strong form technique based on the particular solutions of anisotropic radial basis functions (A-RBFs) is extended to deal with anisotropic problem directly. The A-RBFs are constructed by introducing a special norm associated to the appearing anisotropic tensor and replacing it with the standard Euclidean norm in RBFs. The particular solutions for the anisotropic Laplace operator are approximated explicitly by using the anisotropic radial basis functions. Because of material discontinuity at the interface of anisotropic and free regions, two different sets of basis functions, the approximate particular solutions of the anisotropic generalized multiquadric (A-GMQ) and the standard generalized multiquadric (GMQ) functions are employed to approximate the total electric field inside and the scattered field outside the anisotropic scatterer respectively. Some numerical simulations are implemented to confirm the accuracy and efficiency of the method for anisotropic scatterer with arbitrary cross-sections.

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017

Numerical simulation of 3D nonlinear Schrödinger equations by using the localized method of approximate particular solutions

In this paper, we describe a novel sparse meshless approach to the simulations of three-dimensional time-dependent nonlinear Schrödinger equations. Our procedure is implemented in two successive steps. In the first step, the implicit-Euler scheme is applied for approximating the functional dependence of the solution on the temporal variables. Then, in the second step, the novel localized method of approximate particular solutions (LMAPS) is utilized for highly accurate and efficient numerical approximations of spatial systems. In the implementation of the LMAPS, the closed form particular solutions for the Laplace operator using the Gaussian radial basis function are used. Numerical experiments are provided to verify the stability and efficiency of this method. In summary, the proposed algorithm is efficient and stable, and the magnitude of the error is at about 10−3 for 3D nonlinear Schrödinger problems.

Numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue

This paper investigates the thermal behavior inside skin tissue with the presence of a tumor by solving the Pennes bioheat equation. We apply the method of approximate particular solutions (MAPS) to simulate tumor in 3D. The MAPS is a kind of domain-type meshless collocation methods, which has the merits being free of mesh generation and numerical integration. To ensure the interface conditions were exactly satisfied, an affine space decomposition technique is adopted. After verifying the convergence of the proposed method, we provide various simulations to the thermal effects inside the skin tissue due to an arbitrary-shaped tumor, including the location, geometry and size.

Local Method of Approximate Particular Solutions for Two-Dimensional Unsteady Incompressible Flow

A meshless local method of approximated particular solutions (LMAPS) is used to analyze incompressible fluid flow in a two dimensionalcavity. The method solves the incompressible Navier-Stokes equations in terms of theprimitive variables using the fractional step scheme. The basic equations are derived via interpolation using integrated multiquadrics radial basis functions. Lid-driven cavity benchmark case for various Reynolds numbers is presented in the article. The procedure produces stable solutions with results comparable to those in literature.

Fast Solution of Three-Dimensional Modified Helmholtz Equations by the Method of Fundamental Solutions

AbstractThis paper describes an application of the recently developed sparse scheme of the method of fundamental solutions (MFS) for the simulation of three-dimensional modified Helmholtz problems. The solution to the given problems is approximated by a two-step strategy which consists of evaluating the particular solution and the homogeneous solution. The homogeneous solution is approximated by the traditional MFS. The original dense system of the MFS formulation is condensed into a sparse system based on the exponential decay of the fundamental solutions. Hence, the homogeneous solution can be efficiently obtained. The method of particular solutions with polyharmonic spline radial basis functions and the localized method of approximate particular solutions in combination with the Gaussian radial basis function are employed to approximate the particular solution. Three numerical examples including a near singular problem are presented to show the simplicity and effectiveness of this approach.

Fast simulation of multi-dimensional wave problems by the sparse scheme of the method of fundamental solutions

In this work, a meshless scheme is presented for the fast simulation of multi-dimensional wave problems. The present method is rather simple and straightforward. The Houbolt method is used to eliminate the time dependence of spatial variables. Then the original wave problem is converted into equivalent systems of modified Helmholtz equations. The sparse scheme of the method of fundamental solutions in combination with the localized method of approximate particular solutions is employed for efficient implementation of spatial variables. To demonstrate the effectiveness and simplicity of this new approach, three numerical examples have been assessed with excellent performance.

Fast method of approximate particular solutions using Chebyshev interpolation

The fast method of approximate particular solutions (FMAPS) is based on the global version of the method of approximate particular solutions (MAPS). In this method, given partial differential equations are discretized by the usual MAPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions.

An improved localized method of approximate particular solutions for solving elliptic PDEs

In this paper we improve the localized method of approximate particular solutions (LMAPS) in Yao et al. (2011) by utilizing the polyharmonic splines (PS) radial basis function (RBF) for solving elliptic partial differential equations (PDEs). LMAPS has been widely circulated since it is published in 2010. The multiquadric (MQ) has been considered as the most popular choice among all RBFs. However, adjusting the shape parameter is a critical issue when utilizing the original LMAPS. In this paper, we modified LMAPS by combining conditionally positive definite RBF-PS and an additional low degree of polynomial basis in the localization process. The accuracy of the proposed LMAPS is significantly improved. We can simply increase the order of PS to achieve even higher accuracy. Other than the unexpected high accuracy, there is no need to deal with the difficult issues of choosing optimal shape parameter. This is a huge advantage in the RBF simulations of PDEs. In the numerical experiments, we will present the pros and cons of improved LMAPS (ILMAPS) using PS and some commonly used RBFs (MQ, Matérn, and Gaussian) versus the original LMAPS (OLMAPS).

The localized method of approximated particular solutions for near-singular two- and three-dimensional problems

In this paper, the localized method of approximate particular solutions (LMAPS) using radial basis functions (RBFs) has been simplified and applied to near-singular elliptic problems in two- and three-dimensional spaces. The leave-one-out cross validation (LOOCV) is used in LMAPS to search for a good shape parameter of multiquadric RBF. The main advantage of the method is that a small number of neighboring nodes can be chosen for each influence domain in the discretization to achieve high accuracy. This is especially efficient for three-dimension problems. There is no need to apply adaptivity on node distribution near the region containing spikes of the forcing terms. To examine the performance and limitations of the method, we deliberately push the spike of the forcing term to be extremely large and still obtain excellent results. LMAPS is far superior than the compactly supported RBF (Chen et al. 2003) for such elliptic boundary value problems.

Localized method of approximate particular solutions for solving unsteady Navier–Stokes problem

The localized method of approximate particular solutions (LMAPS) is proposed to solve two-dimensional transient incompressible Navier–Stokes systems of equations in primitive variables. The equations contain the Laplacian operator. In avoiding ill-conditioning problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear system within local supporting domain in which five nearest neighboring points and multiquadrics are used for interpolation. Then local matrices are reformulated in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The method is assessed on driven cavity problem and flow around cylinder. The numerical experiments show that the newly developed LMAPS is suitable for solving incompressible Navier–Stokes equations with high accuracy and efficiency.

Compactly supported kernels method of approximate particular solutions for solving elliptic problems

In this work the method of approximate particular solutions using compactly supported kernels is investigated. In the work of [1] the globally supported radial kernels for the construction of solution are used, and it is observed that for large scaled PDEs the differentiation matrix is ill-conditioned. We extended the work of [1] for compactly supported kernels, in order to solve large-scaled PDEs engineering sciences. The numerical scheme of the present method of approximate particular solutions is very accurate and simple in implementation. Three benchmark problems are solved by the present numerical scheme and the results are compared to other methods in the literature.

The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition

In this paper, we consider the numerical solution of the time-fractional diffusion equation with a non-local boundary condition. The method of approximate particular solutions (MAPS) using multiquadric radial basis function (MQ-RBF) is employed for this equation. Due to the accuracy of the MQ-based meshless methods is severely influenced by the shape parameter, we adopt a leave-one-out cross validation (LOOCV) algorithm proposed by Rippa [34] to enhance the performance of the MAPS. The numerical results obtained show that the proposed numerical algorithm is accurate and computationally efficient for solving time-fractional diffusion equation with a non-local boundary condition.

Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

A combination of the localized method of approximate particular solutions (LMAPS), the implicit Euler method and the Newton’s method is adopted in this paper for transient solutions of two-dimensional velocity–vorticity formulation of the Navier–Stokes equations. The LMAPS, which is truly free from time-consuming mesh generation and numerical quadrature, and the implicit Euler method are, respectively, used for spatial and temporal discretizations of the velocity–vorticity formulation. Using the approximations of particular solutions in every local domain, the derivatives at nodes with respect to space coordinates via the LMAPS can be approximated by linear summations of nearby function values. After the discretizations for space and time derivatives, a system of nonlinear algebraic equations will be yielded at every time step and then the Newton’s method is used for efficiently analyzing these systems. Three numerical examples are provided to validate the accuracy and the simplicity of the proposed scheme and the numerical results are compared well with other numerical and analytical solutions. Besides, the numerical solutions, acquired by using different numbers of total nodes, different numbers of nodes in sub-domain, different shape parameters and different Reynolds numbers, are provided to show the merits of the proposed meshless scheme.

The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.