Numerical solution of differential equations using multiquadric radial basis function networks

Computers & Mathematics with Applications

Tran

et al. 2016

In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first-and second-order derivatives at side nodes of the stencil are included in the computation of first-and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convectiondiffusion, and Navier-Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

Section: Introductionmentioning

confidence: 99%

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Computers & Mathematics with Applications

Tran

et al. 2016

“…Through integration constants, one can impose derivative boundary conditions and the governing equations at the two end points of a grid line in an exact manner. The 1D-IRBFN method is much more efficient than the original IRBFN method reported in [24]. Ngo-Cong et al [25] extended this method to investigate free vibration of composite laminated plates based on first-order shear deformation theory.…”

Section: Introductionmentioning

confidence: 99%

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

Ngo-Cong

Numerical Methods in Fluids

Karunasena

et al. 2012

This paper presents a local moving least square -one-dimensional integrated radial basis function networks (LMLS-1D-IRBFN) method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square (MLS) and one-dimensional integrated radial basis function networks (1D-IRBFN) techniques. The major advantages of the proposed method include: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker-δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local RBF approximations. Several examples including two-dimensional Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method.

“…Since the introduction of the integral RBF collocation approach [19,20], Kansa et al [21], based on the theoretical result of Madych and Nelson [22], have concluded that the decreasing rate of convergence for derivative functions caused by differentiation can be avoided in the integral RBF approach. When applying the integral collocation formulation for the solution of differential equations, with RBFs or Chebyshev polynomials, the constants of integration have been found to be very useful.…”

Section: Introductionmentioning

confidence: 99%

An integral‐collocation‐based fictitious‐domain technique for solving elliptic problems

Commun. Numer. Meth. Engng.

See

Tran-Cong

2007

SUMMARYThis paper presents a new fictitious-domain technique for numerically solving elliptic second-order partial-differential equations (PDEs) in complex geometries. The proposed technique is based on the use of integral collocation schemes and Chebyshev polynomials. The boundary conditions on the actual boundary are implemented by means of integration constants. The method works for both Dirichlet and Neumann boundary conditions. Several test problems are considered to verify the technique. Numerical results show that the present method yields spectral accuracy for smooth (analytic) problems.