The spectral collocation method for efficiently solving PDEs with fractional Laplacian

We derive a spectral collocation approximation to the fractional Laplacian operator based on the Riemann-Liouville fractional derivative operators on a bounded domain Ω = [a, b]. Corresponding matrix representations of (?△) ^α/2 for α ∈ (0,1) and α ∈ (1,2) are obtained. A space-fractional advection-dispersion equation is then solved to investigate the numerical performance of this method under various choices of parameters. It turns out that the proposed method has high accuracy and is efficient for solving these space-fractional advection-dispersion equations when the forcing term is smooth.     

Galerkin approximation for elliptic PDEs on spheres

We discuss a Galerkin approximation scheme for the elliptic partial differential equation -Δu+ω²u=f on SⁿRⁿ⁺¹. Here Δ is the Laplace–Beltrami operator on Sⁿ, ω is a non-zero constant and f belongs to C^2k-2(Sⁿ), where kn/4+1, k is an integer. The shifts of a spherical basis function φ with φH^τ(Sⁿ) and τ>2kn/2+2 are used to construct an approximate solution. An H¹(Sⁿ)-error estimate is derived under the assumption that the exact solution u belongs to C^2k(Sⁿ).     

A numerical study of the accuracy and stability of symmetric and asymmetric RBF collocation methods for hyperbolic PDEs

Differentiation matrices associated with radial basis function (RBF) collocation methods often have eigenvalues with positive real parts of significant magnitude. This prevents the use of the methods for time‐dependent problems, particulary if explicit time integration schemes are employed. In this work, accuracy and eigenvalue stability of symmetric and asymmetric RBF collocation methods are numerically explored for some model hyperbolic initial boundary value problems in one and two dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Applications of RBF collocation method to elliptic PDEs in an arbitrary domain by fictitious domain extensions

A fictitious domain extension approach is introduced to study elliptic PDE's defined in arbitrary domains by the radial basis function (RBF) collocation method. In this approach, arbitrary physical geometries are extended to domains which are topologically rectangular. The solution domain is also extended to the fictitious area and assumed to satisfy the same governing equation in it and on its extended boundaries. The boundary conditions are still specified on the boundaries of the original physical domain. The problem in the extended domain becomes ill-posed. However, it can be easily circumvented by the collocation method. We demonstrate that the solution can be directly obtained without domain decompositions and iterations. The new approach is simple, efficient and accurate. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

Physics-based preconditioners for solving PDEs on highly heterogeneous media

Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials, geological rock properties, and thermal and electrical conductivity. The main objective of this work is to construct a method as algebraic as possible that could efficiently exploit the connectivity of highly heterogeneous media in the solution of diffusion operators. We propose an algebraic way of separating binary-like systems according to a given threshold into high- and low-conductivity regimes of coefficient size O (m) and O (1), respectively where m ≫ 1. The condition number of the linear system depends both on the mesh size and the coefficient size m. For our purposes, we address only the m dependence since the condition number of the linear system is mainly governed by the high-conductivity subblock. Thus, the proposed strategy is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a two-stage preconditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation strategy (if ever needed). The deflation strategies are based on computing near invariant subspaces corresponding to smallest eigenvalues and deflating them by the use of recycled the Krylov subspaces. More detail on the proposed preconditioners can be found in [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solutions based on Chebyshev collocation method for singularly perturbed delay parabolic PDEs

The main purpose of this work is to provide a numerical approach for the delay partial differential equations based on a spectral collocation approach. In this research, a rigorous error analysis for the proposed method is provided. The effectiveness of this approach is illustrated by numerical experiments on two delay partial differential equations. Copyright © 2013 John Wiley & Sons, Ltd.     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients

We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.     

Approximation of parabolic PDEs on spheres using spherical basis functions

In this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres SⁿRⁿ⁺¹ using spherical basis functions. Error estimates in the Sobolev norm are derived. The results presented in this paper are taken from the authors Ph.D. dissertation under supervision of Professor J.D. Ward and Professor F.J. Narcowich at Texas A&M University.AMS subject classification 35K05, 65M70, 46E22     

A collocation method for solving singular integro-differential equations

This paper describes a collocation method for numerically solving Cauchy-type linear singular integro-differential equations. The numerical method is based on the transformation of the integro-differential equation into an integral equation, and then applying a collocation method to solve the latter. The collocation points are chosen as the Chebyshev nodes. Uniform convergence of the resulting method is then discussed. Numerical examples are presented and solved by the numerical techniques.     

Multistep Hermite collocation methods for solving Volterra Integral Equations

In this paper, we propose a new class of multistep collocation methods for solving nonlinear Volterra Integral Equations, based on Hermite interpolation. These methods furnish an approximation of the solution in each subinterval by using approximated values of the solution, as well as its first derivative, in the r previous steps and m collocation points. Convergence order of the new methods is determined and their linear stability is analyzed. Some numerical examples show efficiency of the methods.     

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov-Kuznetsov (ZK) equation and the Kadomtsev-Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.     

The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs

In this paper, the multi-symplectic Fourier pseudospectral (MSFP) method is generalized to solve two-dimensional Hamiltonian PDEs with periodic boundary conditions. Using the Fourier pseudospectral method in the space of the two-dimensional Hamiltonian PDE (2D-HPDE), the semi-discrete system obtained is proved to have semi-discrete multi-symplectic conservation laws and a global symplecticity conservation law. Then, the implicit midpoint rule is employed for time integration to obtain the MSFP method for the 2D-HPDE. The fully discrete multi-symplectic conservation laws are also obtained. In addition, the proposed method is applied to solve the Zakharov–Kuznetsov (ZK) equation and the Kadomtsev–Petviashvili (KP) equation. Numerical experiments on soliton solutions of the ZK equation and the KP equation show the high accuracy and effectiveness of the proposed method.     

A Taylor collocation method for solving delay integral equations

A numerical method based on the use of Taylor polynomials is proposed to construct a collocation solution $u\in S_{m-1}^{(-1)}(\Pi _{N})$ for approximating the solution of delay integral equations. It is shown that this method is convergent. Some numerical examples are given to show the validity of the presented method.     

Error control Gaussian collocation software for boundary value ODEs and 1D time-dependent PDEs

Numerical Algorithms - Error control software packages based on Gaussian collocation have been widely used for the numerical solution of boundary value ODEs (BVODEs) and 1D parabolic time-dependent...     

A multistep collocation method for solving advection equations with delay

Advection equations with delay are appeared in the modeling of the dynamics of structured cell populations. In this article, we construct an efficient two-dimensional multistep collocation method for the numerical solution of a class of advection equations with delay. Equations with aftereffect and equations with both aftereffect and retardation of a state variable are considered. Computability of the algorithm and convergence properties of the proposed numerical method are analyzed for solutions in appropriate Sobolev spaces, and it is shown that the proposed scheme enjoys the spectral accuracy. Numerical examples are given and comparison with other existing methods in the literature is made to demonstrate the efficiency, superiority and high accuracy of the presented method.