Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

A Contact SPH Method with High-Order Limiters for Simulation of Inviscid Compressible Flows

In this paper,we study a class of contact smoothed particle hydrodynamics(SPH)by introducing Riemann solvers and using high-order limiters.In particular,a promising concept ofWENO interpolation as limiter is presented in the reconstruction process.The physical values relating interactional particles used as the initial values of the Riemann problemcan be reconstructed by the Taylor series expansion.The contact solvers of the Riemann problem at contact points are incorporated in SPH approximations.In order to keep the fluid density at the wall rows to be consistent with that of the inner fluid wall boundaries,several lines of dummy particles are placed outside of the solid walls,which are assigned according to the initial configuration.At last,the method is applied to compressible flows with sharp discontinuities such as the collision of two strong shocks and the interaction of two blast waves and so on.The numerical results indicate that the method is capable of handling sharp discontinuity and efficiently reducing unphysical oscillations.

A local refinement purely meshless scheme for time fractional nonlinear Schrodinger equation in irregular geometry region

A local refinement hybrid scheme(LRCSPH-FDM)is proposed to solve the two-dimensional(2D)time fractional nonlinear Schrodinger equation(TF-NLSE)in regularly or irregularly shaped domains,and extends the scheme to predict the quantum mechanical properties governed by the time fractional Gross-Pitaevskii equation(TF-GPE)with the rotating Bose-Einstein condensate.It is the first application of the purely meshless method to the TF-NLSE to the author’s knowledge.The proposed LRCSPH-FDM(which is based on a local refinement corrected SPH method combined with FDM)is derived by using the finite difference scheme(FDM)to discretize the Caputo TF term,followed by using a corrected smoothed particle hydrodynamics(CSPH)scheme continuously without using the kernel derivative to approximate the spatial derivatives.Meanwhile,the local refinement technique is adopted to reduce the numerical error.In numerical simulations,the complex irregular geometry is considered to show the flexibility of the purely meshless particle method and its advantages over the grid-based method.The numerical convergence rate and merits of the proposed LRCSPH-FDM are illustrated by solving several 1D/2D(where 1D stands for one-dimensional)analytical TF-NLSEs in a rectangular region(with regular or irregular particle distribution)or in a region with irregular geometry.The proposed method is then used to predict the complex nonlinear dynamic characters of 2D TF-NLSE/TF-GPE in a complex irregular domain,and the results from the posed method are compared with those from the FDM.All the numerical results show that the present method has a good accuracy and flexible application capacity for the TF-NLSE/GPE in regions of a complex shape.

The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions

In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations.We first approximate the source function by Chebyshev polynomials.We then focus on how to find a polynomial particular solution when the source function is a polynomial.Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined,the coefficients of the particular solution satisfy a triangular system of linear algebraic equations.Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs.The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs.Numerical results show that our approach is efficient and accurate.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fromtheMFS approximation has a block circulant structure,it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space.Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients

A meshless method based on the method of fundamental solutions(MFS)is proposed to solve the time-dependent partial differential equations with variable coefficients.The proposed method combines the time discretization and the onestage MFS for spatial discretization.In contrast to the traditional two-stage process,the one-stage MFS approach is capable of solving a broad spectrum of partial differential equations.The numerical implementation is simple since both closed-form approximate particular solution and fundamental solution are easy to find than the traditional approach.The numerical results show that the one-stage approach is robust and stable.

A Revisit on the Derivation of the Particular Solution for the Differential Operator ∆^(2) ± λ^(2)

In this paper,we applied the polyharmonic splines as the basis functions to derive particular solutions for the differential operator ∆^(2) ± λ^(2).Similar to the derivation of fundamental solutions,it is non-trivial to derive particular solutions for higher order differential operators.In this paper,we provide a simple algebraic factorization approach to derive particular solutions for these types of differential operators in 2D and 3D.The main focus of this paper is its simplicity in the sense that minimal mathematical background is required for numerically solving higher order partial differential equations such as thin plate vibration.Three numerical examples in both 2D and 3D are given to validate particular solutions we derived.

A Novel Method for Solving Time-Dependent 2D Advection-Diffusion-Reaction Equations to Model Transfer in Nonlinear Anisotropic Media

This paper presents a new numerical technique for solving initial and bound-ary value problems with unsteady strongly nonlinear advection diffusion reaction(ADR)equations.The method is based on the use of the radial basis functions(RBF)for the approximation space of the solution.The Crank-Nicolson scheme is used for approximation in time.This results in a sequence of stationary nonlinear ADR equations.The equations are solved sequentially at each time step using the proposed semi-analytical technique based on the RBFs.The approximate solution is sought in the form of the analytical expansion over basis functions and contains free parameters.The basis functions are constructed in such a way that the expansion satisfies the boundary conditions of the problem for any choice of the free parameters.The free parameters are determined by substitution of the expansion in the equation and collocation in the solution domain.In the case of a nonlinear equation,we use the well-known procedure of quasilinearization.This transforms the original equation into a sequence of the linear ones on each time layer.The numerical examples confirm the high accuracy and robustness of the proposed numerical scheme.

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago,the finitedifference time-domain(FDTD)method has been a widely-used technique for solving the time-dependent Maxwell’s equations that has also inspired many other methods.This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability,based on Krylov subspace spectral(KSS)methods.These methods have previously been applied to the variable-coefficient heat equation and wave equation,and have demonstrated high-order accuracy,as well as stability characteristic of implicit timestepping schemes,even though KSS methods are explicit.KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral,rather than physical,domain.We show how they can be generalized to coupled systems of equations,such as Maxwell’s equations,by choosing appropriate basis functions that,while induced by this coupling,still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields.We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators,which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.

A Boundary Meshless Method for Solving Heat Transfer Problems Using the Fourier Transform

Fourier transform is applied to remove the time-dependent variable in the diffusion equation.Under non-harmonic initial conditions this gives rise to a non-homogeneous Helmholtz equation,which is solved by the method of fundamental solutions and the method of particular solutions.The particular solution of Helmholtz equation is available as shown in[4,15].The approximate solution in frequency domain is then inverted numerically using the inverse Fourier transform algorithm.Complex frequencies are used in order to avoid aliasing phenomena and to allow the computation of the static response.Two numerical examples are given to illustrate the effectiveness of the proposed approach for solving 2-D diffusion equations.

Preface Special Issue for SCPDE08: Numerical Methods and Analysis for PDEs and Inverse Problems

The Third International Conference on Scientific Computing and Partial Differential Equations(SCPDE)was held from December 8 to December 12,2008 at China Hong Kong Baptist University.It was a sequel to similar conferences held in Hong Kong region(2002 and 2005).The conference aims to promote research interests in scientific computation.In SCPDE 2008,there were 118 participants from seventeen countries and regions participated in the conference.The Programme included seventeen plenary addresses,thirty invited talks,twenty five contributed talks and seven poster presentations.