The time discretization in classes of integro-differential equations with completely monotonic kernels: Weighted asymptotic stability

We study discretization in classes of integro-differential equations where the functions aj(t),1≤j≤n,are completely monotonic on(0,∞) and locally integrable,but not constant.The equations are discretized using the backward Euler method in combination with order one convolution quadrature for the memory term.The stability properties of the discretization are derived in the weighted l1(ρ;0,∞) norm,where ρ is a given weight function.Applications to the weighted l1 stability of the numerical solutions of a related equation in Hilbert space are given.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

Convolution quadrature time discretization of fractional diffusion-wave equations

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

     

Nonsmooth Data Error Estimates for Approximations of an Evolution Equation with a Positive-Type Memory Term

We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the first- and second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial data and the inhomogeneity are shown for positive times, without assumptions of spatial regularity of the data.

     

Sparse convolution quadrature for time domain boundary integral formulations of the wave equation

S. A. Sauter Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ^{Many important physical applications are governed by the waveequation. The formulation as time domain boundary integral equationsinvolves retarded potentials. For the numerical solution ofthis problem, we employ the convolution quadrature method forthe discretization in time and the Galerkin boundary elementmethod for the space discretization. We introduce a simple apriori cut-off strategy where small entries of the system matricesare replaced by zero. The threshold for the cut-off is determinedby an a priori analysis which will be developed in this paper.This analysis will also allow to estimate the effect of additionalperturbations such as panel clustering and numerical integrationon the overall discretization error. This method reduces thestorage complexity for time domain integral equations from O(M2N)to O(M2N logM), where N denotes the number of time steps andM is the dimension of the boundary element space.}     

Nonlinear analysis of thin rectangular plates on Winkler–Pasternak elastic foundations by DSC–HDQ methods

This article introduces a coupled methodology for the numerical solution of geometrically nonlinear static and dynamic problem of thin rectangular plates resting on elastic foundation. Winkler–Pasternak two-parameter foundation model is considered. Dynamic analogues Von Karman equations are used. The governing nonlinear partial differential equations of the plate are discretized in space and time domains using the discrete singular convolution (DSC) and harmonic differential quadrature (HDQ) methods, respectively. Two different realizations of singular kernels such as the regularized Shannon’s kernel (RSK) and Lagrange delta (LD) kernel are selected as singular convolution to illustrate the present DSC algorithm. The analysis provides for both clamped and simply supported plates with immovable inplane boundary conditions at the edges. Various types of dynamic loading, namely a step function, a sinusoidal pulse, an N-wave pulse, and a triangular load are investigated and the results are presented graphically. The effects of Winkler and Pasternak foundation parameters, influence of mass of foundation on the response have been investigated. In addition, the influence of damping on the dynamic analysis has been studied. The accuracy of the proposed DSC–HDQ coupled methodology is demonstrated by the numerical examples.     

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish L ₂ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.     

The Price of Pessimism for Multidimensional Quadrature

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

1D numerical simulation of charge trapping in an insulator submitted to an electron beam irradiation. Part I: Computation of the initial secondary electron emission yield

The aim of this work is to study by numerical simulation a mathematical modelling technique describing charge trapping during initial charge injection in an insulator submitted to electron beam irradiation. A two-fluxes method described by a set of two stationary transport equations is used to split the electron current j_e(z) into coupled forward j_e+(z) and backward j_e?(z) currents and such that j_e(z) = j_e+(z) ? j_e?(z). The sparse algebraic linear system, resulting from the vertex-centered finite-volume discretization scheme is solved by an iterative decoupled fixed point method which involves the direct inversion of a bi-diagonal matrix.The sensitivity of the initial secondary electron emission yield with respect to the energy of incident primary electrons beam, that is penetration depth of the incident beam, or electron cross sections (absorption and diffusion) is investigated by numerical simulations.     

Some sufficient conditions for the non-negativity preservation property in the discrete heat conduction model

In the course of the numerical approximation of mathematical models there is often a need to solve a system of linear equations with a tridiagonal or a block-tridiagonal matrices. Usually it is efficient to solve these systems using a special algorithm (tridiagonal matrix algorithm or TDMA) which takes advantage of the structure. The main result of this work is to formulate a sufficient condition for the numerical method to preserve the non-negativity for the special algorithm for structured meshes. We show that a different condition can be obtained for such cases where there is no way to fulfill this condition. Moreover, as an example, the numerical solution of the two-dimensional heat conduction equation on a rectangular domain is investigated by applying Dirichlet boundary condition and Neumann boundary condition on different parts of the boundary of the domain. For space discretization, we apply the linear finite element method, and for time discretization, the well-known Θ-method. The theoretical results of the paper are verified by several numerical experiments.     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

Convolution quadrature and discretized operational calculus. II

Summary Operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations: singular integrals (power and logarithmic singularities, finite part integrals), multiple timescale convolution, Volterra integral equations, Wiener-Hopf integral equations. Frequency domain conditions, which determine, the stability of such equations, can be carried over to the discretization.This is Part II to the article with the same title (Part I), which was published in Volume 52, No. 2, pp. 129–145 (1988)     

A Green’s function formulation of nonlocal finite-difference schemes for reaction-diffusion equations

Reaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Green’s function formulation of reaction-diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h²) accuracy and nonlocal approximations in the whole domain (including boundary nodes). By doing so, the approach departs from a Green’s function formulation of the boundary-value problem to pose an approximation problem based on a domain decomposition. Within each subdomain, the corresponding integral equation is forced to have zero residual at given grid points. Different FD schemes are obtained depending on the numerical scheme used for computing the Green’s integral over each subdomain. Dirichlet and Neumann boundary conditions are considered, showing that the FD scheme based on the Green’s function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation.     

Indecomposable modules of the intermediate series over W(a, b)

For any complex parameters a and b,W(a,b)is the Lie algebra with basis{Li,Wi|i∈Z}and relations[Li,Lj]=(j i)Li+j,[Li,Wj]=(a+j+bi)Wi+j,[Wi,Wj]=0.In this paper,indecomposable modules of the intermediate series over W(a,b)are classified.It is also proved that an irreducible Harish-Chandra W(a,b)-module is either a highest/lowest weight module or a uniformly bounded module.Furthermore,if a∈/Q,an irreducible weight W(a,b)-module is simply a Vir-module with trivial actions of Wk’s.     

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szeg?-Radau and Szeg?-Lobatto quadrature formulas on the unit circle. These are (n+m)-point formulas for which m nodes are fixed in advance, with m=1 and m=2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. This means that the free parameters (free nodes and positive weights) are chosen such that the quadrature formula is exact for all powers z^j, −p≤j≤p, with p=p(n,m) as large as possible.     

Product integration of logarithmic singular integrands based on cubic splines

This paper is concerned with the practical evaluation of the product integral ∫¹_{? 1}f(x)k(x)dx for the case when k(x) = In|x - λ|, λ? (?1, +1) and f is bounded in [?1, +1]. The approximation is a quadrature rule

where the weights {w_n,n,i} are chosen to be exact when f is given by a linear combination of a chosen set of functions {φ_n,j}. In this paper the functions {φ_n,j} are chosen to be cubic B-splines. An error bound for product quadrature rules based on cubic splines is provided. Examples that test the performance of the product quadrature rules for different choices of the function are given. A comparison is made with product quadrature rules based on first kind Chebyshev polynomials.     

Enhanced consistency of the Resampled Convolution Particle Filter

Among the convolution particle filters for discrete-time dynamic systems defined by nonlinear state space models, the Resampled Convolution Filter is one of the most efficient, in terms of estimation of the conditional probability density functions (pdf’s) of the state variables and unknown parameters and in terms of implementation. This nonparametric filter is known for its almost sure L₁-convergence property. But contrarily to the other convolution filters, its almost sure punctual convergence had not yet been established. This paper is devoted to the proof of this property.     

The space fractional diffusion equation with Feller’s operator

This paper investigates the space fractional diffusion equation with fractional Feller’s operator. The Green’s function is obtained by using Fourier transform, and the analytical solutions of some space fractional diffusion equations with initial (or initial and boundary) condition are obtained in terms of Green’s function. In addition, numerical simulations are discussed. The results indicate that the effect range of skewness parameter θ has more effect on probability density than that of parameter α. The results also explain the property of the skewness and long tail in the asymmetry diffusion process.     

A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A ₁ A ₂?A ₅ has an interior weighted Fermat-Torricelli point A ₀ with respects to the weights B _i and let α _i0j =∠A _i A ₀ A _j . Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A ₀, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case.     

A numerical comparison of Chebyshev methods for solving fourth order semilinear initial boundary value problems

In solving semilinear initial boundary value problems with prescribed non-periodic boundary conditions using implicit-explicit and implicit time stepping schemes, both the function and derivatives of the function may need to be computed accurately at each time step. To determine the best Chebyshev collocation method to do this, the accuracy of the real space Chebyshev differentiation, spectral space preconditioned Chebyshev tau, real space Chebyshev integration and spectral space Chebyshev integration methods are compared in the L² and W^2,2 norms when solving linear fourth order boundary value problems; and in the L^∞([0,T];L²) and L^∞([0,T];W^2,2) norms when solving initial boundary value problems. We find that the best Chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space Chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to Fourier spectral discretization.