On a Steffensen-Hermite method of order three

In this paper we study a third order Steffensen type method obtained by controlling the interpolation nodes in the Hermite inverse interpolation polynomial of degree 2. We study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution; it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors. The numerical examples confirm the advantage of considering these error bounds.     

一类常微分方程的非多项式样条函数逼近解

本文提出一种新的数值方法来获得三阶带障碍边值问题的常微分方程的数值解,这类方法的基函数包括三角函数、指数函数、多项式函数。本文将给出一数值例子说明这种方法优于其他差分法、配置法、多项式样条函数法。     

Approximations of a Continuous Time Filter. Application to Optimal Allocation Problems in Finance

Abstract

In this article, we study a continuous time optimal filter and its various numerical approximations. This filter arises in an optimal allocation problem in the particular context of a non-stationary economy. We analyse the rates of convergence of the approximations of the filter when the model is misspecified and when the observations can only be made at discrete times. We give bounds that are uniform in time. Numerical results are presented.     

Error estimates for approximations of a gradient dynamics for phase field elastic bending energy of vesicle membrane deformation

In this paper, we study the numerical approximations of a gradient flow associated with a phase field bending elasticity model of a vesicle membrane with prescribed volume and surface area. A spatially semi‐discrete scheme based on a mixed finite element formulation and a fully discrete in space and time scheme are analyzed. Optimal order error estimates are rigorously derived for these numerical schemes without any a priori assumption. Copyright © 2013 John Wiley & Sons, Ltd.     

A parameter robust numerical method for a two dimensional reaction-diffusion problem

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

     

Incremental unknowns for solving partial differential equations

Summary Incremental unknowns have been proposed in [T] as a method to approximate fractal attractors by using finite difference approximations of evolution equations. In the case of linear elliptic problems, the utilization of incremental unknown methods provides a new way for solving such problems using several levels of discretization; the method is similar but different from the classical multigrid method.In this article we describe the application of incremental unknowns for solving Laplace equations in dimensions one and two. We provide theoretical results concerning two-level approximations and we report on numerical tests done with multi-level approximations.     

Numerical identification of the leading coefficient of a parabolic equation

For a multidimensional parabolic equation, we study the problem of finding the leading coefficient, which is assumed to depend only on time, on the basis of additional information about the solution at an interior point of the computational domain. For the approximate solution of the nonlinear inverse problem, we construct linearized approximations in time with the use of ordinary finite-element approximations with respect to space. The numerical algorithm is based on a special decomposition of the approximate solution for which the transition to the next time level is carried out by solving two standard elliptic problems. The capabilities of the suggested numerical algorithm are illustrated by the results of numerical solution of a model inverse two-dimensional problem.     

Numerical verification of solutions for variational inequalities

In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities. This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical example for an obstacle problem is presented. Received October 28,1996 / Revised version received December 29,1997     

Numerical Solution of a Scalar One-Dimensional Monotonicity-Preserving Nonlocal Nonlinear Conservation Law

In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local conservation laws in the form of partial differential equations. We discuss convergent numerical approximations that preserve similar properties on the discrete level. We also present numerical experiments to study various limiting behavior of the numerical solutions.     

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.     

An a priori error analysis of a type III thermoelastic problem with two porosities

In this work, we study, from the numerical point of view, a type III thermoelastic model with double porosity. The thermomechanical problem is written as a linear system composed of hyperbolic partial differential equations for the displacements and the two porosities, and a parabolic partial differential equation for the thermal displacement. An existence and uniqueness result is recalled. Then, we perform its a priori error numerical analysis approximating the resulting variational problem by using the finite element method and the implicit Euler scheme. The linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are shown to demonstrate the accuracy of the approximations and the dependence of the solution on a coupling coefficient.     

On some iterative numerical methods for a Volterra functional integral equation of the second kind

In this paper, we study an iterative numerical method for approximating solutions of a certain type of Volterra functional integral equations of the second kind (Volterra integral equations where both limits of integration are variables). The method uses the contraction principle and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates for our approximations. We also included a numerical example which illustrates the fast approximations.     

On an efficient implementation and mass boundedness conditions for a discrete Dirichlet problem associated with a nonlinear system of singular partial differential equations

In this work, we propose an efficient implementation of a finite-difference method employed to approximate the solutions of a system of partial differential equations that appears in the investigation of the growth of biological films. The associated homogeneous Dirichlet problem is discretized using a linear approach. This discretization yields a positivity- and boundedness-preserving implicit technique which is represented in vector form through the multiplication by a sparse matrix. A straightforward implementation of this methodology would require a substantial amount of computer memory and time, but the problem is conveniently coded using a continual reduction of the zero sub-matrices of the representing matrix. In addition to the conditions that guarantee the positivity and the boundedness of the numerical approximations, we establish some parametric constraints that assure that the same properties for the discrete total mass at each point of the mesh-grid and each discrete time are actually satisfied. Some simulations are provided in order to illustrate both the performance of the implementation, and the preservation of the positivity and the boundedness of the numerical approximations.     

Positivity-Preserving Local Discontinuous Galerkin Method for Pattern Formation Dynamical Model in Polymerizing Actin Flocks

In this paper, we apply local discontinuous Galerkin (LDG) methods for pattern formation dynamical model in polymerizing actin flocks. There are two main difficulties in designing effective numerical solvers. First of all, the density function is non-negative, and zero is an unstable equilibrium solution. Therefore, negative density values may yield blow-up solutions. To obtain positive numerical approximations, we apply the positivity-preserving (PP) techniques. Secondly, the model may contain stiff source. The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method. However, for problems with stiff source, such time discretizations may require strictly limited time step sizes, leading to large computational cost. Moreover, the stiff source any trigger spurious filament polarization, leading to wrong numerical approximations on coarse meshes. In this paper, we combine the PP LDG methods with the semi-implicit Runge-Kutta methods. Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.     

Corrected phase-type approximations of heavy-tailed risk models using perturbation analysis

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phase-type approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.     

Exponential spline solutions for a class of two point boundary value problems over a semi-infinite range

In this paper, we developed numerical methods of order O(h ²) and O(h ⁴) based on exponential spline function for the numerical solution of class of two point boundary value problems over a Semi-infinite range. The present approach gives better approximations over all the existing finite difference methods. Properties of the infinite linear system are established. Convergence analysis and a bound on the approximate solution are discussed. Test problem with various kinds of boundary conditions is included to illustrate the practical usefulness and superiority of our methods.     

Determination of material functions through second sound measurements in a hyperbolic heat conduction theory

In a previous paper [1], numerical solutions to initial-boundary value problems for a semi-empirical model of heat conduction were compared with available experimental results.

In the present paper, we modify the model by introducing more realistic approximations of constitutive functions, based on measured heat conductivities and second sound speeds for NaF at low temperatures (10…20° K). We achieve good accordance between measured second sound pulses and numerical solutions in the temperature range covered by experiments, and reasonable behaviour even beyond this interval. Especially, a passage to the diffusive regime of the classical Fourier law is possible.     

A parameter-uniform numerical method for a Sobolev problem with initial layer

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a linear Sobolev or pseudo-parabolic equation with initial jump. In order to obtain an efficient method, to provide good approximations with independence of the perturbation parameter, we have developed a numerical method which combines a finite difference spatial discretization on uniform mesh and the implicit rule on Shishkin mesh(S-mesh) for the time variable. The fully discrete scheme is shown to be convergent of order two in space and of order one expect for a logarithmic factor in time, uniformly in the singular perturbation parameter. Some numerical results confirming the expected behavior of the method are shown.      

Dynamic contact of a beam against rigid obstacles: Convergence of a velocity-based approximation and numerical results

Motivated by the study of vibrations due to looseness of joints, we consider the motion of a beam between rigid obstacles. Due to the non-penetrability condition, the dynamics is described by a hyperbolic fourth order variational inequality. We build a family of fully discretized approximations of this problem by combining some classical space discretizations with velocity based time-stepping algorithms for discrete mechanical systems subjected to unilateral constraints. We prove the stability and the convergence of these numerical methods. Finally we propose some examples of implementation using either Hermite or B-spline finite element approximations.