Restoring Orders of Accuracy for Multilevel Schemes for Non-linear Hyperbolic Systems in Many Space Variables

In this paper boundary procedures are discussed for a new secondorder accurate method, developed in Morris (1972) and Zwas,Eilon & Gottlieb (1972), for non-linear hyperbolic systemsin two space variables. This method is a multilevel scheme ofthe same type as those of Strang, (1964, 1968). It is shownthat the straightforward method of incorporating boundary datagives, in general, only locally first order accurate values.A boundary procedure which preserves local second order accuracyis developed. The method is also extended to systems in manyspace variables. The results of some numerical experiments arereported.     

Modified Locally One Dimensional Methods for Parabolic Partial Differential Equations in Two Space Variables

In this note we consider the locally one dimensional methodas discussed in Gourlay & Mitchell (1969, 1972) and indicatehow this scheme can be used to solve accurately a parabolicpartial differential equation in two space variables.     

A family of numerical methods for diffusion and reaction–diffusion equations

A family of methods is developed for the numerical solution of second-order parabolic partial differential equations in one space dimension. The methods are second-, third-, or fourth-order accurate in time; five of them are seen to be L₀-stable in the sense of Gourlay and Morris, while the sixth is seen to be A₀-stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction–diffusion problem.     

On two step Lax-Wendroff methods in several dimensions

A version of Richtmyer's two step Lax-Wendroff scheme for solving hyperbolic systems in conservation form, is considered. This version uses only the nearest points, has second order accuracy at every time cycle and allows a time step which is larger by a factor of than Richtmyer's, whered is the number of spatial dimensions. The scheme appears to be competitive with the optimal stability schemes proposed by Strang and carried out by Gourlay and Morris.     

On the Parametric Cubic Spline Approach for Solving Second Order Boundary Value Problems

In this article some comments on the paper “parametric cubic spline approach to the solution of a system of second order boundary value problems” in (Khan and Aziz, J. Optim. Theory Appl. 118:45–54, 2003) are given. This paper concerns with a numerical method for solving a second order boundary value problem associated with obstacle, unilateral and contact problems. Corrections are given for the convergence analysis of the numerical method and the computational experiments.     

A mixed formulation for the direct approximation of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.     

On a Moving Boundary Problem from Biomechanics

A moving boundary problem arising in biomechanical diffusiontheory which has previously been investigated by Crank &Gupta (1972a, b) is studied using a different method of solution.The method is based on an integral equation for the functiondefining the position of the moving boundary and an integralformula for the concentration. The integral equation is solvedasymptotically for small times and numerically during the entiremotion of the boundary. The concentration is estimated asymptoticallyfor small times and computed by numerical quadrature at laterinstants. The results are compared with those of Crank &Gupta. In most cases the agreement is fair.     

The Solution of the Heat Equation in Two Space Variables using Chebyshev Series

A numerical solution of the heat conduction equation within a closed curve, is obtained in the form of a combinedFourier/Chebyshev series. The method is an extension of themethods for the one-dimensional heat equation given by Knibb& Scraton (1971) and Dew & Scraton (1972).     

Projection method III: Spatial discretization on the staggered grid

In E & Liu (SIAM J Numer. Anal., 1995), we studied convergence and the structure of the error for several projection methods when the spatial variable was kept continuous (we call this the semi-discrete case). In this paper, we address similar questions for the fully discrete case when the spatial variables are discretized using a staggered grid. We prove that the numerical solution in velocity has full accuracy up to the boundary, despite the fact that there are numerical boundary layers present in the semi-discrete solutions.

     

Chebyshev Methods for the Numerical Solution of Parabolic Partial Differential Equations in Two and Three Space Variables

Chebyshev methods for the numerical solution of parabolic partialdifferential equations in a region which can be transformedto either a square or a circular cylinder are developed. Theseprocedures are an extension of the method of Knibb & Scraton(1971). To illustrate the technique the solution of the heatconduction equation within an elliptical region is consideredin detail. The Chebyshev method given for this problem requiresconsiderably less computer time than the method of Dew &Scraton (1973). In the case when the space operators commutea highly efficient alternating direction Chebyshev method isgiven.     

Stability of Huang's update for the conjugate gradient method

The conditions under which Huang's conjugate gradient method generates descent directions are given and discussed. Bounds for the condition number of the inverse Hessian matrix are estimated for the case of a symmetric update.The author is much indebted to Professor C. G. Broyden of the Essex University Computing Center, for valuable advice and criticism; he is also grateful to Drs. J. Greenstadt and A. R. Gourlay for having sent copies of their unpublished papers.     

Heat Transfer with a Free Boundary Moving Within a Concentrated Thermal Capacity

A finite-difference scheme is described for the numerical solutionof plane-parallel and axially symmetric problems of heat transferwith a free boundary moving within concentrated thermal capacities.The relevant existence and uniqueness theorems are proved inFasano, Primicerio & Rubinstein (1980) and Rubinstein (1980).Numerical illustrations are offered.     

Finding the unknown boundary in the initial boundary-value problem for the heat equation

The article considers the determination of the boundary of a two-dimensional region in which an initial boundary-value problem for the heat equation is defined, given the solution of the problem for all time instants at some points of the region. The direct problem is reduced to an integral equation, and numerical solutions of the inverse problem are obtained for the case when the boundary is an ellipse. We investigate the sensitivity of the observed variables to the location (relative to the boundary) of the point where the right-hand side of the equation is specified. Translated from Prikladnaya Matematika i Informatika, No. 30, 2008, pp. 18–24.     

Stable Procedures for the Inversion of Abel's Equation

Because the inversion of Abel's equation often arises in practicalcontexts, numerous numerical methods for accomplishing it havebeen proposed. The most successful to-date have been the pseudo-analyticmethods of the type proposed by Minerbo & Levy, and Piessens& Verbaeten, as they are simple to implement and conditionallystable. In this paper, we propose methods based on the evaluationof the known inversion formulas using spectral differentiation.We show that they perform as well as the pseudo-analytic methodswhen the latter behave stably, and that there exist data forwhich the former yield a satisfactory solution while the latterfail. However, weighing up all numerical considerations, wepropose for general implementation an algorithm (viz., ProcedureIII) which uses a pseudo-analytic method to generate a firstapproximation to the solution, and a spectral differentiationprocedure (viz., Procedure II) to compute the correction tothis approximation.     

The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Applications

The aim of this paper is to discuss the numerical performanceof the Galerkin method for the approximate solution of severaltwo-dimensional Fredholm integral equations of the first kindwith logarithmic kernel, and for the approximation of linearfunctionals of the solution. Predicted rates of convergenceare obtained from the theory in Sloan & Spence (1987), andthese are compared with the numerical rates for the case ofpiecewise constant approximation over equal subintervals. Thephenomenon of ‘superconvergence’ is analysed indetail and some examples are given which attain remarkably highrates of convergence.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

阻尼边界条件散射问题的数值解法

该文研究了光滑区域上二维Helmholtz方程阻尼边界条件外问题的数值解法, 应用单双层位势组合来逼近散射场, 因此积分方程中含有超奇异算子. 给出了超奇异算子的离散化方法, 在Holder空间中给出了误差估计和解析边界的收敛性分析. 最后针对该方法给出数值实例, 以表明该方法的有效性.     

Numerical simulation of the boundary exact control for the system of linear elasticity

The problem of computing numerically the boundary exact control for the system of linear elasticity in two dimensions is addressed. A numerical method which has been recently proposed in [P. Pedregal, F. Periago, J. Villena, A numerical method of local energy decay for the boundary controllability of time-reversible distributed parameter systems. Stud. Appl. Math. 121 (1) (2008) 27–47] is implemented. Two cases are considered: first, a rectangular domain with Dirichlet controls acting on two adjacent edges, and secondly, a circular domain with Neumann controls distributed along the whole boundary.     

Stability and Accuracy of a Class of Numerical Boundary Conditions for the Advection Equation

Elvius & Sundstrm (1973) have used a stable time averagedboundary condition with the leap-frog scheme for the numericalsolution of hyperbolic partial differential equations. Gary(1978) generalized the time averaged boundary condition by includinga scalar parameter. This paper examines the stability and accuracyof the more general boundary condition. A stability intervalis found for the scalar parameter and an optimum value of theparameter is obtained which minimizes the boundary errors. Numericalexperiments are described which support the theoretical predictions.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.