Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.

     

A fourth order ADI method for semidiscrete parabolic equations

A fourth order fourstep ADI method is described for solving the systems of ordinary differential equations which are obtained when a (nonlinear) parabolic initial-boundary value problem in two dimensions is semi-discretized. The local time-discretization error and the stability conditions are derived. By numerical experiments it is demonstrated that the (asymptotic) fourth order behaviour does not degenerate if the time step increases to relatively large values. Also a comparison is made with the classical ADI method of Peaceman and Rachford showing the superiority of the fourth order method in the higher accuracy region, particularly in nonlinear problems.     

Parallel algorithms for a singularly perturbed parabolic problem

This article deals with iterative algorithms for domain decomposition applied to the solution of a singularly perturbed parabolic problem. These algorithms are based on finite difference domain decomposition methods and are suitable for parallel computing. Convergence properties of the algorithms are established. Numerical results for test problems are presented. © 1999 Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 389–405, 1999     

SCHWARZ TYPE DOMAIN DECOMPOSITION ALGORITHMS FOR PARABOLIC EQUATIONS AND ERROR ESTIMATES

1.IntroductionTheSchwarzalternatingmethodwasintroducedbyH.A.SchWarz(1870)120yearsagoasatechniqueforprovingtheexistenceofsolutionstocertainellipticproblemsdefinedonadomainofcomplicatedgeometries.Sincethenthemethodhasbeenextendedtononlinearproblemsandhasbeenproventobeasuitabledivideandconquertechniquetosolveawideclassofproblems,forinstance,see[2,3,8--10].Inthispaperweareinterestedinsolvingtheparabolicproblemsbyusingimplicitschemes,suchasbackwardEulerschemeinthetimevariable.Ateachfixedtimeleve…     

A delay differential equation solver based on the parallel Adams algorithms

IntroductionWith the rapid development of science and technology, a lot of delay-dynamic systems invarious fields of natural and social science have arisentl]. The presented delay systems are oftenof both multiparameter and largescale, that is, so-called large systems. Considering scientificcomputation and real-time simulation, the classical serial algorithmsIZ--4] have not fulfilled therequirements for solying this kind of systems. So, it becomes quite emergent to construct thefast algorithm…     

Numerical experiments with ABS algorithms for linear systems on a parallel machine

Numerical results are obtained on sequential and parallel versions of ABS algorithms for linear systems for both full matrices andq-band matrices. The results using the sequential algorithm on full matrices indicate the superiority of a particular implementation of the symmetric algorithm. The condensed form of the algorithm is well suited for implementation in a parallel environment, and results obtained on the IBM 4381 system favor a synchronous implementation over the asynchronous one. Results are obtained from sequential implementations of theLU, Cholesky, and symmetric algorithms of the ABS class forq-band matrices able to reduce memory storage. A simple parallelization of do-loops for calculating components gives interesting performances.This work has been developed in the framework of a collaboration between IBM-ECSEC, Rome, Italy, and the Department of Mathematics of the University of Bergamo, Bergamo, Italy.The author is grateful to Prof. J. Abaffy (University of Economics, Budapest), Prof. L. Dixon (Hatfield Polytechnic), and Prof. E. Spedicato (Department of Mathematics, University of Bergamo) for useful suggestions.     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

Numerical method of lines for parabolic functional differential equations

Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the numerical method of lines are given. An explicit Euler method is proposed for the numerical solution of systems thus obtained. This leads to difference scheme for the original problem. A complete convergence analysis for the method is given.     

A comparative study of ADI splitting methods for parabolic equations in two space dimensions

The main purpose of the paper is a numerical comparison of three integration methods for semi-discrete parabolic partial differential equations in two space variables. Linear as well as nonlinear,equations are considered. The integration methods are the well-known ADI method of Peaceman and Rachford, a global extrapolation scheme of the classical ADI method to order four and a fourth order, four-step ADI splitting method.     

Spectral methods based on Hermite functions for linear hyperbolic equations

We consider the approximation by spectral and pseudo‐spectral methods of the solution of the Cauchy problem for a scalar linear hyperbolic equation in one space dimension posed in the whole real line. To deal with the fact that the domain of the equation is unbounded, we use Hermite functions as orthogonal basis. Under certain conditions on the coefficients of the equation, we prove the spectral convergence rate of the approximate solutions for regular initial data in a weighted space related to the Hermite functions. Numerical evidence of this convergence is also presented. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

IBLU decompositions based on Padé approximants

In this paper, we introduce a new class of frequency‐filtering IBLU decompositions that use continued‐fraction approximation for the diagonal blocks. This technique allows us to construct efficient frequency‐filtering preconditioners for discretizations of elliptic partial differential equations on domains with non‐trivial geometries. We prove theoretically for a class of model problems that the application of the proposed preconditioners leads to a convergence rate of up to 1?O(h^1/4) of the CG iteration. Copyright © 2008 John Wiley & Sons, Ltd.     

A parareal approach of semi‐linear parabolic equations based on general waveform relaxation

We present a parareal approach of semi‐linear parabolic equations based on general waveform relaxation (WR) at the partial differential equation (PDE) level. An algorithm for initial‐boundary value problem and two algorithms for time‐periodic boundary value problem are constructed. The convergence analysis of three algorithms are provided. The results show that the algorithm for initial‐boundary value problem is superlinearly convergent while both algorithms for the time‐periodic boundary value problem linearly converge to the exact solutions at most. Numerical experiments show that the parareal algorithms based on general WR at the PDE level, compared with the parareal algorithm based on the classical WR at the ordinary differential equations (ODEs) level (the PDEs is discretized into ODEs), require much fewer number of iterations to converge.     

Finite element formulation based on proper orthogonal decomposition for parabolic equations

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method. This work was supported by National Natural Science Foundation of China (Grant Nos. 10871022, 10771065, and 60573158) and Natural Science Foundation of Hebei Province (Grant No. A2007001027)     

On finite-time blow-up for a nonlocal parabolic problem arising from shear bands in metals

Results on finite-time blow-up of solutions to the nonlocal parabolic problem

are established. They extend some known results to higher dimensions.

     

A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature

We consider the discretization in time of an inhomogeneous parabolicequation in a Banach space setting, using a representation ofthe solution as an integral along a smooth curve in the complexleft half-plane which, after transformation to a finite interval,is then evaluated to high accuracy by a quadrature rule. Thisreduces the problem to a finite set of elliptic equations withcomplex coefficients, which may be solved in parallel. The paperis a further development of earlier work by the authors, wherewe treated the homogeneous equation in a Hilbert space framework.Special attention is given here to the treatment of the forcingterm. The method is combined with finite-element discretizationin spatial variables.     

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.     

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Email: bahaa_gm{at}hotmail.com Received on December 6, 2005; Accepted on December 7, 2006 Optimal control problems of systems governed by parabolic equationswith an infinite number of variables and with additional equalityconstraints are considered. The extremum principle, as wellas sufficient condition of optimality, is formulated for theNeumann problem by using certain extensions of Dubovitskii–Milyutinmethod.     

Algorithm composition scheme for problems in composite domains based on the difference potential method

An algorithm composition scheme for the numerical solution of boundary value problems in composite domains is proposed and illustrated using an example. The scheme requires neither difference approximations of the boundary conditions nor matching conditions on the boundary between the subdomains. The scheme is suited for multiprocessor computers.     

Time-optimal control problem for parabolic equations with control constraints and infinite number of variables

A time optimal control problem for parabolic equations withan infinite number of variables is considered. A time optimalcontrol problem is replaced by an equivalent one with a performanceindex in the form of integral form. Constraints on controlsare assumed. To obtain the optimality conditions for the Neumannproblem, the generalization of the Dubovitskii–Milyutintheorem given by Walczak (1984, Acta Universitatis LodziensisFolia Mathematica, 187–196; 1984, J. Optim. Theor. Appl.,42, 561–582) was applied.