S. Daoud

A splitting up algorithm for the determination of the control parameter in multi dimensional parabolic problem

One of the global approaches for solving the two dimensional inverse parabolic problem is the predictor corrector which takes place for evaluating the pair (u,p) and adjusting the evaluation for the desired accuracy. For general class of higher dimensional parabolic problems the ADI or the FS methods have been considered with an advantage of reducing or splitting the problem, in accordance to the time fractions, into one dimensional dependent problems and they are a non-parallel type of splittings. In 1992 Lu et al. proposed an additive parallel type of splitting method such that the splitting is defined in accordance to the spatial variables to solve multi dimensional parabolic problem. In this work we will present a new algorithm for solving two or higher dimensional inverse control problem. The algorithm is a parallel predictor corrector type of method such that the solution and the predictor and corrector schemes are defined by the parallel splitting up method. Some numerical results from the solution of two model problems are considered to demonstrate the accuracy of the presented algorithm.

On the numerical solution of multi-dimensional parabolic problem by the additive splitting up method

Several numerical methods arising from the difference methods for certain classes of two dimensional parabolic equation are based on an operator slitting. From the theoretical point of view the success and the evaluation of the splitting approach is primarily determined by the accuracy and the stability constrained. Most of the splitting methods defined in the past decades were of multiplicative non-parallel types with respect to the spatial variables. In 1994 Lui, Tai and Neittaanmaki presented a parallelisable implicit-splitting type of methods of different order of approximation and its of additive type with regard to the solution at the advanced time step. In this paper the stability analysis will be presented for the implicit splitting methods by Lu et al. and also we presented a parallelisable explicit splitting algorithm. Several model problems are solved by the splitting up algorithms to enhance the theoretical results and the concluding remarks.

Overlapping Schwarz wave form relaxation for the solution of coupled and decoupled system of convection diffusion reaction equation

In this article we study the convergence and the error bound for the solution of the convection diffusion reaction equation using overlapping Schwarz wave form relaxation method combined with the first order fractional splitting method (Strang’s splitting) as basic solver. We extended the study to solve decoupled and coupled system of equations of same class in order to demonstrate the effect of the coupling in the system, through the reaction term, on the convergence and error decay. The accuracy and the efficiency of the methods are investigated through the solution of different model problems of scalar, coupled and decoupled systems of convection diffusion reaction equations.

On the preconditioning conjugate gradient method for the solution of 9-point elliptic difference equations

The Preconditioned Conjugate Gradient (PCG) method is implemented to solve the system with 9 diagonal entries generated from a 9-point finite difference approximation of the self-adjoint elliptic partial differential equation using an incomplete matrix decomposition. A simpler matrix decomposition for such a linear system is also proposed with a main advantage that it preserves the symmetry of the original matrix, and is easy to implement. Results of the numerical experiments and comparison with other iterative methods are presented.

Stability of the Parareal Time Discretization for Parabolic Inverse Problems

The practical aspect in the parareal algorithm that it consist of using two solvers over different time stepping, the coarse and fine solvers to produce a rapid convergent iterative method for multi processors computations. The coarse solver solve the equation sequentially on the coarse time step while the fine solver use the information from the coarse solution to solve, in parallel, over the fine time steps. In this work we discussed the stability of the parareal-inverse problem algorithm for solving the parabolic inverse problem given by

Time lagging and explicit interface prediction for nonoverlapping domain decomposition with parallel additive splitting method for multi-dimensional parabolic problem

Two different interface predictions are considered, in this work, for the solution of multi-dimensional parabolic problem defined over nonoverlapping subdomains. These interface boundary conditions are predicted by the explicit prediction (EP) and the time lagging (TL) methods with δx and 2δx mesh spacings away from the interface lines. We considered the additive splitting up method with respect to the spatial variable to solve the multi-dimensional parabolic problem over each subdomains. For more accurate solution we present the correction for the explicit interface prediction and implement the proposed methods for the solution of two different model problems.

A Fractional Splitting Algorithm for Non-overlapping Domain Decomposition

In this paper we study the convergence of the non overlapping domain decomposition for solving large linear system arising from semi discretization of two dimensionalinitial value problem with homogeneous boundary conditions, and solved by implicittime stepping using first and, two alternatives of second order FS-methods. The interface values along the artificial boundary condition line are found using explicit forward Eulers method for the first order FS-method, and for the second order FS-method to use extra polation procedure for each spatial variable individually. The solution by the non overlapping domain decomposition with FS-method is applicable to problems that requires the solution on non uniform meshes for each spatial variables which will unable us to use different time stepping over different sub domains, and with the possibility of extension to three dimensional problem.

On the stationary and non stationary extrapolation iterative method

In this article theoretical results are presented on the extrapolation of the first order iterative method to estimate the optimal value of the extrapolation parameter with the rate of convergence which is found to be depends inversely on the p-condition number of the errors operator matrix.A further extension is considered to estimate the extrapolation parameters for the nonstationary extrapolation method and the rate of convergence is found to be depend inversely on . The stationary and non stationary extrapolation methods are considered to define the relevant extrapolation versions for the Jacobi and SOR methods

A Parallel Splitting up Algorithm for the Determination of an Unknown Coefficient in Multi Dimensional Parabolic Problem

One of the global approach for solving the two dimensional inverse parabolic problem is the predictor corrector which takes place for evaluating the pair (u,p) and adjusting the evaluation for the desired accuracy. In this work we will present a new parallel algorithm(of non iterative type) for solving two or higher dimensional inverse control problem.

Monotone iterative methods and Schwarz methods for nonlinear parabolic PDE with time delay

Its well known that the Schwarz alternating method proved to be feasible and powerful approach to solve elliptic or parabolic PDEs over multi overlapped sub domains . Recently Schwarz method proved to be very effective method when embedded within a well established solution methods such as monotone iterative method( or the method of lower and upper solution method). In this work we present the proofs of convergence of additive and multiplicative Schwarz alternating method for non linear parabolic equation where the reaction function involves a time delay function.