A MODIFIED UPWIND DIFFERENCE SCHEME FOR NONLINEAR PARABOLIC EQUATIONS IN A VARIABLE DOMAIN

The nonlinear parabolic equations in a variable domain are considered. A modified up-wind difference scheme is given in the variable domain. Stability in l norm and error estimate in norm are obtained.     

Lie‐group method for solving the problem of fission product behavior in nuclear fuel

Calculation of the gas atom concentration is an important feature of all physical models of fission gas release. We apply Lie‐group method for determining symmetry reductions to the diffusion equation describing the fission gas release from nuclear fuel. The resulting nonlinear ordinary differential equation is solved numerically using nonlinear finite difference method. Effects of the dimensionless group constant, the time, and the grain radius on the concentration diffusion function have been studied, and the results are plotted. It is found that the concentration of gas atoms increases as the dimensionless group constant, the power index, and the time increase, and it decreases with increase of the grain radius. Copyright © 2013 John Wiley & Sons, Ltd.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A method for solving a boundary problem for a nonlinear controlled system

An algorithm is proposed for constructing a control function that transfers a wide class of nonlinear systems of ordinary differential equations from an initial state to an arbitrarily small neighborhood of a given terminal state. The algorithm is convenient for numerical implementation. Taking into account the restrictions on the control and phase coordinates, a constructive criterion is obtained for choosing terminal states for which this transfer is possible. The problem of an interorbital flight is considered and modeled numerically.     

A non-local boundary value problem method for parabolic equations backward in time

The ill-posed parabolic equation backward in time

     

Parallel iterative methods using factorized preconditioning matrices for solving elliptic equations on triangular grids

Parallel analogs of the variants of the incomplete Cholesky-conjugate gradient method and the modified incomplete Cholesky-conjugate gradient method for solving elliptic equations on uniform triangular and unstructured triangular grids on parallel computer systems with the MIMD architecture are considered. The construction of parallel methods is based on the use of various variants of ordering the grid points depending on the decomposition of the computation domain. Results of the theoretic and experimental studies of the convergence rate of these methods are presented. The solution of model problems on a moderate number processors is used to examine the efficiency of the proposed parallel methods.     

The use of linear approximation scheme for solving the Stefan problem

This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.     

An upwind center difference parallel method and numerical analysis for the displacement problem with moving boundary

A nonlinear coupled mathematical system of two‐phase seepage flow displacement is discussed in this paper including an elliptic equation for the pressure and a convection‐dominated diffusion equation for the saturation. In fact, the boundary of an underground region where the fluid flows through is nonstationary. So a moving boundary should be considered. The saturation equation is convection‐dominated, therefore the method of upwind finite difference is introduced for the accurate computation. The upwind approximation could eliminate numerical oscillation and strong stability is shown. Since the computational work of saturation is larger than the pressure, the authors apply a parallel method, decomposing the whole domain into several nonoverlapping subdomains, to simplify the computation. A domain decomposition method coupled with upwind differences is presented for the saturation. The pressure equation is discretized by a five‐point center finite difference method. By using a transformation and defining new inner products and norms, error estimates in l² norm is discussed. Finally, two experimental tests are given to illustrate the efficiency and accuracy of the parallel algorithm.     

A variable step implicit block multistep method for solving first-order ODEs

A new four-point implicit block multistep method is developed for solving systems of first-order ordinary differential equations with variable step size. The method computes the numerical solution at four equally spaced points simultaneously. The stability of the proposed method is investigated. The Gauss-Seidel approach is used for the implementation of the proposed method in the PE(CE)^m mode. The method is presented in a simple form of Adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients. Numerical examples are given to illustrate the efficiency of the proposed method.     

A new semi-analytical method for phase transformations in binary alloys

In the present paper, a new semi-analytical method is developed to cover a wide range of phase transformation problems and their practical applications. The solution procedure consists of two parts: first, determination of the position of the moving boundary named the homogenous part and second, determination of the concentration named the non-homogenous part. The homogenous part leads to a system of homogenous linear equations, based on the mathematical fact that a homogenous system has a non-trivial solution if the determinant of the coefficient matrix equals zero. This determinant leads to an ordinary differential equation for the moving boundary, and its solution leads to a closed form formula for the position of the moving boundary. The non-homogenous part transforms the governing equations to a non-homogenous linear system of equations, having three unknowns that appear in the concentration profile assumed in the beginning of the proposed method. Solution of the non-homogenous system leads to a value of these unknowns. Once these unknowns are computed, the concentration at any time and at any point can be found easily.     

A nonstandard finite difference method for solving a mathematical model of HIV-TB co-infection

We design and analyze an unconditionally convergent nonstandard finite-difference method to study transmission dynamics of a mathematical model of HIV-TB co-infection. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves positivity of the solution which is one of the essential requirements when modelling epidemic diseases. Furthermore, we show that the numerical method is unconditionally stable. Competitive numerical results confirming theoretical investigations are provided. Comparisons are also made with other conventional approaches that are routinely used to solve these types of problems.     

Cost-efficient numerical algorithm for solving the linear inverse problem of finding a variable magnetization

The paper is devoted to developing an original cost-efficient algorithm for solving the inverse problem of finding a variable magnetization in a rectangular parallelepiped. The problem is ill-posed and is described by the integral Fredholm equation. It is shown that after discretization of the area and approximation of the integral operator, this problem is reduced to solving a system of linear algebraic equations with the Toeplitz-block-Toeplitz matrix. We have constructed the memory efficient variant of the stabilized biconjugate gradient method BiCGSTABmem. This optimized algorithm exploits the special structure of the matrix to reduce the memory requirements and computing time. The efficient implementation is developed for multicore CPU and GPU. A series of the model problems with synthetic and real magnetic data are solved. Investigation of efficiency and speedup of parallel algorithm is performed.     

A direct variable step block multistep method for solving general third-order ODEs

This paper discusses a direct three-point implicit block multistep method for direct solution of the general third-order initial value problems of ordinary differential equations using variable step size. The method is based on a pair of explicit and implicit of Adams type formulas which are implemented in PE(CE) ^t mode and in order to avoid calculating divided difference and integration coefficients all the coefficients are stored in the code. The method approximates the numerical solution at three equally spaced points simultaneously. The Gauss Seidel approach is used for the implementation of the proposed method. The local truncation error of the proposed scheme is studied. Numerical examples are given to illustrate the efficiency of the method.     

Constructive method for solving a linear minimax problem of optimal control

In this paper, we propose a constructive method for solving a linear minimax problem of optimal control. Following the Gabasov-Kirillova approach, we introduce the concept of so-called support control. After establishing an optimality criterion for the support control, we describe a scheme for reducing the initial infinite-dimensional problem to a finite-dimensional one, which can be solved numerically by the methods of linear programming. At the end, we give an illustrative example.     

On an efficient splitting‐based method for solving the diffusion equation on a sphere

A novel numerical approach for solving the diffusion problem on a sphere is suggested. By using operator splitting, we develop a new method that allows constructing finite difference schemes of the second and fourth approximation orders in the spatial variables. Both schemes properly ensure the balance of mass and the energy dissipation in the L₂ ‐norm. The schemes are very cheap from the computational standpoint. Numerical results demonstrate the skillfulness of the approach in describing the diffusion dynamics on a sphere. It is shown the method can directly be extended to nonlinear diffusion problems.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 331–352, 2012     

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.     

A modified SQP-filter method for nonlinear complementarity problem

The nonlinear complementarity problem can be reformulated as a nonlinear programming. For solving nonlinear programming, sequential quadratic programming (SQP) type method is very effective. Moreover, filter method, for its good numerical results, are extensively studied to handle nonlinear programming problems recently. In this paper, a modified quadratic subproblem is proposed. Based on it, we employ filter technique to tackle nonlinear complementarity problem. This method has no demand on initial point. The restoration phase, which is always used in traditional filter method, is not needed. Global convergence results of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.     

On an initial‐boundary value problem for a wide‐angle parabolic equation in a waveguide with a variable bottom

We consider the third‐order Claerbout‐type wide‐angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There is strong indication that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well‐posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second‐order boundary condition, derived by assuming that the standard (narrow‐angle) PE holds on B, and establish a priori H² estimates for the solution of the resulting initial‐boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second‐order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge‐type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson–Kreiss boundary condition in the wide‐angle case. Copyright © 2008 John Wiley & Sons, Ltd.     

Wavelet method for solving the unsteady porous-medium flow problem with discontinuous coefficients

A method is proposed for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data. The method is based on the general scheme for finite-dimensional approximation in Tikhonov regularization and on multiresolution analysis with basis functions defined as the product of one-dimensional Daubechies wavelets.