Leak identification in a saturated unsteady flow via a Cauchy problem

In this initial study, we propose a numerical method for identifying multiple leak zones in a saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modeled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes because the leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we modify it and employ a previously proposed iterative regularizing method. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint to obtain a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a finite element method and the numerical results indicate that the leak zones can be identified with the proposed method.     

A meshless procedure for transient heat conduction in functionally graded materials

A meshless numerical procedure is developed for analyzing the transient heat conduction problem in non-homogeneous functionally graded materials. In the proposed method the time derivative of temperature is approximate by the finite difference. At each time step the original nonlinear boundary value problem is transform into a hierarchy of non-homogeneous linear problem by used the homotopy analysis method. In this method a sought solution is presented by using a finite expansion in Taylor series, which consecutive elements are solutions of series linear value problems defining differential deformations. Each of linear boundary value problems with the corresponding boundary conditions is solved by using the method of fundamental solutions and radial basis functions which are used for interpolation of the inhomogeneous term. The accuracy of the obtained approximate solution is controlled by the number of components of the Taylor series, while the convergence of the process is monitored by an additional parameter of the method. Numerical experiments demonstrate the efficiency and accuracy of the present scheme in the solution of the heat conduction problem in nonlinear functionally graded materials. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of a quasilinear parabolic problem

A combined approach of linearisation techniques and finite difference method is presented for obtaining the numerical solution of a quasilinear parabolic problem. The given problem is reduced to a sequence of linear problems by using the Picard or Newton methods. Each problem of this sequence is approximated by Crank-Nicolson difference scheme. The solutions of the resulting system of algebraic equations are obtained by using Block-Gaussian elimination method. Two numerical examples are solved by using both linearisation procedures to illustrate the method. For these examples, the Newton method is found to be more effective, especially when the given nonlinear problem has steep gradients.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.     

On the optimal control of the Schlögl-model

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation. The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.     

A new approach to the weighted peak-constrained least-square error FIR digital filter optimal design problem

This paper deals with the design of linear-phase finite impulse response (FIR) digital filters using weighted peak-constrained least-squares (PCLS) optimization. The PCLS error design problem is formulated as a quadratically constrained quadratic semi-infinite programming problem. An exchange algorithm with a new exchange rule is proposed to solve the problem. The algorithm provides the approximate optimal solution after a finite number of iterations. In particular, the subproblem solved at each iteration is a quadratically constrained quadratic programming. We can rewrite it as a conic optimization problem solvable in polynomial time. For illustration, numerical examples are solved using the proposed algorithm.     

A mixed collocation–finite difference method for 3D microscopic heat transport problems

Three-dimensional time-dependent initial-boundary value problems of a novel microscopic heat equation are solved by the mixed collocation–finite difference method in and on the boundaries of a particle when the thickness is much smaller than both the length and width. The collocation method on fixed grid size is used to approximate the space operator, whereas the finite difference scheme is used for time discretization. This new mixed method is applied to a novel heat problem in a particle, in order to compute the temperature distribution in and on the particle's surface. The second derivatives of the basis functions for the spectral approximation are derived. Direct substitution of derivatives in the model transforms the differential equation into a linear system of equations that is solved by the specific preconditioned conjugate gradient method. The high-order accuracy and resolution achieved by the proposed method allows one to obtain engineering-accuracy solution on coarse meshes. The consistency, stability and convergence analysis are provided and numerical results are presented.     

Numerical Solution of the Incompressible Navier-Stokes Equations with Singular Perturbation Considerations

A numerical method for coarse grids is proposed for the numerical solution of the incompressible Navier-Stokes equations. From singular perturbation considerations, we obtain partial differential equations and boundary conditions for the outer solution and the boundary layer correction. The former problem is solved with the finite difference method and the latter with the approximate method. Numerical experiments show that accurate outer flow and boundary flux result with little computational effort.     

On a free interface problem for linear ordinary differential equations and the one-phase Stefan problem

A numerical method for the solution of the one-phase Stefan problem is discussed. By discretizing the time variable the Stefan problem is reduced to a sequence of free boundary value problems for ordinary differential equations which are solved by conversion to initial value problems. The numerical solution is shown to converge to the solution of the Stefan problem with decreasing time increments. Sample calculations indicate that the method is stable provided the proper algorithm is chosen for integrating the initial value problems.     

Numerical solution for the anisotropic Willmore flow of graphs

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.     

Evaluating American put options on zero-coupon bonds by a penalty method

In this paper, American put options on zero-coupon bonds are priced under a single factor model of short-term rate. The linear complementarity problem of the option value is solved numerically by a penalty method, by which the problem is transformed into a nonlinear PDE by adding a power penalty term. The solution of the penalized problem converges to that of the original problem. A numerical scheme is established by using the finite volume method and the corresponding stability and convergence are discussed. Numerical results are presented to show the usefulness of the method.     

A Linear Complementarity Approach for the Non-convex Seismic Frictional Interaction between Adjacent Structures under Instabilizing Effects*

The paper deals with a numerical treatment of the dynamic hemivariational inequality problem concerning the elastoplastic-fracturing unilateral contact with friction between neighboring structures under second-order geometric effects during earthquakes. The numerical procedure is based on an incremental problem formulation and on a double discretization, in space by the finite element method and in time by the Houbolt method. The generally nonconvex constitutive contact laws are piece-wise linearized, and in each time-step a nonconvex linear complementarity problem is solved with a reduced number of unknowns.     

A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics

This study deals with obtaining numerical solutions of two-dimensional (2D) fractional cable equation in neuronal dynamics by using a recently introduced meshless method. In solution process at first stage, time derivatives that are appeared in the considered problem are discretized by using finite difference method. Then a meshless method based on hybridization of Gaussian and cubic kernels is developed in local fashion. The problem is solved both on regular and irregular domians. L_∞ and RMS error norms are calculated and compared with other numerical methods in literature as well as exact solutions. Also, obtained condition numbers are monitored. Numerical simulations show that local hybrid kernel meshless method is a thriving method for solving 2D fractional cable equation on regular and irregular domians.     

Numerical analysis of evolution problems in nonlinear small strains elastoviscoplasticity

Summary The present paper deals with the mathematical and numerical analysis of evolution problems in nonlinear small strains viscoelasticity of Burger's type. After a brief review of the mechanical model, the viscoelastic problem to be solved is written as an abstract evolution problem. The associated operator is proved to be maximal monotone, thus implying existence and uniqueness of solutions. This problem is then solved numerically by a backward Euler discretization in time, a finite element approximation in space and by using a preconditioned conjugate gradient algorithm for solving the resulting nonlinear algebraic systems. Numerical results are finally presented to illustrate the solution procedure.     

自然对流换热问题基于混合元法的差分格式及其数值模拟

研究自然对流换热问题,通过对于空间变量采用有限元离散而对于时间变量用差分离散,导出一种基于混合有限元法的最低阶的差分格式,这种格式可以同时求出流体的速度、温度和压力的数值解,并给出了模拟方腔流的自然换热的数值例子。     

Analysis of traffic flow by the finite element method

A finite elernent methodology is developed for the numerical solution of traffic flow problems encountered in arterial streets. The simple continuum traffic flow model consisting of the equation of continuity and an equilibrium flow-density relationship is adopted. A Galerkin type finite element method is used to formulate the problem in discrete form and the solution is obtained by a step-by-step time integration in conjunction with the Newton-Raphson method. The proposed finite element methodology, which is of the shock capturing type, is applied to flow traffic problems. Two numerical examples illustrate the method and demonstrate its advantages over other analytical or numerical techniques.     

外推瀑布多网格法(EXCMG)——-大规模求解椭圆问题的新算法

基于有限元的渐近展开式,导出了新的外推公式,它们更精确地逼近密网上的有限元解(而不是微分方程的解).提出了新的外推瀑布型多网格法(EXCMG),采用新外推公式及其二次插值提供密网上的好初值.数值实验表明,新方法有很高的精度和效率.最后在PC机上求解了大规模二维椭圆问题.     

A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media

In this paper, we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media. The flow equation is solved by the mixed finite element method, and the transport equation is approximated by a discontinuous Galerkin method. The scheme is continuous in time and a priori hp error estimates is presented.     

基于非均匀参数化的自由终端时间最优控制问题求解

针对自由终端时间最优控制问题,提出了一种基于非均匀控制向量参数化的数值解法.将控制时域离散化为不同长度的时间段,各时间段长度作为新的控制变量.通过引入标准化的时间变量,原问题转化为均匀参数化的固定终端时间最优控制问题.建立目标和约束函数的Hamilton函数,通过求解伴随方程获得目标和约束函数的梯度,采用序列二次规划(SQP)获得数值解.针对两个经典的化工过程自由终端时间最优控制问题进行仿真研究,验证了所提出算法的可行性和有效性.