Numerical solutions of a two-phase membrane problem

In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.     

A reliable algorithm for solving fourth-order boundary value problems

In this paper, we present an efficient numerical algorithm for approximate solutions of fourth-order boundary values problems with twopoint boundary conditions. The Adomian decomposition method and a modified form of this method are applied to construct the numerical solution. The scheme is tested on one linear problem and two nonlinear problems. The obtained results demonstrate the applicability and efficiency of the proposed scheme.     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

Numerical Approximation Of The Phase-Field Transition System With Non-Homogeneous Cauchy-Neumann Boundary Conditions In Both Unknown Functions Via Fractional Steps Method

The paper concerns with the proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system endowed with non-homogeneous Cauchy-Neumann boundary conditions, in both unknown functions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic system. On the basis of this approach, a numerical algorithm in 2D case is introduced and an industrial implementation is made.     

Fractional Steps Scheme to Approximate the Phase-Field Transition System with Non-Homogeneous Cauchy–Neumann Boundary Conditions

The paper presents a proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system (a nonlinear parabolic system) with non-homogeneous Cauchy–Neumann boundary conditions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of a nonlinear parabolic system. On the basis of this approach, a numerical algorithm in the two dimensional case is introduced and an industrial implementation is made.     

热方程非线性边界条件问题的有限差分方法

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

A FINITE DIFFERENCE METHOD FOR THE HEAT EQUATION WITH A NONLINEAR BOUNDARY CONDITION

This article is concerned with the numerical solution to a parabolic equation with a kind of nonlinear boundary conditions. A difference scheme is constructed by the method of reduction of order on uniform mesh to solve the problem. It is proved that the difference scheme is uniquely solvable and uncon-ditionaUy convergent with the convergence order 2 in both space and time in an energy norm. An effective iterative algorithm is given and a numerical example is presented to demonstrate the theoretical results.     

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.     

An alternating method for an inverse Cauchy problem

In this paper, an iterative boundary element method based on our relaxed algorithm introduced in [8] is used to solve numerically a class of inverse boundary problems. A dynamical choice of the relaxation parameter is presented and a stopping criterion based on our theoretical results is used. The numerical results show that the algorithm produces a reasonably approximate solution and improves the rate of convergence of Kozlov's scheme [10]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this study, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model comprises a Cahn–Hilliard equation, Navier–Stokes equation, and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is employed to transform the governing system into an equivalent form, thereby allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at the fluid–solid interface. A pressure stabilization method is used to decouple the velocity and pressure computations. Some subtle implicit–explicit treatments are employed to deal with convention and stress terms. We establish a rigorous proof of the energy stability for the proposed time-marching scheme. A finite difference method based on staggered grids is then used to spatially discretize the constructed time-marching scheme. We also prove that the fully discrete scheme satisfies the discrete energy dissipation law. Our numerical results demonstrate the accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics based on a shear flow-driven droplet sliding case. Three-dimensional droplet spreading is also investigated based on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolution and it successfully reproduces the expected phenomena where an oil droplet contracts inward on a hydrophobic zone and then spreads outward rapidly on a hydrophilic zone.     

An iterative method for the reconstruction of a stationary flow

In this article, an iterative algorithm based on the Landweber‐Fridman method in combination with the boundary element method is developed for solving a Cauchy problem in linear hydrostatics Stokes flow of a slow viscous fluid. This is an iteration scheme where mixed well‐posed problems for the stationary generalized Stokes system and its adjoint are solved in an alternating way. A convergence proof of this procedure is included and an efficient stopping criterion is employed. The numerical results confirm that the iterative method produces a convergent and stable numerical solution. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

An Unconditionally Stable Scheme for the Numerical Solution of Finite Difference Forms of Poisson's Equation

A procedure for an unconditionally stable non-iterative step-by-stepscheme for the solution of second order ordinary boundary valuedifference equations is extended to Poisson's equation withDirichlet boundary conditions. It is shown that owing to theparticular method of solution rounding error from a particulargrid point has practically no effect on other grid points, andit is absolutely stable for all finite difference forms of Poisson'sequation. The extension to other two dimensional partial differentialequations is discussed. It is also shown that the scheme isparticularly valuable in problems in which a high degree ofaccuracy is demanded and it is recommended for numerical weatherexperiments.     

A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.     

A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs

In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations.     

Application of numerical schemes with singling out the boundary layer for the computation of turbulent flows using eddy-resolving approaches on unstructured grids

The use of eddy-resolving approaches to solving problems on arbitrary unstructured grids is investigated. The applications of such approaches requires the use of low dissipation numerical schemes, which can lead to numerical oscillations of the solution on unstructured grids. Numerical oscillations typically occur in domains with large gradients of velocities, in particular, in the near-wall layer. It is proposed to single out the boundary layer and use a numerical scheme with increased numerical dissipation in it. The algorithm for singling out the boundary layer uses a switching function to change the parameters of the numerical scheme. This algorithm is formulated based on the BCD scheme from the family NVD. Its validity and advantages are investigated using the zonal RANS–LES approach for solving some problems of turbulent flow of incompressible fluids.     

A non-overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two-dimension

We apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two-dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non-overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.     

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.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Time step selection for the numerical solution of boundary value problems for parabolic equations

An algorithm is proposed for selecting a time step for the numerical solution of boundary value problems for parabolic equations. The solution is found by applying unconditionally stable implicit schemes, while the time step is selected using the solution produced by an explicit scheme. Explicit computational formulas are based on truncation error estimation at a new time level. Numerical results for a model parabolic boundary value problem are presented, which demonstrate the performance of the time step selection algorithm.