Finite-difference Solution Of Equations Research Articles

An enthalpy method for modeling dendritic growth in a binary alloy

An enthalpy fixed grid method is developed for modeling dendritic growth in an under-cooled binary alloy. The proposed numerical method couples explicit finite difference solutions of equations expressing the conservation of enthalpy and solute to an iterative procedure that enforces node by node consistency between enthalpy, solute, liquid fraction, and interface under-cooling. Calculations made with the scheme, are consistent with previously reported work, agree well with limit analytical solutions, approach the correct steady-state tip operating conditions, show grid size independence, are relatively free of grid anisotropy, and can be obtained with a low CPU cost.

On the approximation of the solutions of stochastic equations with θ-integrals

The paper investigates the problem of approximation of stochastic θ-integrals and the solutions of stochastic differential equations. The complete classification of the methods of approximation of stochastic θ-integrals in the convolution algebra is proposed. It is proved that the solutions of stochastic integral equations with θ-integral can be approximated by the solutions of finite-difference equations with averaging.

A comment on the solution of finite-difference equations

A comment on the solution of finite-difference equations

The solution of finite-difference equations

In every textbook which deals with second order linear finite-difference equations, methods will be found for solving such equations when the coefficients are constants. In the author’s experience it is unknown for textbooks to treat second order linear difference equations with variable coefficients. Yet such equations certainly occur in practical applications. It is the present intention to remedy this defect in the literature. It will be seen that, by the methods discussed here, we can always reduce the solution of a linear second order difference equation to that of a first order equation.

Finite difference approximations to time-dependent multigroup transport equations in slab geometry

Abstract In this article, we consider a monotone iterative scheme to find the solution of a system of finite difference equations approximation to the time-dependent multigroup transport equations in slab geometry. The iterative scheme, coupled with the upper and lower solutions generates two sequences which converge monotonically to the solution of the finite difference equations. It is shown that this monotone convergence leads to the convergence of the solutions of finite difference equations to the solutions of the continuous equations as the mesh length of the space, angle and time variable approaches zero. Numerical results are given for the time-dependent two group transport equations.

Consistent finite difference modelling of Maxwell's equations with lossy symmetrical condensed TLM node

AbstractA propagator integral interpretation for general TLM processes is presented and applied to discretized Maxwell's equations. The approach provides exact algebraic solutions of finite difference equations along a quite universal scheme.Specifically, Johns's process based on the symmetrical condensed node is reconstructed and generalized as a clear‐cut FD algorithm. The fields computed are exact time‐domain solutions of the FD equations, provided that total voltages and currents are evaluated and stub quantities are not externally excited. Losses are very naturally incorporated into the TLM algorithm.The structural similarity of the TLM process (properly operated) with the propagator integral appears just as fundamental as general and should have further applications.

Simultaneously Developing, Laminar Flow, Forced Convection in the Entrance Region of Parallel Plates

This paper reports on a numerical study of simultaneously developing flow in the entrance region of a cascade of thin parallel horizontal plates with a uniform flow at upstream infinity. Results obtained by ADI solution of finite difference equations for the Navier-Stokes and energy partial differential equations are extrapolated to zero mesh size with extended Richardson extrapolation. Nusselt numbers, incremental heat transfer number, and thermal entrance length are presented for Reynolds numbers between 40 and 2000; Pr = 0.2, 0.7, 2, 7, 10 and 100; and constant wall temperature and heat flux boundary conditions.

Continuous dependence and differentiation of solutions of finite difference equations

Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m + n) = f(m, u(m), u(m + 1), …, u(m + n − 1)), m ∈ ℤ.

Generation of solution-adaptive computational grids using optimization

A new method for generating solution-adaptive computational grids is presented that builds on the requirement that the adapted grid retains maximum possible smoothness and local orthogonality. The approach taken is one of nonlinear optimization, where an objective function combining measures of grid smoothness, local orthogonality, and cell volume control is minimized using a fast iterative scheme. The method is multidimensional by construction, and accepts any arbitrary grid as input, even a grid that is initially overlapped. Several applications to published test problems allow comparisons with some of the existing adaptive grid generation methods. An example of dynamic adaptation to the solution of finite difference equations is also given that demonstrates the error reduction capabilities of the new adaptive grid generation and optimization method, and suggests further applications to practical engineering problems.

Propagation and stability of wavelike solutions of finite difference equations with variable coefficients

An asymptotic approach is used to analyze the propagation and dissipation of wavelike solutions to finite difference equations. It is shown that to first order the amplitude of a wave is convected at the local group velocity and varies in magnitude if the coefficients of the finite difference equation vary. Asymptotic boundary conditions coupling the amplitudes of different wave solutions are also derived. Equations are derived for the motion of wavepackets and their interaction at boundaries. Comparison with numerical experiments demonstrates the success and limitations of the asymptotic approach. Finally, a global stability analysis is developed.

A Technique for Calculating Shunt Leakage and Cell Currents in Bipolar Stacks Having Divided or Undivided Cells

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Reassessment of solutions of finite difference equations in fluid mechanics

Reassessment of solutions of finite difference equations in fluid mechanics

Reassessment of solutions of finite difference equations in fluid mechanics

Reassessment of solutions of finite difference equations in fluid mechanics

Assessment of solutions of finite difference equations in fluid mechanics

Assessment of solutions of finite difference equations in fluid mechanics

The Design of Axisymmetric Ducts for Incompressible Flow

SummaryAn algorithm is described for solving the partial differential equations governing inviscid, incompressible, irrotational and axisymmetric flow in order to obtain duct wall shapes from prescribed wall velocity distributions. Equations of motion are formulated with distances from and along the axis of symmetry as dependent variables. A velocity potential and Stokes stream function are employed as independent variables. A numerical method based on the iterative solution of finite-difference equations on a non-uniform mesh is presented. The residuals obtained from calculations with differing mesh sizes, for the two examples described, indicate that the proposed method gives satisfactory solutions to the specified design problem.

Periodic solutions of finite difference equations

Periodic solutions of finite difference equations

Numerical investigations of the thermally stressed state of hollow eccentric cylinders

In the present article, an investigation is made of the thermally stressed state of a thick-walled cylinder with an eccentrically arranged opening, with a nonaxisymmetric distribution of the temperatures. The article gives the determination of the temperature stresses in these cylinders as a result of the solution of finite-difference equations, obtained from approximate expressions for the energy of the deformation, set up by a variational method [i]. The cross section of the cylinder under investigation was represented in the form of an eccentric ring (Fig. I).

A Generalization of the Additive Correction Methods for the Iterative Solution of Matrix Equations

A general formulation of the additive correction methods of Poussin [4] and Watts [7] is presented. The methods are applied to the solution of finite difference equations resulting from elliptic and parabolic partial differential equations. A new method is developed for anisotropic and heterogeneous problems. For such difficult problems the method presented here is comparable with Stone’s strongly implicit method, while all other methods tested require much greater computational effort. The correction approach discussed here can be easily applied with any iterative method.

On matrix methods for the solution of partial differential equations

Bickley and McNamee (1) describe techniques for obtaining the solution of finite difference equations, arising from partial differential equations, making extensive use of matrix methods. In all cases solutions are obtained by solving algebraic equations as distinct from differential equations. For example, in order to solvethe second space derivative is replaced by finite differences and the time derivative is replaced either by substituting the backward finite difference form or by using the Laplace transformation.

Qualitative Investigation of the Trajectories of Difference Equations Near a Fixed Point

T SOLUTION of the problem of the stability of intermittent control systems plays an important role in modern technology (l). The works of Tsypkin (2), Perron (3), Bellman (4), Skalkina (5) and KovaF (6) have been devoted to an investigation of the stability of the solutions of finite difference equations, one of the basic mathematical devices used to compute intermittent control systems. The qualitative investigation of systems described by nonlinear difference equations is a problem that is interesting, but which has been only slightly developed. Let us examine a system of difference equations