Neumann boundary conditions schemes

Using the example of a boundary value problem for a singularly perturbed ordinary differential equation with Neumann boundary conditions, we discuss some principles of constructing special finite difference schemes. These principles will be used in Section III.D to construct special finite difference schemes for singularly perturbed equations of the parabolic type. 

The above scheme has first been applied by Sadygov and Yarkony in a 2D study on HeH2 and more recently by Abrol and Kuppermann in a 3D study of H3. In the latter work the whole domain of nuclear configurations relevant for reactive scattering has been treated and various boundary conditions have been compared. In this way the smallness of the residual (transverse) couplings in the diabatie basis could be established and a combination of Neumann and Dirichlet boundary conditions be shown to be optimum for this purpose. 

The treatment of boundary conditions can be incorporated in the EMM scheme easily. Periodic boundary conditions as well as Dirichlet, Neumann, and mixed conditions can be accounted for. The EMM approach has been shown to be more efficient than the Ewald summation method (see the next... 

Note, in particular, one feature in the behavior of the approximate solutions of boundary value problems with a concentrated source. It follows from the results of Section II that, in the case of the Dirichlet problem, the solution of the classical finite difference scheme is bounded 6-uniformly, and even though the grid solution does not converge s-uniformly, it approximates qualitatively the exact solution e-uniformly. But now, in the case of a Dirichlet boundary value problem with a concentrated source, the behavior of the approximate solution differs sharply from what was said above. For example, in the case of a Dirichlet boundary value problem with a concentrated source acting in the middle of the segment D = [-1,1], when the equation coefficients are constant, the right-hand side and the boundary function are equal to zero, the solution is equivalent to the solution of the problem on [0,1] with a Neumann condition at x = 0. It follows that the solution of the classical finite difference scheme for the Dirichlet problem with a concentrated source is not bounded e-uniformly, and that it does not approximate the exact solution uniformly in e, even qualitatively. 

Numerical stability. The stability of a scheme can be determined by a relatively simple von Neumann stability test (Carnahan, Luther, and Wilkes, 1969 Richtmyer and Morton, 1957). This test, we emphasize, is only qualitatively accurate, since it does not account for the detailed effects of boundary and initial conditions, and for the role of heterogeneities when variable coefficients are present in the given equation. Note that a stable finite difference scheme does not necessarily converge to solutions of the PDE even if Ar, At are vanishingly small. This subject is treated in advanced courses. 

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