Dirichlet boundary condition schemes

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. 

Now we return to the study of the normalized diffusion fluxes for boundary value problems with Dirichlet boundary condition. In Section II.D the e-uniformly convergent finite difference schemes (2.74), (2.76) and (2.67), (2.72) were constructed for the Dirichlet problems (2.12), (2.13) and (2.14), (2.15), respectively. For these problems, we now construct and analyze the approximations of the normalized diffusion fluxes. We consider the normalized diffusion fluxes for problem (2.14), (2.15) in the form... 

The exact form of the matrices Qi and Q2 depends on the type of partial differential equations that make up the system of equations describing the process units, i.e., parabolic, elliptic, or hyperbolic, as well as the type of applicable boundary conditions, i.e., Dirichlet, Neuman, or Robin boundary conditions. The matrix G contains the source terms as well as any nonlinear terms present in F. It may or may not be averaged over two successive times corresponding to the indices n and n + 1. The numerical scheme solves for the unknown dependent variables at time t = (n + l)At and all spatial positions on the grid in terms of the values of the dependent variables at time t = nAt and all spatial positions. Boundary conditions of the Neuman or Robin type, which involve evaluation of the flux at the boundary, require additional consideration. The approximation of the derivative at the boundary by a finite difference introduces an error into the calculation at the boundary that propagates inward from the boundary as the computation steps forward in time. This requires a modification of the algorithm to compensate for this effect. 

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. 

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