Uniform approximation finite difference schemes

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 above examples illustrate the fact that, in the case of singularly perturbed elliptic and parabolic equations, the use of classical finite difference schemes does not enable us to find the approximate solutions and the normalized diffusion fluxes with e-uniform accuracy. To find approximate solutions and normalized fluxes that converge e-uniformly, it is necessary to develop special numerical methods, in particular, special finite difference schemes. 

In Section I we obtained an intuitive impression of the numerical problems appearing when one uses classical finite difference schemes to solve singularly perturbed boundary value problems for ordinary differential equations. In this section, for a parabolic equation, we study the nature of the errors in the approximate solution and the normalized diffusion flux for a classical finite difference scheme on a uniform grid and also on a grid with an arbitrary distribution of nodes in space. We find distributions of the grid nodes for which the solution of the finite difference scheme approximates the exact one uniformly with respect to the parameter. The efficiency of the new scheme for finding the approximate solution will be demonstrated with numerical examples. 

Recall that we are interested in the behavior of the error in the approximate solution for various values of the parameter e. To compute the solution of the boundary value problem (2.16), (2.18), we use a classical finite difference scheme. We now describe this scheme. On the set G the uniform rectangular grid... 

Thus, by numerical experiments we verify that the approximate solution of the Dirichlet problem (2.16), found by the classical finite difference scheme (2.28), (2.27), and the computed normalized diffusion fiux converge for N, Nq respectively, to the solution of the boundary value problem and the real normalized diffusion flux for fixed e. However, we can also see that they do not converge e-uniformly. The solution of the grid problem approaches the solution of the boundary value problem uniformly in e qualitatively well. The normalized flux computed according to the solution of the difference problem does not approach e-uniformly the real normalized flux (i.e., the flux related to the solution of the boundary value problem) even qualitatively. Nevertheless, if the solution of the singularly perturbed boundary value problem is smooth and e-uniformly bounded, the approximate solution and the computed normalized flux converge e-uniformly (when N, Nq oo) to the exact solution and flux. 

Thus, we have constructed the grid on which the solution of the finite difference scheme (2.46), (2.59) converges e-uniformly. For the approximate solution the estimate (2.61) for < i(2.43) and the esti-... 

For this problem, we use a finite difference scheme on a grid with an arbitrary distribution of N -I-1 nodes and with fixed N. We show that the nodes of this grid can be distributed in such a way that the error in the approximate solution tends to zero e-uniformly when N increases. 

Thus, we see that the newly constructed finite difference schemes are indeed effective and that they allow us to approximate the solution and the normalized diffusion fluxes g-uniformly for both Dirichlet and Neumann boundary value problems with singular perturbations. 

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. 

Note that, in the case of problem (5.3), an interior layer appears as the parameter e tends to zero. Boundary layers do not appear. In the problem under consideration the parameter e is characterized by some value (). It is required to decide whether eg is a small value, and, therefore, the special finite difference scheme should be applied, or the value eg is not too small, and, therefore, we can use a finite difference scheme on a uniform grid. The answer depends on N, that is, the number of nodes of the space grid. According to the results in Section IV (see Section FV.C), if Ne l, we should expect large errors to occur in the case of uniform grids. If Ne 1, then good accuracy of the approximate solution on uniform grids can be expected. 

In Section V.B.l, we describe heat transfer in hot die-forming and give a mathematical formulation of the problem. In Section V.B.2, we construct a special finite difference scheme. This scheme allows us to approximate e-uniformly both the problem solution and the amount of heat that passes through the die-body interface. In Section V.B.3 we numerically analyze the problem of heat exchange between the die and the hot body with technological parameters typical for die-forming. 

We describe a finite difference scheme, the solution of which approximates the solution of the boundary value problem -uniformly on the whole grid set in G. 

Note that the approximate solution j,)(x, /), which is obtained with the classical finite difference scheme on the grid with an arbitrary distribution of nodes, does not converge e-uniformly. This is due to the unsatisfactory approximation of the function U x, t) by the function z x, t) in the neighborhood of the boundary layer. 

Multistep methods can be used for integration of the system of equations in time, such as Runge-Kutta, and the central finite difference scheme for the spatial approximation of first-order derivatives for each grid point (i,j, k) as shown in Figure 6.4 for uniform meshes by simplicity... 

The stability analysis of the dispersion-correction numerical scheme shows that the stability criterion is < 0.67 and —0.125 < 7 < 0.083, where C/ is the Courant number. The lower limit of 7 imposes a limitation on the ratio of grid size to water depth as 1.27. This means that the uniform grid size greater than approximately 1.27 times of a local water depth must be employed for the stability of the present dispersion-correction scheme. To satisfy the stability criterion of the dispersion-correction finite-difference model, the range of applicable water depth is limited if a uniform grid size is used for varying water depth. For practical purposes, this stability criterion can be solved by imposing intentionally a limitation on the... 

The solution of problem was carried out by the numerical finite-difference method briefly presented in [1], The calculating domain was covered by uniform grid with 76x46 mesh points. A presence of two-order elliptic operators in all equations of the mathematical model allow us to approximate each equation by implicit iterative finite-difference splitting-up scheme with stabilizing correction. The scheme in general form looks as follows ... 

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