Diffusion flux classical finite difference schemes

Numerical Experiments with the Classical Finite Difference Scheme Principles for Constructing Special Finite Difference Schemes Special Finite Difference Schemes for Problems (2.12), 2.13) and (2.14), (2.15) Numerical Experiments with the Special Difference Scheme Numerical Solutions of the Diffusion Equation with Prescribed Diffusion Fluxes on the Boundary... 

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. 

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. 

In this section, we consider singularly perturbed diffusion equations when the diffusion flux is given on the domain boundary. We show (see Section III.B) that the error in the approximate solution obtained by a classical finite difference scheme, depending on the parameter value, can be many times greater than the magnitude of the exact solution. For the boundary value problems under study we construct special finite difference schemes (see Sections III.C and III.D), which allow us to find the solution and diffusion flux. The errors in the approximate solution for these schemes and the computed diffusion flux are independent of the parameter value and depend only on the number of nodes in the grid. 

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