Normalized diffusion flux schemes

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. 

The behavior of the errors in the normalized flux is more complicated in the case of problem (2.25). It is similar to the behavior of the errors in the normalized diffusion flux in the case of boundary value problem (1.16) for an ordinary differential equation, when the difference scheme... 

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 Section III we will see that the new finite difference schemes allow us to find the normalized diffusion fluxes with an e-uniform accuracy. 

In the case of problem (3.6), we are interested in the approximation of the normalized diffusion flux P(x, t). It is known from the theory of finite difference schemes that, for a fixed value of the parameter e and N, the discrete normalized diffusion flux P 3 ig)(x, t) converges to... 

In Table XXIII we give the errors in the normalized diffusion fluxes Qi024,i024(< ) and Qio24,io24( ) for problem (3.17) computed with the finite difference scheme (3.21), (3.16) for various values of the parameter s and the number of nodes N with = N. One can see from the table that, as N increases, the error G1024,1024( 5 tends to zero (while the... 

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... 

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. 

Thus, for singularly perturbed diffusion equations with mixed boundary conditions and concentrated sources, we have constructed new finite difference schemes that enable us to find solutions and normalized diffusion fluxes to an e-uniform accuracy. 

The Tl-variables represent the generalized diffusion conductance and are related to the diffusive fluxes through the grid cell surfaces. In order to approximate these terms the gradients of the transported properties and the diffusion coefficients T are required. The property gradients are normally approximated by the central difference scheme. In a uniform grid the diffusion coefficients are obtained by linear interpolation from the node values (i.e., using arithmetic mean values) ... 

The realizable schemes developed in the preceding section are applied to the moment distributions shown in Eigure 8.2. The normalized moments are defined by dividing their values by the initial values at x = 0 (1, 10, 200, 6000, 240 000). Eor the cases with constant diffusivity, we set M = 20, while for those with Stokes-Einstein diffusivity we use 0 = 1, and set M = 100 to improve the representation of the volume-dependent diffusive fluxes. 

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