Boundary value problems finite difference schemes

The new conjugation boundary-value problem (3.87) that consist of (3.116) and of the equations and boundary conditions from the right column of (3.85) and (3.86) was also solved numerically by the same finite-difference scheme. Again, the problem led to an equivalent transcendental equation... 

In section 3.1.4, an analytical series solution using the matrizant was developed for the case where the coefficient matrix is a function of the independent variable. This methodology provides series solutions for Boundary value problems without resorting to any conventional series solution technique. In section 3.1.5, finite difference solutions were obtained for linear Boundary value problems as a function of parameters in the system. The solution obtained is equivalent to the analytical solution because the parameters are explicitly seen in the solution. One has to be careful when solving convective diffusion equations, since the central difference scheme for the first derivative produces numerical oscillations. 

The finite difference scheme is one of the direct methods to solve the initial and boundary value problem of a diffiision controlled adsorption model. To minimize the numerical efforts in solving the present complex problem with several independent physical parameters, it is efficient to use dimensionless variables. The following transforms are used for the present transport problem, which was defined in chapter4. 

The numerical solution of Eq. (54) as an initial-boundary-value problem, specified to the spatial relaxation problem in uniform electric fields, can be obtained (Sigeneger and Winkler, 1996) by using a finite-difference approach according to the well-known Crank Nicholson scheme for parabolic equations. 

The material balance condition must apply at the nodes of the network such that no adsorption or reaction occurs there, that is, no accumulation at the nodes. For each pore a boundary value problem has to be solved. Adopting the methods of Rieckmann and Keil, it was necessary to discretize each pore using a finite difference scheme. This led to a very large system of nonlinear equations, which were solved using the Schur complement method, described by Rieckmann and Keil [44]. 

For the boundary value problem (1.13) we obtain a difference scheme (i.e., the combination of the grid sets D, depending on the value of N, and the system of difference equations defined on these sets). The system of finite difference equations is as follows ... 

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... 

It is known (see, e.g., [1]) from the theory of finite difference schemes that for N, Nq °o, and for a fixed value of the parameter e, the solution of the finite difference scheme (2.28), (2.27) [or, as is usually said in short, the finite difference scheme (2.28), (2.27)] converges to the solution of the boundary value problem. In the case of sufficiently smooth solutions for problem (2.16), the following estimate is fulfilled... 

They are calculated for the finite difference scheme (2.34), (2.27) corresponding to the boundary value problem (2.26) for different values of e and N, with Nq = N. For values of e smaller than 2 the errors io24.io24( > become equal to zero up to the computer accuracy. From... 

In this section, using an example of a boundary value problem for a singularly perturbed ordinary differential equation, we discuss some principles for constructing special finite difference schemes. In Section II.D, these principles will be applied to the construction of special schemes for singularly perturbed equations of the parabolic type. 

In the case of the boundary value problem (2.37) the solution of the finite difference scheme (2.46), (2.44) converges c-uniformly for Af— if the estimate... 

From the results in Section I we know that even when the coefficient c(x) and the right-hand side f(x) are constant, the solution of the finite difference scheme on a uniform grid [namely, scheme (1.17), (1.18)] does not converge e-uniformly to the solution of the boundary value problem (1.16). Therefore, we will try to find a distribution of the nodes of the grid D/,(2.44) so that inequality (2.53) is valid for the function ijix). 

From the results given in Tables XIV-XVI it follows that the error in the solution of the finite difference scheme (2.78), (2.77) does not exceed the sum of the errors in the solutions of the finite difference schemes (2.79), (2.77) (2.80), (2.77), and (2.81), (2.77). Therefore, the solution of the special scheme (2.78), (2.77) converges e-uniformly to the solution of the boundary value problem (2.16). Thus, numerical experiments illustrate the efficiency of the constructed finite difference scheme. 

In Section II (see Sections II.B, II.D) we considered finite difference schemes in the case when the unknown function takes given values on the boundary. The boundary value problem for the singularly perturbed parabolic equation on a rectangle, that is, a two-dimensional problem, is described by Eqs. (2.12), while the boundary value problem on a segment, that is, a one-dimensional problem, is described by equations (2.14). In Section II.B classical finite difference schemes were analyzed. It was shown that the error in the approximate solution, as a function of the perturbation parameter, is comparable to the required solution for any fine grid. For the above mentioned problems special finite difference schemes were constructed. The error in the approximate solution obtained by the new scheme does not depend on the parameter value and tends to zero as the number of grid nodes increases. 

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. 

We say that the finite difference scheme solves the Neumann problem (3.5), if the grid solution converges to the solution of the boundary value problem, and for problem (3.6), if, in addition, the grid solution allows... 

From the theory of finite difference schemes it is known that, for a fixed value of the parameter, the solution of the difference scheme (3.17), (3.16) converges to the solution of the boundary value problem... 

In the case of Dirichlet problem (2.37) considered in Section 2.3, the unknown solution and the solution of the finite difference scheme take the same given values on the boundary. Thus, in the Dirichlet problem no error appears on the boundary. The error is generated only by an error in the approximation of the differential equation by the difference equation. 

Thus, for the Neumann boundary value problem (3.23), a special finite difference scheme has been constructed. Its solution z(x),xE.D,, and the function xED allow us to approximate the solution of 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... 

Now we study the efficiency of the special finite difference scheme in the case of the Neumann boundary value problem (3.7), (3.9). On the set G(3 g) we introduce the special grid... 

Finally, we have also analyzed the convergence of the discrete fluxes for the Dirichlet boundary value problem (2.16). Here we used the special finite difference scheme. The grid fluxes were considered separately for each component in the representation (2.24). In Tables XXX and XXXI, we give the errors Q(e,N),Q(N) computed from the solutions of the difference schemes (2.79), (2.77) and (2.80), (2.77) corresponding to problems (2.20) and (2.25). Table XXXII shows the... 

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. 

---