Diffusion flux schemes

Based on a higher order scheme (indicated by superscript H), compute anti-diffusive fluxes ... 

When the transport equation for c is solved with a discretization scheme such as upwind, artificial diffusive fluxes are induced, effecting a smearing of the interface. When these diffusive fluxes are significant on the time-scale of the simulation, the information on the location of different fluid volumes is lost. The use of higher order discretization schemes is usually not sufficient to reduce the artificial smearing of the interface to a tolerable level. Hence special methods are used to guarantee that a physically reasonable distribution of the volume fraction field is maintained. 

The mathematical models corresponding to the schemes in Fig. 7.4 contain the diffusive fluxes between the phases, which depend both on fluid dynamics (through... 

According to the scheme of the compartments in the liquid membrane system in Figure 13.5b, all local diffusion fluxes of M species from ktok+1 compartment can be defined by a phenomenological Equation 13.41 corresponding with the first Pick s law for diffusion ... 

One such approach is based on the concepts of non-linear flux limiters introduced by van Leer [193] and Boris and Book [13]. The work of Boris and Book [13] and Zalesak [213] determine the basis for a group of methods called flux correction transport (FCT) schemes. The schemes of Smolarkiewicz [175] is representative for this group. In the FCT schemes a first order accurate monotone scheme is converted to a high resolution scheme by adding limited amounts of an anti-diffusive flux. The work of van Leer [193, 195], on the other hand, represents an extension of the ideas of Gudunov [64] to higher order 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. 

The realizable scheme for solving Eq. (8.125) is the same as in Eq. (8.89) (without diffusive fluxes), but with a slightly different definition for the numerical fluxes ... 

In general, boundary conditions are difficult to specify and oftentimes difficult to incorporate into the numerical scheme. Typical boundary conditions used are given in [92, 97, 129, 130, 136]. Boundary conditions for the mass continuity Eq. (22) specify a zero electron density at the wall, or an electron flux equal to the local thermal flux multiplied by an electron reflection coefficient. The ion diffusion flux is set to... 

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. 

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 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. 

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... 

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. 

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