A. Klar, Asymptotic--Induced Domain Decomposition Methods for Kinetic and Drift-- Diffusion Semiconductor Equations, to appear in SIAM Scientific Comput. (1996).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....method that couples the di#usion equation with the transport equation. Domain decomposition methods matching kinetic and hydrodynamic or di#usion models have received a lot of attention in the past 15 years. Some of the methods have been proposed in [3] 7] 14] 15] 18] 19] 25] [26], 28] 29] 33] 34] 37] Typically a domain decomposition is done by an iteration procedure at each time step in which the di#usion and the transport equation are solved alternately until convergence of the successive approximation is reached, or through an interface condition which ....

....3. Numerical Method. In this section we introduce a new (spatially discrete) numerical method for the coupling problem. In fact, this numerical scheme can be used for a discretization of the transport equation with di#erent order of magnitude in #, in the spirit of asymptotic preserving method [17, 20, 21, 22, 23, 24, 26, 30, 31, 32] that works uniformly with respect to the mean free path. However, this new asymptotic preserving spatial discretization method has not been reported in the literature. 3.1. Parity formulation. We explain the new scheme using the transport equation with isotropic scattering (2.4) It is based on ....

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A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-di#usion semiconductor equations, SIAM J. Sci. Comput., 19 (

.... at the kinetic level that will deliver the expected boundary conditions at the fluid level (see Markowich [M1] and Markowich, Ringhofer and Schmeiser [MRS] and references therein) Some recent numerical and analytical work on drift diffusion equations has been done by Golse and Klar[GK] and Klar [Kl]; however they use different boundary conditions from those proposed here. Some of the results worked out here have been presented in a short communication (see [CGL] For simplicity sake, we shall only consider electrons, since it is clear what modifications are needed when holes are to be taken ....

A. Klar, Asymptotic--Induced Domain Decomposition Methods for Kinetic and Drift-- Diffusion Semiconductor Equations, to appear in SIAM Scientific Comput. (1996).

....is solved in regions with strong kinetic effects and a macroscopic model is used whereever it is accurate enough. Then the two models need to be coupled at the kinetic fluid interface. Coupling conditions connecting the Boltzmann equation and the standard drift diffusion model have been derived in [10]. An extension of these results to other macroscopic models is carried out in this work. The basic assumption in the following is that elastic scattering of electrons with lattice defects is the dominating effect in macroscopic regions. This means that the energy gained or lost by an electron in ....

....from those for the SHE model by integrations. Interface conditions for fluid kinetic coupling are presented in section 5, where the fluid model can be the SHE model, the ET model or the DD model. This is a straightforward application of the boundary layer analysis and an adaption of the ideas in [10]. The formulation of the coupling conditions again requires the solution of certain half space problems which can be efficiently carried out by a recently developped iteration method [8] The method is outlined in section 6, and the resulting approximations are given. 2 The Hilbert Expansion To ....

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A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion semiconductor equations, preprint, Univ. Kaiserslautern, 1996.

.... F 0 i x (0; t) O(ff) Therefore condition (4.3) requires (4.9) while (4.4) requires (4.10) We now proceed to the derivation of second order transmission conditions. The forthcoming procedure is inspired from the now classical method of [2] already applied to semiconductors in [7] [13], to derive higher order boundary conditions and kinetic fluid coupling algorithms. We start again with the Hilbert expansion (away from the heterojunction) like in (2.5) but retaining an additional term: f ff i = F 0 i ff F 1 i Gamma i;x F 0 i x ff 2 f 2 i ff ....

A. KLAR: "Asymptotic induced domain decomposition methods for kinetic and drift diffusion semiconductor equation", preprint, Univ. Kaiserslautern, 1996.

....element methods are used for the macroscopic equations and particle methods for the kinetic equations. The coupling is performed by interface conditions. In the semiconductor field this approach has been carried out for a particular application in [4] Theoretical investigations can be found in [3]. The basic idea is that the kinetic model should be solved in the whole domain, however, the computationally much less expensive macroscopic model is sufficiently accurate in regions where scattering processes dominate. An alternative approach used in the semiconductor field is to extend the ....

....= e Gamma2M oe 0 2c 1 and an appropriately chosen value of c. 7 Implementation and Numerical Results Our main motivation for considering moment methods are problems with both kinetic and macroscopic regions. Recently, strategies for the numerical solution of such problems have been developped [3], 4] where interfaces between macroscopic and kinetic regions are introduced. Then the discretized drift diffusion equation is solved in the macroscopic regions and a particle method is used in the kinetic regions. A major issue in this approach is the derivation of appropriate coupling ....

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion semiconductor equations, preprint, Univ. Kaiserslautern, 1996.

....and expensive discretization depending on the mean free path has to be used for standard finite difference or particle methods due to the stiffness of the equations. This makes these schemes extremely time consuming. One way to handle the problem are domain decomposition techniques, see, e.g. [14, 16] for the semiconductor case. The basic idea is to use the computationally much cheaper limit equation where and whenever it gives a good approximation. In all other cases the kinetic equation is used. In this approach kinetic semiconductor and macroscopic equations are solved simultaneously on ....

....solution of the half space problem, we define q(x; v; t) x (0; v; t) x 2 Omega ; v Delta n 0: It is obviously not reasonable to determine the outgoing function by solving the halfspace problem. This would need too much computing time. Here a fast approximate scheme as in [7] or [14] is needed. For example, a first approximation is given by choosing simply an approximation x ( Gamma1; v; t) of the asymptotic value x ( Gamma1; v; t) of the halfspace problem as the outgoing function: q(x; v; t) x ( Gamma1; v; t) x 2 Omega ; v Delta n 0: 4.3) NUMERICAL ....

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A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, to appear in SIAM J. Sci. Comp.

.... condition for T 0 at x: T 0 (x) Gamma fflff(x)n Delta r x T 0 (x) T 0 b (1; x) O(ffl 2 ) 27) To obtain explicit conditions from (27) one has to solve the above half space problems in order to determine ff and T 0 b (1; x) This can be achieved using methods as in [10] 4] [5]. A simpler approximation is shown in the next section. 10 3.7 Approximate boundary conditions Here we explain a simple approach to obtain boundary conditions for the diffusion equation based on the assumption of the equality of half range fluxes. One uses the boundary condition given by (27) ....

....from (42) and (43) We remark, that solving the full half space problem, for example, by a standard discretization procedure would need a lot of computing time, in particular, since it has to be solved at each point of the interface. Instead, one uses an approximation procedure, as in [10] 4] [5], leading to easy to evaluate, however, accurate explicit coupling conditions. See Section 5, where the results of such a procedure are presented. One obtains in this way higher order accuracy than using the following simple conditions. 4.6 Approximate coupling conditions In the following simple ....

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A. Klar. Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations. to appear in SIAM J. Sci. Comp.

....First one has to choose suitable codes for Boltzmann and Euler equations. Second, the regions, where the fluid dynamic equations can be used, have to be determined. Once this is done the third problem is the matching of the Boltzmann code with the Euler or Navier Stokes solver. We refer to [18, 17, 5, 4, 21, 11, 13, 14, 28] for different domain decomposition approaches. In this paper Boltzmann and Euler equations are solved by particle methods. Numerical codes for the Boltzmann equation are usually based on particle methods, see [1, 2, 3, 23] Although for the Euler equations a variety of other methods exist, we did ....

A. Klar. Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations. to appear in SIAM J. Sci. Comp.

....= supp f . Applying the Hlawka Muck transformation means, to solve the nonlinear system of equations T 1 (P 1 ) R 1 T 2 (P 1 ; P 2 ) R 2 Delta Delta Delta T k (P 1 ; P k ) R k where R = R 1 ; R k ) 2 [0; 1] For a detailled discussion of the Hlawka Muck transformation see [5]) Example 18 Consider the density f(x 1 ; x 2 ) on [0; 1] 2 given by f(x 1 ; x 2 ) 4 5 (1 x 1 x 2 ) Then T 1 (x 1 ) x 1 Z 0 1 Z 0 4 5 (1 x 1 x 2 )dx 2 dx 1 = 4 5 x 1 1 5 x 2 1 and first we have to solve the equation T 1 (x 1 ) 4 5 x 1 1 5 x 2 1 = r 1 for given r ....

....The Mathematical Theory of Dilute Gases, Springer, 1994 [3] L. DE CLERCK, A Method for Exact Calculation of the Stardiscrepancy of Plane Sets Applied to the Sequences of Hammersley, Monatshefte Math. 101, 261 278, 1986 [4] L. DEVROYE, Non Uniform Random Variate Generation, Springer, 1986 [5] M.HACK, Construction of Particlesets to Simulate Rarefied Gases, AGTM Report Nr. 89, 1993 [6] H. NIEDERREITER, Random Number Generation and Quasi Monte Carlo Methods, SIAM, 1992 [7] H. NEUNZERT, Die Darstellung von Funktionen mehrerer Variablen durch Punktmengen, KFA Julich, Nr. 996, 1975 [8] ....

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A. KLAR, Asymptotic-Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations, preprint, Dept. of Mathematics, University Kaiserslautern, to appear in SIAM J. Sci. Comp.

....way the correct boundary value. For k = k(x; t) independent of v we get q(x; v; t) k(x; t) v Delta n 0. It is obviously not reasonable to determine the outgoing function by solving the halfspace problem. This would need too much computing time. Here a fast approximate scheme as in [11] or [17] is needed to determine the outgoing function. For example a first approximation is given by choosing simply an approximation x ( Gamma1; t) of the asymptotic value x ( Gamma1; t) of the halfspace problem as the outgoing function: q(x; v; t) x ( Gamma1; t) x 2 Omega ; v ....

.... Delta n 0: 3.7) The simplest approximation of x ( Gamma1; t)is given by equalizing the half range fluxes of the halfspace problem at 0 and 1: x ( Gamma1; t) R v Deltan 0 v Delta nk(x; v; t)dv R v Deltan 0 v Delta ndv : 3. 8) A more sophisticated approximation for q, see [17], is given by x ( Gamma1; t) R v Deltan 0 v Delta nk(x; v; t)dv R v Deltan 0 v Delta ndv (3.9) 1 D 1 4 Z v Deltan 0 (v Delta n) 2 k(x; v; t) Gamma R v Deltan 0 v Delta nk(x; v; t)dv R v Deltan 0 v Delta ndv # dv and q(x; v; t) x ( Gamma1; t) ....

A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, to appear in SIAM J. Sci. Comp.

....way the correct boundary value. For k = k(x, t) independent of v we get q(x, v, t) k(x, t) v n 0. It is obviously not reasonable to determine the outgoing function by solving the half space problem. This would need too much computing time. Here a fast approximate scheme as in [11] or [17] is needed to determine the outgoing function. For example, a first approximation is given by simply choosing an approximation # x ( #, t) of the asymptotic value # x ( #, t) of the half space problem as the outgoing function, q(x, v, t) # x ( #, t) x # ## , v n 0. 11) The ....

....= # x ( #, t) x # ## , v n 0. 11) The simplest approximation of # x ( #, t)is given by equalizing the half range fluxes of the half space problem at 0 and #, # x ( #, t) R v n 0 v nk(x, v, t)dv R v n 0 v ndv . 12) A more sophisticated approximation for q, see [17], is given by # x ( #, t) R v n 0 v nk(x, v, t)dv R v n 0 v ndv (13) 1 D 1 4# Z v n 0 (v n) 2 k(x, v, t) R v n 0 v nk(x, v, t)dv R v n 0 v ndv # dv and q(x, v, t) # x ( #, t) 1 4# Z w n 0 w n (w v) n s(v, w) w ....

A. KLAR, Asymptotic-induced domain decomposition methods for kinetic and drift di#usion semiconductor equations, SIAM J. Sci. Comput., to appear.

....halfspace problem yields coupling conditions for the domain decomposition problem. Such conditions and suitable approximations can be found in [3] We mention that for kinetic semiconductor equations a domain decomposition approach based on an analysis of kinetic interface layers can be found in [1]. 4. Numerical Results In this section we describe shortly the results of a simulation of an example appearing in glass manufacturing processes. A cylindrical solid is considered. The material is glass. One is interested in the development of the temperature of the piece of glass during the ....

Klar,A.: Asymptotic Induced Domain Decomposition Methods for Kinetic and Drift Diffusion Semiconductor Equations, to appear in SIAM J. Sci. Comput.

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A. Klar, Asymptotic--Induced Domain Decomposition Methods for Kinetic and Drift-- Diffusion Semiconductor Equations, to appear in SIAM Scientific Comput. (1996).

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A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-di#usion semiconductor equations, SIAM J. Sci. Comput. 19 (1998.

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