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A new scaling method for non-symmetric matrices is presented. The scaling is based on purely mathematical considerations. It is highly adapted to the demands of incomplete LU factorization preconditioners with numerical dropping strategy. These preconditioners are the best state-of-the-art instruments for solving ill-conditioned linear systems with very large numbers of equations. The new scaling method causes a considerable improvement in the capabilities of these preconditioners, both in execution speed and robustness.

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Pseudomorphic submicron High Electron Mobility Transistors (HEMT) have conquered a broad field of application because of their highfrequency performance. The DC characteristics of a 0.23pm gate length transistor have been calculated by our recently developed device simulator using a hydrodynamic model (HD) which accounts for carrier heating effects in the short channel region. A block iterative scheme combined with a full Newton method is applied to improve the convergence performance, robustness anc! stability of the HD model. Furthermore, an extended Scharfetter-Gummel scheme was usedto account forthe spatialvariationof material properties such as band edge energy and effective density of states. HETEROJUNCTIONSThe multi-layer structure of HEMTs requires a special treatment of heterojunctions. As a result of an idealization process an abrupt variation of parameters on interfaces is assumed and is modeled by special interface conditions. This leads to a better condition of the linearized systemcomparedto amethodwhich simply reduces grid spacing in the vicinity of heterojunctions. The HD equations are discretizedby anextendedscharfetter-gumel scheme i [ 1 1 [ 2 ] ) which additionally accounts for a spatial variation of material properties such as effective density of states and band edge energy. ITERATION SCHEMEThe well known moderate convergence performance of the HD model has been improvedby a block iterative scheme. Considericg the partial differential equations div(5 grad$) = -pan aP at div J, = +q-+ qRzJJ div Jp = -9--qReff at (0-7803-3062-5) 590 where ( 6 ) (E) Jp = q p P p grad ($ -4 ) ptpkNi, grad (7) and 5 kTn 5 kTp 2q 2q S, = -rc,gradTn --J,, S, = -~,gradT, + -Jp,the set of Possion's equation and continuity equations (1) (2) (3) is solved alternatelywith the set of continuity equations and energy balance equations (2) (3) (4) ( 5 ) . Each set is solved iteratively using Newton's method until the norm of the updates remains under a certain value. At last, for the full equation set Newton's method ([3]) is invoked to obtain the desired final accuracy. Therefore, this method combines both a block iterative scheme for effectiveness and a full Newton method for accuracy. After about 60 iterations the norm has decrceased by a factor of M and the scheme can be switched to full Newton. It is remarkable that, even for high bias conditions convergence is achieved from an initial equilibrium state without stepping the applied voltages, whereas with a full-Newton only strategy convergence is possible only for a small initial bias.. RESULTSThe simulated characteristics obtained by the HD model are fairly similar to the results of a drift-diffusion model, even though the velocities along the channel differ significantly (Fig. 1, the gate starts at x = 0.585pm and ends at x = 0.815pm). This is explained by the fact that the channel charge is nearly identical for both models, since it is mainly determined by the gate voltage, whereas the carrier concentrationvaries along the channel according tothe carrier velocity. F...

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