Analysis of a new implicit solver for a semiconductor model

Convergence rates of the NLS method with and without acceleration.
The relative residual versus iteration for the base NLS method (blue curve) and its accelerated variants with a Knudsen number of 5E-4. Anderson acceleration (AA) and drift-diffusion synthetic acceleration (DDSA) greatly improve efficiency when the Knudsen number is small. Using AA and DDSA together (red curve) reduce the iteration count by three orders of magnitude in this example.


We developed a new implicit solver for the simulation of a simplified Boltzmann-Poisson system which is used to model electron transport in semiconductor devices. The new method, called nonlinear sweeping (NLS), improves upon recently developed iterative methods for the same system by reducing the work per iteration by around a factor of three in our experiments. We derived analytic estimates for the convergence rate of the new method which are verified numerically, and successfully employed Anderson acceleration and low order models to improve the convergence rate.

Significance and Impact

Implicit methods robustly handle the wide range of scales featured in semiconductor devices. This robustness comes at a cost since implicit methods require solving a large nonlinear system for each timestep. By developing more efficient solvers, we can both reduce the overall runtime of simulations and increase temporal accuracy by taking smaller timesteps.

Research Details

The basis of the new solver is a fixed-point formulation which is posed on a lower dimensional space than that of the Boltzmann-Poisson solution. This enables the storage of more fixed-point evaluations which in turn allows us to employ Anderson acceleration (AA) to reduce the total number of iterations.

If the solution is in the drift-diffusion regime rather than the kinetic regime, the spectral radius of the fixed-point method approaches one. As a result, the iterative solver requires many iterations whether or not AA is applied. Fortunately, the equations for the drift-diffusion limit are inexpensive to solve, and they can be used as a low order correction to the NLS iterates. This correction, which we call drift-diffusion synthetic acceleration (DDSA), reduces the total number of iterations. We show that AA and DDSA can be used simultaneously for even faster convergence (see image).

Last Updated: January 14, 2021 - 7:24 pm