An implicit LU‐SGS spectral volume method for the moment models in device simulations: Formulation in 1D and application to a p‐multigrid algorithm

Abstract

SUMMARYA high-order spectral volume (SV) method was implemented for solving the steady-state moment models, such as the hydrodynamic (HD) models and the energy transport (ET) models for semiconductor device simulations (in 1D). The first derivative inviscid/convective fluxes are handled using an approximate Riemann flux and the second derivative diffusive fluxes are discretized using the local discontinuous Galerkin formulation (LDG). The LDG method is also used for discretizing the potential equation. An implicit pre-conditioned LU-SGS p-multigrid method developed for the SV Navier-Stokes (NS) solver by Kannan and Wang is adopted here for time marching. The entire formulation is compact and hence can be easily parallelized. A n + -n-n + diode was assumed for simulation purposes and the results are compared with the existing discontinuous Galerkin simulation results. In general, the numerical results are very promising and indicate that the approach has a great potential for higher-dimensional device problems.

High-Order and High Accurate CFD Methods and Their Applications for Complex Grid Problems

“…The results indicate that GMRES can considerably improve the convergence rate of WCNSs for the flows they simulated. Implicit LU-SGS is also successfully used for SD and SV, such as the methods in [107,108]. P-multigrid approach, where p is the order of polynomial degree, is widely employed in DG [111,112], SV [108], SD [113], SE [114], and some similar schemes to accelerate convergence rate.…”

“…Implicit LU-SGS is also successfully used for SD and SV, such as the methods in [107,108]. P-multigrid approach, where p is the order of polynomial degree, is widely employed in DG [111,112], SV [108], SD [113], SE [114], and some similar schemes to accelerate convergence rate. It is worth to note that Kannan et al [107,108,113] have successfully blended p-multigrid approach with pre-conditions or implicit LU-SGS, and the convergence rate is drastically improved for bad and skewed unstructured grids.…”

“…P-multigrid approach, where p is the order of polynomial degree, is widely employed in DG [111,112], SV [108], SD [113], SE [114], and some similar schemes to accelerate convergence rate. It is worth to note that Kannan et al [107,108,113] have successfully blended p-multigrid approach with pre-conditions or implicit LU-SGS, and the convergence rate is drastically improved for bad and skewed unstructured grids. For more details about p-multigrid methods, please refer to [108,[111][112][113][114] and the references therein.…”

“…It is worth to note that Kannan et al [107,108,113] have successfully blended p-multigrid approach with pre-conditions or implicit LU-SGS, and the convergence rate is drastically improved for bad and skewed unstructured grids. For more details about p-multigrid methods, please refer to [108,[111][112][113][114] and the references therein.…”

An implicit LU‐SGS spectral volume method for the moment models in device simulations III: accuracy enhancement using the LDG2 flux formulation for non‐uniform grids

“…In the earlier papers of this series, Kannan et al . used the LDG, penalty, and the BR2 formulations to discretize the diffusive flux. Unlike the penalizing schemes, the LDG2 method requires no length scales and is more symmetrical than the traditional LDG method.…”