System in package (SiP) is mainstream technology in the design of electronics system. Numerical simulation plays an important role in SiP applications. However, due to the specific complexity of SiP applications, existing algorithms for linear systems arising from time-harmonic Maxwell's equations are faced with great challenges, which become a bottleneck restricting efficiency of large-scale numerical simulations. In this paper, we review algorithms for time-harmonic Maxwell's equations in SiP applications. Based on capability assessment of existing algorithms for realistic SiP models, we propose a preconditioning strategy, and show its feasibility and efficiency. Furthermore, we analyze impact of such applications on performance behavior of current algorithms and the challenges we faced with.

We present a Lax-Wendroff discontinuous Galerkin (LWDG) method combining with adaptive mesh refinement (AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method (RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.

Four semi-implicit discretization schemes are used to construct preconditioners.And preconditioned Jacobian-free Newton-Krylov (JFNK) are presented to solve one-dimensional problems.Numerical results show that the preconditioning methods improve the convergence behavior of JFNK method dramatically.

We consider solution of diffusion equations using structured adaptive mesh refinement(SAMR).In SAMR hierarchy, each level is organized as a union of uniform rectangular patches.An implicit time-integration algorithm with temporal refinement strategy is shown.In the algorithm,timestepping advances from the coarsest level to the finest level sequentially,and a multilevel synchronization process is required for fixing fluxes dismatch at coarse-fine interface.A criterion for algorithm complexity is introduced. Numerical results show validation and performance of the algorithm.Finally,the algorithm is applied to radiation hydrodynamics simulations,where nonlinear non-equilibrium radiation diffusion equations are solved.Simulation result shows that,compared with uniform refinement mesh,performance of the method is improved by 33 times.

Jacobian.free Newton-Krylov(JFNK) method is analyzed and improved by using physical constraints in iteration. Consequently.physical constraints are always satisfied in iteration process of the improved JFNK method.Non。physical phenomenon is avoided.In particular.there is no negative temperature as the method is used for 2-D 3-T energy equations.Robustness of JFNK method is improved.

A two-level iterative method is proposed for linear systems discretizated from two-dimensional(2-D) radiative diffusion equations with photon, electron,ion temperatures(3-T).The main idea is to decouple one temperature from other two by a special coarsening strategy.Variables related to electron temperature are forced to be selected as coarse points and photon and ion temperatures are forced to be fine points.Several single temperature equations instead of coupled linear systems need to be solved by a classical-AMG method.The method is applied to the JFNK framework for preconditioning.Numerical results show effectiveness of the method.

As 2-D 3-T heat conduct equations are discretized in a fully implicit method,it is very difficult to solve the nonlinear algebraic equations obtained due to strong nonlinearity.Efficient initial guesses for iterative solutions of discretized nonlinear algebraic equations are presented.Numerical results for two media with different properties show that the proposed initial guesses improve computational efficiency and reduce the influence of nonlinear solver on the time step as well.