Central Schemes for Balance Laws of Relaxation Type

Abstract

Several models in mathematical physics are described by quasi-linear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the Nessyahu-Tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations. A second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.

Exploitation of the Maximum Entropy Principle in Mathematical Modeling of Charge Transport in Semiconductors

“…The physical parameters of the device are reported in Table 1. The numerical solution has been obtained by resorting to an extension [51] of the traditional central differencing scheme for one-dimensional balance laws with (possibly stiff) source terms, which has been developed on the basis of the Nessyhau and Tadmor scheme [52] for homogeneous hyperbolic systems. The bias voltage at the contacts is the superposition of the thermal equilibrium boundary potential (the built in potential V bi ) and the applied potential V a .…”

Boundary conditions for Grad's 13 moment equations

“…In order to solve them with slip and jump conditions for velocity, temperature and normal stress we use a method provided by Liotta et al [31]. These authors extended the Nessyahu-Tadmor scheme for hyperbolic conservation laws [32] to balance laws of relaxation type.…”

2D Simulation of a Silicon MESFET with a Nonparabolic Hydrodynamical Model Based on the Maximum Entropy Principle

“…The central schemes known in the literature deal almost exclusively with homogeneous systems. In [5,6] a suitable extension for one-dimensional balance laws with (possibly stiff ) source terms has been developed on the basis of the Nessyahu and Tadmor scheme [3] for homogeneous hyperbolic system. It has been applied in [22,23] to parabolic band hydrodynamical models of semiconductors.…”

“…In order to get a full second-order scheme we combine the relaxation and convective steps in the same way as proposed in [5,6] for the one-dimensional case. Note that the analysis of the splitting accuracy does not really depend on the dimension of the space.…”

“…A numerical scheme for problems of this kind based on central differencing has been developed in [5,6,22] for the one-dimensional case and applied in [7,9] to bulk silicon and to an n + -n-n + silicon diode (for the parabolic band case see [23]). The obtained results are rather encouraging regarding the accuracy of the model and the performance of the numerical scheme.…”