Manuel A. Huerta

E-CUSP scheme for the equations of ideal magnetohydrodynamics with high order WENO Scheme

An E-CUSP (energy-convective upwind and split pressure) scheme is developed to solve the equations of magnetohydrodynamics. Fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. A characteristic speed of sound by averaging the fast wave speed and the acoustic speed of sound is suggested to evaluate the Mach number, which will yield robust and accurate solutions. The numerical experiments have demonstrated the accuracy and the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

Entropy, information theory, and the approach to equilibrium of coupled harmonic oscillator systems

Finite segments of infinite chains of classical coupled harmonic oscillators are treated as models of thermodynamic systems in contact with a heat bath, i.e., canonical ensembles. The Liouville functionρ for the infinite chain is reduced by integrating over the “outside” variables to a functionρN of the variables of theN-particle segment that is the thermodynamic system. The reduced Liouville functionρN which is calculated from the dynamics of the infinite chain and the statistical knowledge of the coordinates and momenta att = 0, is a time-dependent probability density in the 2N-dimensional phase space of the system. A Gibbs entropy defined in terms ofρN measures the evolution of knowledge of the system (more accurately, the growth of missing pertinent information) in the sense of information theory. As ¦t ¦ → ∞, energy is equipartitioned, the entropy evolves to the value expected from equilibrium statistical mechanics, and ρN evolves to an equilibrium distribution function. The simple chain exhibits diffusion in coordinate space, i.e., Brownian motion, and the diffusivity is shown to depend only on the initial distribution of momenta (not of coordinates) in the heat bath. The harmonically bound chain, in the limit of weak coupling, serves as an excellent model for the approach to equilibrium of a canonical ensemble of weakly interacting particles.

Stationary striations in an argon plasma as a bifurcation phenomenon

The formation of stationary striations in an argon plasma due to the development of an instability is studied using the equations for the competing plasma population species. The nonlinear dynamic stability of the system is analyzed using a two‐time perturbation procedure. The unstable uniform steady state bifurcates to a new state with a sinusoidal density variation in space leading to the formation of balls of glowing plasma. This is analogous to phenomena of structure formation that appear in chemical reactions and biological processes.

Exact Equilibration of Harmonically Bound Oscillator Chains

The approach to equilibrium of a finite segment of an infinite chain of harmonically coupled, harmonically bound oscillators is treated exactly, both when the initial description of the rest of the chain is canonical and when it is Gaussian. The necessary mathematical properties of the bound oscillator functions are developed and used to demonstrate exact equipartition of energy. The entropy of the finite segment, or system, is shown to evolve to a time‐independent equilibrium state that is, in the limit of weak coupling, the correct one for a system of noninteracting harmonic oscillators.

The surface potentials produced by electric sources in stratified spherical and prolate spheroidal volume conductors

The exact Greens function solution for the surface potential produced by a point dipole current source immersed in a stratified prolate spheroidal or spherical volume conductor is given. The transmission matrix approach used allows a systematic calculation for any number of stratifications or dipole locations. Numerical results are given for prolate spheroidal and spherical models of the human head.

Information Theory and the Approach to Equilibrium

After a general discussion of thermodynamic equilibrium and the information-theoretic formulation of equilibrium statistical mechanics, illustrative calculations are presented of the evolution to equilibrium of a finite segment (the system) of an infinite coupled harmonic-oscillator chain, most of which is regarded as the heat bath. The reduced Liouville function ρN is used to define the information-theoretic version of the Gibbs entropy as SN = −kB  ∫  ρN ln(hρNN)dΓN. This entropy evolves to a proper equilibrium value as |t| → ∞ from time-reversible dynamics, because ρN spreads from an initially sharp distribution to a diffuse one characteristic of the heat bath in equilibrium. The approach is regarded as generally valid, in principle, although the procedure is most easily carried out in the treatment of coupled harmonic-oscillator systems.

E-CUSP Scheme for the Equations of Magnetothydrodynamics with High Order WENO Scheme

An E-CUSP scheme is developed to solve the equations of magnetothydrodynamics. A fifth order WENO reconstructions are employed to calculate the fluxes in order to achieve high order spacial accuracy. Four standard testing cases, including two one-dimensional problems, the 2D Kelvin-Helmholtz instability and the Orszag-Tang MHD turbulence problem, are solved to validate the accuracy and robustness of the scheme. The 1D Brio-Wu shock tube problem is used to show the capability of capturing the compound waves in MHD. The other 1D problem tested is to demonstrate the robustness for high Mach number flow in MHD. The simulations of two 2D cases have demonstrated the capability of the new scheme to capture complex interactions of multiple shocks and vortices.

Rotated Hybrid Low Diffusion ECUSP-HLL scheme and Its Applications to Hypersonic Flows

In this paper, a hybrid numerical flux scheme of low diffusion ECUSP (LDE) and HLL scheme is proposed by decomposing the cell-face normal vector. In the direction normal to shocks, the HLL scheme is applied to suppress carbuncles, in the other direction, the ECUSP scheme is implemented to keep low numerical diffusion. The new scheme with fifth-order WENO reconstruction is tested by using several benchmark cases, and then is applied to simulate a three dimensional double cone hypersonic flow of Mach number 12.43. The numerical simulation results agree very well with experiments. Nomenclature ∗ Research Scientist † Associate Professor ‡ Professor 1 20th AIAA Computational Fluid Dynamics Conference 27 30 June 2011, Honolulu, Hawaii AIAA 2011-3545 Copyright

Steady detonation waves with losses

The Euler equations for a reacting polytropic gas with model losses, applied to unsupported steady detonation waves, lead to fast and slow detonation speeds for a given loss. The reaction rate is taken to have the Arrhenius form. The differential equations for the Mach number squared m and the reaction progress variable λ are integrated numerically using a fourth‐order Runge–Kutta–Fehlberg method. The calculations are presented as plots of trajectories in the λ‐log (m) plane. The main parameters that are varied are the activation temperature, the heat of reaction, the order of the reaction, and the polytropic exponent γ. The onset of detonation failure, and a continuum of solutions that appears in some parameter ranges, are explored in detail. The results are summarized in plots of the detonation Mach number squared m0 versus the logarithm of the loss parameter.

Approach to equilibrium of coupled harmonic oscillator systems. II

The approach to equilibrium of a finite segment of an infinite chain of harmonically coupled masses is studied in several variations. The chain is taken as completely free, or it is bound atx0=0; ordinary coordinates and momenta or Schrödinger variables are used to treat the dynamics; and the inital distribution of heat-bath variables is chosen to be canonical or noncanonical. Equipartition of energy is found in all cases. Brownian drifts are obtained for the free chain with ordinary variables, but when this is excluded, the equilibrium entropy is found to be canonical and extensive when the initial heat bath is canonical, but less than canonical and slightly nonextensive when the initial heat bath is noncanonical. The modifications of the entropy do not contribute to the heat capacity of the system.