Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme

Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives

A time-domain numerical modeling of Biot poroelastic waves is presented. The viscous dissipation occurring in the pores is described using the dynamic permeability model developed by Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce order 1/2 shifted fractional derivatives involving a convolution product. Based on a diffusive representation, the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. Thanks to the dispersion relation, the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest. A splitting strategy is then applied numerically: the propagative part of Biot-JKD equations is discretized using a fourth-order ADER scheme on a Cartesian grid, whereas the diffusive part is solved exactly. Comparisons with analytical solutions show the efficiency and the accuracy of this approach.Comment: arXiv admin note: substantial text overlap with arXiv:1210.036

On the application and extension of Harten's high resolution scheme

Extensions of a second order high resolution explicit method for the numerical computation of weak solutions of one dimensonal hyperbolic conservation laws are discussed. The main objectives were (1) to examine the shock resoluton of Harten's method for a two dimensional shock reflection problem, (2) to study the use of a high resolution scheme as a post-processor to an approximate steady state solution, and (3) to construct an implicit in the delta-form using Harten's scheme for the explicit operator and a simplified iteration matrix for the implicit operator

Analysis of the effects of baffles on combustion instability

An analytical model has been developed for predicting the effects of baffles on combustion instability. This model has been developed by coupling an acoustic analysis of the wave motion within baffled chambers with a model for the oscillatory combustion response of a propellant droplet developed by Heidmann. A computer program was developed for numerical solution of the resultant coupled equations. Diagnostic calculations were made to determine the reasons for the improper prediction. These calculations showed that the chosen method of representing the combustion response was a very poor approximation. At the end of the program, attempts were made to minimize this effect but the model still improperly predicts the stability trends. Therefore, it is recommended that additional analysis be done with an improved approximation

Atmospheric Heat Redistribution on Hot Jupiters

Infrared lightcurves of transiting hot Jupiters present a trend in which the atmospheres of the hottest planets are less efficient at redistributing the stellar energy absorbed on their daysides---and thus have a larger day-night temperature contrast---than colder planets. No predictive atmospheric model has been published that identifies which dynamical mechanisms determine the atmospheric heat redistribution efficiency on tidally locked exoplanets. Here we present a two-layer shallow water model of the atmospheric dynamics on synchronously rotating planets that explains the observed trend. Our model shows that planets with weak friction and weak irradiation exhibit a banded zonal flow with minimal day-night temperature differences, while models with strong irradiation and/or strong friction exhibit a day-night flow pattern with order-unity fractional day-night temperature differences. To interpret the model, we develop a scaling theory that shows that the timescale for gravity waves to propagate horizontally over planetary scales, t_wave, plays a dominant role in controlling the transition from small to large temperature contrasts. This implies that heat redistribution is governed by a wave-like process, similar to the one responsible for the weak temperature gradients in the Earth's tropics. When atmospheric drag can be neglected, the transition from small to large day-night temperature contrasts occurs when t_wave ~ sqrt(t_rad/Omega), where t_rad is the radiative relaxation time and Omega is the planetary rotation frequency. Alternatively, this transition criterion can be expressed as t_rad ~ t_vert, where t_vert is the timescale for a fluid parcel to move vertically over the difference in day-night thickness. These results subsume the commonly used timescale comparison for estimating heat redistribution efficiency between t_rad and the global horizontal advection timescale, t_adv.Comment: Accepted to ApJ with minor edits compared to version 1; 17 pages, 11 figure