A Hybrid Scheme for Gas-Dust Systems Stiffly Coupled via Viscous Drag

We present a stable and convergent method for studying a system of gas and dust, coupled through viscous drag in both non-stiff and stiff regimes. To account for the effects of dust drag in the update of the fluid quantities, we employ a fluid description of the dust component and study the modified gas-dust hyperbolic system following the approach in Miniati & Colella (2007). In addition to two entropy waves for the gas and dust components, respectively, the extended system includes three waves driven partially by gas pressure and partially by dust drift, which, in the limit of vanishing coupling, tend to the two original acoustic waves and the unhindered dust streaming. Based on this analysis we formulate a predictor step providing first order accurate reconstruction of the time-averaged state variables at cell interfaces, whence a second order accurate estimate of the conservative fluxes can be obtained through a suitable linearized Riemann solver. The final source term update is carried out using a one-step, second order accurate, L-stable, predictor corrector asymptotic method (the alpha-QSS method suggested by Mott et. al. 2000). This procedure completely defines a two-fluid method for gas-dust system. Using the updated fluid solution allows us to then advance the individual particle solutions, including self-consistently the time evolution of the gas velocity in the estimate of the drag force. This is done with a suitable particle scheme also based on the alpha-QSS method. A set of benchmark problems shows that our method is stable and convergent. When dust is modeled as a fluid (two-fluid) second order accuracy is achieved in both stiff and non-stiff regimes, whereas when dust is modeled with particles (hybrid) second order is achieved in the non-stiff regime and first order otherwise.Comment: 41 pages, 3 figures, 14 tables, accepted to J. Comp. Phys

A Hybrid Godunov Method for Radiation Hydrodynamics

From a mathematical perspective, radiation hydrodynamics can be thought of as a system of hyperbolic balance laws with dual multiscale behavior (multiscale behavior associated with the hyperbolic wave speeds as well as multiscale behavior associated with source term relaxation). With this outlook in mind, this paper presents a hybrid Godunov method for one-dimensional radiation hydrodynamics that is uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds that the technique preserves certain asymptotic limits. The method incorporates a backward Euler upwinding scheme for the radiation energy density and flux as well as a modified Godunov scheme for the material density, momentum density, and energy density. The backward Euler upwinding scheme is first-order accurate and uses an implicit HLLE flux function to temporally advance the radiation components according to the material flow scale. The modified Godunov scheme is second-order accurate and directly couples stiff source term effects to the hyperbolic structure of the system of balance laws. This Godunov technique is composed of a predictor step that is based on Duhamel's principle and a corrector step that is based on Picard iteration. The Godunov scheme is explicit on the material flow scale but is unsplit and fully couples matter and radiation without invoking a diffusion-type approximation for radiation hydrodynamics. This technique derives from earlier work by Miniati & Colella 2007. Numerical tests demonstrate that the method is stable, robust, and accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61 pages, 15 figures, 11 table

Hybrid Entropy Stable HLL-Type Riemann Solvers for Hyperbolic Conservation Laws

It is known that HLL-type schemes are more dissipative than schemes based on characteristic decompositions. However, HLL-type methods offer greater flexibility to large systems of hyperbolic conservation laws because the eigenstructure of the flux Jacobian is not needed. We demonstrate in the present work that several HLL-type Riemann solvers are provably entropy stable. Further, we provide convex combinations of standard dissipation terms to create hybrid HLL-type methods that have less dissipation while retaining entropy stability. The decrease in dissipation is demonstrated for the ideal MHD equations with a numerical example.Comment: 6 page

Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area