Smadar Karni

Hybrid Multifluid Algorithms

Extensions of many successful single-component schemes to compute multicomponent gas dynamics suffer from oscillations and other computational inaccuracies near material interfaces that are caused by the failure of the schemes to maintain pressure equilibrium between the fluid components. A new algorithm based on the compressible Euler equations for multicomponent fluids augmented by the pressure evolution equation is presented. The extended set of equations offers two alternative ways to update the pressure field: (i) using the equation of state or (ii) using the pressure evolution equation. In a numerical implementation, these two procedures generally yield different answers. The former is a standard conservative update, but may produce oscillations near material interfaces; the latter is nonconservative, but becomes exact near interfaces and automatically maintains pressure equilibrium. A hybrid scheme which selects from the two pressure update procedures is presented. The scheme perfectly conserves total mass and momentum and conserves total energy everywhere except at a finite (very small) number of grid cells. Computed solutions exhibit oscillation-free interfaces and have {\em negligible} relative conservation errors in total energy even for very strong shocks. The proposed hybrid approach and switching strategies are independent of the numerical implementation and may provide a simple framework within which to extend ones favourite scheme to solve multifluid dynamics.

Short Note: A comment on the computation of non-conservative products

We are interested in the solution of non-conservative hyperbolic systems, and consider in particular the so-called path-conservative schemes (see e.g. [2,3]) which rely on the theoretical work in [1]. The example of the standard Euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach.

Two-Layer Shallow Water System: A Relaxation Approach

The two-layer shallow water system is an averaged flow model. It forms a nonconservative system which is only conditionally hyperbolic. The coupling between the layers, due to the hydrostatic pressure assumption, does not provide explicit access to the system eigenstructure, which is inconvenient for Riemann solution based numerical schemes. We consider a relaxation approach which offers greater decoupling and accessible eigenstructure. The stability of the model is discussed. Numerical results are shown for unsteady flows as well as for smooth and nonsmooth steady flows.

CRASH: A BLOCK-ADAPTIVE-MESH CODE FOR RADIATIVE SHOCK HYDRODYNAMICS-IMPLEMENTATION AND VERIFICATION

We describe the Center for Radiative Shock Hydrodynamics (CRASH) code, a block-adaptive-mesh code for multi-material radiation hydrodynamics. The implementation solves the radiation diffusion model with a gray or multi-group method and uses a flux-limited diffusion approximation to recover the free-streaming limit. Electrons and ions are allowed to have different temperatures and we include flux-limited electron heat conduction. The radiation hydrodynamic equations are solved in the Eulerian frame by means of a conservative finite-volume discretization in either one-, two-, or three-dimensional slab geometry or in two-dimensional cylindrical symmetry. An operator-split method is used to solve these equations in three substeps: (1) an explicit step of a shock-capturing hydrodynamic solver; (2) a linear advection of the radiation in frequency-logarithm space; and (3) an implicit solution of the stiff radiation diffusion, heat conduction, and energy exchange. We present a suite of verification test problems to demonstrate the accuracy and performance of the algorithms. The applications are for astrophysics and laboratory astrophysics. The CRASH code is an extension of the Block-Adaptive Tree Solarwind Roe Upwind Scheme (BATS-R-US) code with a new radiation transfer and heat conduction library and equation-of-state and multi-group opacity solvers. Both CRASH and BATS-R-US are part of the publicly available Space Weather Modeling Framework.

Viscous shock profiles and primitive formulations

Weak solutions of hyperbolic systems in primitive (nonconservative) form for which a consistent conservation form exists are considered. It is shown that for primitive formulations, shock relations are not uniquely defined by the states to either side of the shock, but also depend on the viscous path connecting the two. Consistent viscous shock profiles are enforced by adding scheme-dependent small viscous perturbations that account for leading order conservation errors. The resulting primitive algorithm is conservative to the order of the approximation. One-dimensional Euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms. While not eliminating conservation errors entirely, it is found that for a wide range of problems, both conservative and primitive flow calculations are of comparable quality.

Far-field filtering operators for suppression of reflections from artificial boundaries

The concern of this paper is the treatment of far-field artificial boundaries for first order hyperbolic systems, where the aim is to suppress nonphysical reflections of outgoing waves. We construct far-field filtering operators of two types: (1) slowing-down operators and (2) damping operators. The operators are activated in anarrow sponge layer near the far-field boundary and have the following properties: (a) the operators act only on the outgoing part of the solution leaving the incoming part unharmed, (b) the passage of waves across the modified layer is reflection-free, and (c) the implementation of the operators amounts to a small modification of the governing equations in the far-field. In either case, the treatment of the boundary itself is greatly simplified, since the outgoing waves either do not reach the boundary or have practically zero strength when they do. Well-posedness of the modified equations and stability of the discretization scheme are established. We also show that such reflection...

Numerical computation of two-dimensional unsteady detonation waves in high energy solids

We are concerned with theoretical modelling of unsteady, two-dimensional detonation waves in high energy solids. A mathematical model and a numerical method to solve the associated hyperbolic system of equations are presented. The model consists of the Euler equations augmented bby extra conservation laws and source terms to accoxint for chemical reaction and tracking of materials. Both the thermodynamics and the chemistry are treated in a simple way. Using a detonation analogue due to Fickett, we test several numerical methods and assess their performance in modelling the essential features of detonation waves. The numerical method selected for the full model is an extension of the conservative, shock capturing technique of Roe, together with an adaptive mesh refinement procedure that allows the resolution of fine features such as reaction zones. Results for some typical tests problems are presented. Starting in 1946 as the College of Aeronautics, the Cranfield Institute of Technology was granted university status in 1969. In 1993 it changed its name to Cranfield University.

Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted intoLloc∞ estimates, following theLip′ convergence theory developed by Tadmor et al. Comparisons between the local truncation error and theLloc∞-error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].

A Hybrid Algorithm for the Baer-Nunziato Model Using the Riemann Invariants

The paper considers the Baer-Nunziato model for two-phase flow in porous media, with discontinuous porosity. Computing solutions of the Riemann problem rests on capturing the jump in the solution across the porosity jump. A recent study (Lowe in J. Comput. Phys. 204:598–632, 2005) showed that numerical discretizations may fail to correctly capture the jump conditions across the so-called compaction wave, and yield incorrect solutions. We have formulated the Baer-Nunziato system using the Riemann invariants across the porosity jump, and propose a hybrid algorithm that uses the Riemann invariants formulation across the compaction wave, and the conservative formulation away from the compaction wave. The paper motivates and describes the hybrid scheme and present numerical results.

A new level set model for multimaterial flows

We propose a new level set model for representing multimaterial flows in multiple space dimensions. Instead of associating a level set function with a specific fluid material, the function is associated with a pair of materials and the interface that separates them. A voting algorithm collects sign information from all level set functions and determines material designation. To represent a general M-material configuration, M(M-1)/2 level set functions need to be accounted for; problems of practical interest use far fewer functions, as not all pairs of materials share an interface, and level-set functions that coincide are grouped together. Under this model, regions of potential material ambiguity, i.e. overlaps or vacuum, are markedly reduced in size: in 2D, ambiguous regions are points, as opposed to lines in material-based level set models; in 3D, they are lines as opposed to surfaces. The model produces excellent results without the need for reinitialization, thereby avoiding additional computational costs and preventing excessive numerical diffusion.