A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws

Sod, Gary A.

doi:10.1016/0021-9991(78)90023-2

Cited by 2,260 publications

(1,046 citation statements)

References 16 publications

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“…We have performed standard Sod (1978) shock tube tests used for the original TreeSPH (Hernquist & Katz 1989). We find the best results with the pairwise viscosity of equation 7 which is marginally better than the bulk viscosity formulation for this test .…”

Section: Shocks: Spherical Adiabatic Collapsementioning

confidence: 96%

“…The ongoing challenge is to see if better load balance through more complex work division offsets increases in communication and other parallel overheads. Sod (1978) shock tube test results with Gasoline for density (left) and velocity (right). This a three dimensional test using glass initial conditions similar to the conditions in a typical simulation.…”

Section: Performancementioning

confidence: 99%

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Gasoline: a flexible, parallel implementation of TreeSPH

2004

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The key algorithms and features of the Gasoline code for parallel hydrodynamics with self-gravity are described. Gasoline is an extension of the efficient Pkdgrav parallel N -body code using smoothed particle hydrodynamics. Accuracy measurements, performance analysis and tests of the code are presented. Recent successful Gasoline applications are summarized. These cover a diverse set of areas in astrophysics including galaxy clusters, galaxy formation and gas-giant planets. Future directions for gasdynamical simulations in astrophysics and code development strategies for tackling cutting edge problems are discussed.

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Section: Shocks: Spherical Adiabatic Collapsementioning

confidence: 96%

Section: Performancementioning

confidence: 99%

Gasoline: a flexible, parallel implementation of TreeSPH

2004

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“…The last shock tube problem to be considered is Sod's problem [61]. For this problem, (i) at t = 0, Fig.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The last shock tube problem to be considered is Sod's problem [61]. For this problem, (i) at t = 0, encountered, it is seen that the numerical results generated by both schemes match very well with the exact solution.…”

Section: Nasa/tm-2005-213868 25mentioning

confidence: 85%

“…A comparion of these results and those shown in Fig. 6 reveals that the Mach number sensitive scheme is indeed Mach number sensitive while the Mach number insensitive scheme is indeed Mach number insensitive.The last shock tube problem to be considered is Sod's problem [61]. For this problem, (i) at t = 0, encountered, it is seen that the numerical results generated by both schemes match very well with the exact solution.…”

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confidence: 85%

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Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Chang¹

2005

41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference &Amp;amp; Exhibit

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IntrodutionThe space-time conservation element and solution element (CE/SE) method is a high-resolution and genuinely multidimensional method for solving conservation laws . Its nontraditional features include: (i) a unified treatment of space and time; (ii) the introduction of conservation elements (CEs) and solution elements (SEs) as the vehicles for enforcing space-time flux conservation; (iii) a novel time marching strategy that has a space-time staggered stencil at its core and, as such, fluxes at an interface can be evaluated without using any interpolation or extrapolation procedure (which, in turn, leads to the method's ability to capture shocks without using Riemann solvers); (iv) the requirement that each scheme be built from a non-dissipative core scheme and, as a result, the numerical dissipation can be controlled effectively; and (v) the fact that mesh values of the physical dependent variables and their spatial derivatives are considered as independent marching variables to be solve for simultaneously. Note that CEs are nonoverlapping space-time subdomains introduced such that (i) the computational domain can be filled by these subdomains; and (ii) flux conservation can be enforced over each of them and also over the union of any combination of them. On the other hand, SEs are space-time subdomains introduced such that (i) the boundary of each CE can be divided into several component parts with each of them belonging to a unique SE; and (ii) within a SE, any physical flux vector is approximated using simple smooth functions. In general, a CE does not coincide with a SE.Without using flux-splitting or other special techniques, since its inception [1] the unstructured-mesh compatible CE/SE method has been used to obtain numerous accurate 1D, 2D and 3D steady and unsteady flow solutions with Mach numbers ranging from 0.0028 to 10 [42]. The phyical phenomena modeled include traveling and interacting shocks, acoustic waves, shedding vortices, viscous flows, detonation waves, cavitation, flows in fluid film bearings, heat conduction with melting and/or freezing, electrodynamics, MHD vortex, hydraulic jump, crystal growth, and chromatographic problems . In particular, the rather unique capability of the CE/SE method to resolve both strong shocks and small disturbances (e.g., acoustic waves) simultaneously [11,13,14] makes it an effective tool for attacking computational aeroacoustics (CAA) problems. Note that the fact that second-order CE/SE schemes can solve CAA problems accurately is an exception to the commonly-held belief that a second-order scheme is not adequate for solving CAA problems. Also note that, while numerical dissipation is needed for shock capturing, it may also result in annihilation of small disturbances. Thus a solver that can handle both strong shocks and small disturbances simultaneously must be able to overcome this difficulty.In spite of its past successes, there is still room for improving the CE/SE method. An example is the fact that, in a CE/SE simulation with a fixed t...

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Fluid-membrane interaction based on the material point method

York

Sulsky

Schreyer

2000

Int. J. Numer. Meth. Engng.

140

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SUMMARYThe material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, #uids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-in#ated airbag.

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A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws

Cited by 2,260 publications

References 16 publications

Gasoline: a flexible, parallel implementation of TreeSPH

Gasoline: a flexible, parallel implementation of TreeSPH

Courant Number and Mach Number Insensitive CE/SE Euler Solvers

Fluid-membrane interaction based on the material point method

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