An arbitrary Lagrangian–Eulerian method with adaptive mesh refinement for the solution of the Euler equations

Anderson, Robert; Elliott, N. S.; Pember, Richard B.

doi:10.1016/j.jcp.2004.02.021

Cited by 90 publications

(80 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The smeared interface generates significant numerical errors in these latter two quantities that propagate away from the interface along the forward and backward characteristic waves, u ± c, where c is the sound speed for the appropriate material state. The numerical error shown in Figure 1 is typical for any standard conservative, shock-capturing scheme, such as Godunov's method, applied to (11). The error may be explained by considering one step in the numerical method from a uniform-pressure-velocity (UPV) state, as in the example above.…”

Section: A Shock-capturing Methods With Energy Correctionmentioning

confidence: 99%

“…The governing equations for each constituent are solved on either side of the interface with the aid of "ghost-fluid" points which are constructed across the interface. ALE methods [10,11] advect the material interface and use boundary-conforming grids for their discretization. Finally, the volume of fluid methods [12,13] reconstruct a material interface from advected volume fractions at every step and thus maintain sharp material interfaces.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

Banks

Schwendeman

Kapila

et al. 2007

Journal of Computational Physics

108

View full text Add to dashboard Cite

A numerical method is described for inviscid, compressible, multi-material flow in two space dimensions. The flow is governed by the multi-material Euler equations with a general mixture equation of state. Composite overlapping grids are used to handle complex flow geometry and block-structured adaptive mesh refinement (AMR) is used to locally increase grid resolution near shocks and material interfaces. The discretization of the governing equations is based on a high-resolution Godunov method, but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard, conservative shock-capturing schemes. The energy correction is constructed based on a uniform pressure-velocity flow and is significant only near the captured interface. A variety of two-material flows are presented to verify the accuracy of the numerical approach and to illustrate its use. These flows assume an equation of state for the mixture based on JonesWilkins-Lee (JWL) forms for the components. This equation of state includes a mixture of ideal gases as a special case. Flow problems considered include unsteady one-dimensional shock-interface collision, steady interaction of an planar interface and an oblique shock, planar shock interaction with a collection of gas-filled cylindrical inhomogeneities, and the impulsive motion of the two-component mixture in a rigid cylindrical vessel.

show abstract

Section: A Shock-capturing Methods With Energy Correctionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

Banks

Schwendeman

Kapila

et al. 2007

Journal of Computational Physics

108

View full text Add to dashboard Cite

show abstract

“…Certainly for Eulerian schemes [2,3], all waves are captured and so the connection is clear. Lagrangian schemes [4][5][6] are often used to circumvent the issues associated with captured linear jumps by performing computations in the frame of the fluid, but most interesting simulations require, at the very least, mesh remap which results in the so-called arbitrary-Lagrangian-Eulerian (ALE) schemes [7]. The use of such remap has the potential to cause ALE schemes suffer the same fate as purely Eulerian schemes although the details may depend on the frequency at which remapping is performed.…”

Section: Introductionmentioning

confidence: 99%

On sub-linear convergence for linearly degenerate waves in capturing schemes

Banks

Aslam

Rider

2008

Journal of Computational Physics

View full text Add to dashboard Cite

A common attribute of capturing schemes used to find approximate solutions to the Euler equations is a sub-linear rate of convergence with respect to mesh resolution. Purely nonlinear jumps, such as shock waves produce a first-order convergence rate, but linearly degenerate discontinuous waves, where present, produce sub-linear convergence rates which eventually dominate the global rate of convergence. The classical explanation for this phenomenon investigates the behavior of the exact solution to the numerical method in combination with the finite error terms, often referred to as the modified equation. For a first-order method, the modified equation produces the hyperbolic evolution equation with second-order diffusive terms. In the frame of reference of the traveling wave, the solution of a discontinuous wave consists of a diffusive layer that grows with a rate of t 1/2 , yielding a convergence rate of 1/2. Self-similar heuristics for higher order discretizations produce a growth rate for the layer thickness of ∆t 1/(p+1) which yields an estimate for the convergence rate as p/(p+1) where p is the order of the discretization. In this paper we show that this estimated convergence rate can be derived with greater rigor for both dissipative and dispersive forms of the discrete error. In particular, the form of the analytical solution for linear modified equations can be solved exactly. These estimates and forms for the error are confirmed in a variety of demonstrations ranging from simple linear waves to multidimensional solutions of the Euler equations.

show abstract

“…The ALE-AMR multi-physics hydrocode has been developed as a predictive modeling tool to identify and assess sources of shrapnel and debris and the hazards they pose so these may be mitigated [1][2][3][4][5][6]. Such simulations generally require large computational domains in which to track fragment formation and trajectories.…”

Section: Introductionmentioning

confidence: 99%

Laser ray tracing in a parallel arbitrary Lagrangian-Eulerian adaptive mesh refinement hydrocode

Masters

Kaiser

Anderson

et al. 2010

J. Phys.: Conf. Ser.

View full text Add to dashboard Cite

An arbitrary Lagrangian–Eulerian method with adaptive mesh refinement for the solution of the Euler equations

Cited by 90 publications

References 19 publications

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

On sub-linear convergence for linearly degenerate waves in capturing schemes

Laser ray tracing in a parallel arbitrary Lagrangian-Eulerian adaptive mesh refinement hydrocode

Contact Info

Product

Resources

About