Central WENO schemes for hyperbolic systems of conservation laws

Levy, Doron; Puppo, Gabriella; Russo, Giovanni

doi:10.1051/m2an:1999152

Cited by 385 publications

(353 citation statements)

References 32 publications

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“…where C l , and β are free parameters, which for example can be taken from Levy et al (1999). This are the so-called weighted essentially non-oscillatory (WENO) schemes.…”

Section: Eulerian (Grid)mentioning

confidence: 99%

Simulation Techniques for Cosmological Simulations

et al. 2008

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Modern cosmological observations allow us to study in great detail the evolution and history of the large scale structure hierarchy. The fundamental problem of accurate constraints on the cosmological parameters, within a given cosmological model, requires precise modelling of the observed structure. In this paper we briefly review the current most effective techniques of large scale structure simulations, emphasising both their advantages and shortcomings. Starting with basics of the direct N -body simulations appropriate to modelling cold dark matter evolution, we then discuss the direct-sum technique GRAPE, particlemesh (PM) and hybrid methods, combining the PM and the tree algorithms. Simulations of baryonic matter in the Universe often use hydrodynamic codes based on both particle methods that discretise mass, and grid-based methods. We briefly describe Eulerian grid methods, and also some variants of Lagrangian smoothed particle hydrodynamics (SPH) methods.

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“…where C l , and β are free parameters, which for example can be taken from Levy et al (1999). This are the so-called weighted essentially non-oscillatory (WENO) schemes.…”

Section: Eulerian (Grid)mentioning

confidence: 99%

Simulation Techniques for Cosmological Simulations

et al. 2008

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“…In this work, several Runge-Kutta time integrators were assessed -although none was categorized as strong-stability preserving (SSP). The work by Levy, Puppo and Russo [26] considered a weighted ENO (WENO) reconstruction in conjunction with Runge-Kutta time integrators for third and fourth-order accurate schemes -albeit in only one space dimension.…”

Section: Central Schemesmentioning

confidence: 99%

“…In this work, the KT algorithm is implemented with both third and fourth-order Runge-Kutta time integrators and applied to problems with both convex and non-convex flux functions. A third-order semi-discrete version of the KT method was outlined by Kurganov and Levy [17] using the WENO reconstruction from Levy, et al [26].…”

Section: Central Schemesmentioning

confidence: 99%

“…Far away from the vortex, the pressure and density are p ∞ = 1.0, and ρ ∞ = 1.0. The initial conditions are given in terms of x, y, and r = √ x 2 + y 2 as 26) where β = 5.0 is the vortex strength and u ∞ and v ∞ are the far-field x and y-velocities, respectively, and T ∞ is the ambient temperature. The domain is initialized using cell averages.…”

Section: The Isentropic Vortexmentioning

confidence: 99%

See 1 more Smart Citation

An assessment of semi-discrete central schemes for hyperbolic conservation laws.

Christon¹,

Robinson²,

Ketcheson³

2003

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High-resolution finite volume methods for solving systems of conservation laws have been widely embraced in research areas ranging from astrophysics to geophysics and aero-thermodynamics. These methods are typically at least second-order accurate in space and time, deliver non-oscillatory solutions in the presence of near discontinuities, e.g., shocks, and introduce minimal dispersive and diffusive effects. High-resolution methods promise to provide greatly enhanced solution methods for Sandia's mainstream shock hydrodynamics and compressible flow applications, and they admit the possibility of a generalized framework for treating multi-physics problems such as the coupled hydrodynamics, electro-magnetics and radiative transport found in zpinch physics. In this work, we describe initial efforts to develop a generalized "black-box" conservation law framework based on modern high-resolution methods and implemented in an object-oriented software framework. The framework is based on the solution of systems of general non-linear hyperbolic conservation laws using Godunov-type central schemes. In our initial 1 Keywords and Phrases: high-resolution, hyperbolic conservation laws, Godunov, semidiscrete, central schemes 2 Authors listed in alphabetic order.3 efforts, we have focused on central or central-upwind schemes that can be implemented with only a knowledge of the physical flux function and the minimal/maximal eigenvalues of the Jacobian of the flux functions, i.e., they do not rely on extensive Riemann decompositions. Initial experimentation with high-resolution central schemes suggests that contact discontinuities with the concomitant linearly degenerate eigenvalues of the flux Jacobian do not pose algorithmic difficulties. However, central schemes can produce significant smearing of contact discontinuities and excessive dissipation for rotational flows. Comparisons between "black-box" central schemes and the piecewiseparabolic method (PPM), which relies heavily on a Riemann decomposition, shows that roughly equivalent accuracy can be achieved for the same computational cost with both methods. However, PPM clearly outperforms the central schemes in terms of accuracy at a given grid resolution and the cost of additional complexity in the numerical flux functions. Overall we have observed that the finite volume schemes, implemented within a well-designed framework, are extremely efficient with (potentially) very low memory storage. Finally, we have found by computational experiment that second and third-order strong-stability preserving (SSP) time integration methods with the number of stages greater than the order provide a useful enhanced stability region. However, we observe that non-SSP and non-optimal SSP schemes with SSP factors less than one can still be very useful if used with time-steps below the standard CFL limit. The "well-designed" integration schemes that we have examined appear to perform well in all instances where the time step is maintained below the standard physical CFL limit. Acknowled...

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“…The NT scheme has smaller numerical viscosity and resolves shock waves, rarefactions and contact discontinuities much better than the first-order LxF scheme. Higher-order generalizations of the NT scheme were introduced in [3,17,21]. We refer the reader to [24,25] for an alternative staggered approach, and to [1,2,9,18,19] for examples of central schemes in the multidimensional case.…”

Section: Introductionmentioning

confidence: 99%

Central schemes and contact discontinuities

Kurganov

Petrova

2000

ESAIM: M2AN

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Abstract. We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes, proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition, suggested by Nessyahu and Tadmor [J. Comput. Phys. 87 (1990) , which is efficiently applied in the context of the new schemes. The method is tested on the one-dimensional Euler equations, subject to different initial data, and the results are compared to the numerical solutions, computed by other second-order central schemes. The numerical experiments clearly illustrate the advantages of the proposed technique.Mathematics Subject Classification. 65M10, 65M05.

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Central WENO schemes for hyperbolic systems of conservation laws

Cited by 385 publications

References 32 publications

Simulation Techniques for Cosmological Simulations

Simulation Techniques for Cosmological Simulations

An assessment of semi-discrete central schemes for hyperbolic conservation laws.

Central schemes and contact discontinuities

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