High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation Laws

Jiang, Guang-Shan; Levy, Doron; Lin, Chin-Teng; Osher, Stanley; Tadmor, Eitan

doi:10.1137/s0036142997317560

Cited by 189 publications

(171 citation statements)

References 29 publications

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“…In order to test our schemes in practice, we compared them with two other schemes, the first order Engquist-Osher scheme proposed in [6] and a central scheme which is an adaptation of schemes presented in [10]. We have no convergence proofs for these schemes.…”

Section: Numerical Examplesmentioning

confidence: 99%

Convergent difference schemes for the Hunter–Saxton equation

2007

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Abstract. We propose and analyze several finite difference schemes for the Hunter-Saxton equationThis equation has been suggested as a simple model for nematic liquid crystals. We prove that the numerical approximations converge to the unique dissipative solution of (HS), as identified by Zhang and Zheng. A main aspect of the analysis, in addition to the derivation of several a priori estimates that yield some basic convergence results, is to prove strong convergence of the discrete spatial derivative of the numerical approximations of u, which is achieved by analyzing various renormalizations (in the sense of DiPerna and Lions) of the numerical schemes. Finally, we demonstrate through several numerical examples the proposed schemes as well as some other schemes for which we have no rigorous convergence results.

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Section: Numerical Examplesmentioning

confidence: 99%

Convergent difference schemes for the Hunter–Saxton equation

2007

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“…This is of importance for a construction of higher-order non-staggered central schemes which does not follow the mentioned recipe presented in [7]. The construction of such schemes is surely an issue for future developments because non-staggered grids are advantageous if complex geometries and boundary conditions come into play.…”

Section: Conclusive Remarksmentioning

confidence: 99%

“…The investigations within this paper show, that one needs to be very careful when constructing a central scheme, especially on a non-staggered grid. One may understand some parts of our investigations as a justification of the approach presented in [7]: there, non-staggered central schemes are derived by integration using staggered grids followed by an averaging projection onto the initial non-staggered cells.…”

Section: Introductionmentioning

confidence: 99%

An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws

Breuß

2005

ESAIM: M2AN

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Abstract.We discuss the occurrence of oscillations when using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered and non-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema. The dependence of oscillatory properties on the numerical viscosity coefficient is investigated rigorously for the LFt schemes, illuminating also the properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient are prone to oscillations at data extrema. For all LFt schemes except for the classical Lax-Friedrichs method, occurring oscillations are damped in the course of a computation. This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggered version may yield oscillatory results, while it can be shown rigorously that the staggered NT scheme does not produce oscillations when using the classical minmod-limiter under a restriction on the time step size. Note that this restriction is not the same as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.

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“…Jiang, et al [14] later extended the NT algorithm to multiple space dimensions and demonstrated that the central scheme does not require a spatial splitting. Subsequently, Jiang, et al [14] introduced a non-staggered central scheme that retains the "Riemann-solver-free" aspect of the original staggered NT algorithm.…”

Section: Central Schemesmentioning

confidence: 99%

“…Subsequently, Jiang, et al [14] introduced a non-staggered central scheme that retains the "Riemann-solver-free" aspect of the original staggered NT algorithm. Here, it was first suggested that central schemes could form the basis for a robust generalized computational framework for systems of conservation laws.…”

Section: Central Schemesmentioning

confidence: 99%

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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High-Resolution Nonoscillatory Central Schemes with Nonstaggered Grids for Hyperbolic Conservation Laws

Cited by 189 publications

References 29 publications

Convergent difference schemes for the Hunter–Saxton equation

Convergent difference schemes for the Hunter–Saxton equation

An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws

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

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