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576
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
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Cited by 32 (5 self)
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We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
Comparison of several difference schemes on 1D and 2D test problems for the . . .
, 2001
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ADER schemes for threedimensional nonlinear hyperbolic systems
 J. Comput. Phys
, 2005
"... In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order o ..."
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Cited by 30 (6 self)
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In this paper we carry out the extension of the ADER approach to multidimensional nonlinear systems of conservation laws. We implement nonlinear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order of accuracy and nonoscillatory properties of the new schemes. Compared to the stateofart finitevolume WENO schemes the ADER schemes are faster, more accurate and need less computer memory. Key words: highorder schemes, weighted essentially nonoscillatory, ADER, generalized Riemann problem, three space dimensions. 1 1
Denlinger (2004), Granular avalanches across irregular threedimensional terrain: 2. Experimental tests
 J. Geophys. Res
"... [1] To establish a theoretical basis for predicting and interpreting the behavior of rapid mass movements on Earth’s surface, we develop and test a new computational model for gravitydriven motion of granular avalanches across irregular, threedimensional (3D) terrain. The principles embodied in t ..."
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Cited by 28 (0 self)
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[1] To establish a theoretical basis for predicting and interpreting the behavior of rapid mass movements on Earth’s surface, we develop and test a new computational model for gravitydriven motion of granular avalanches across irregular, threedimensional (3D) terrain. The principles embodied in the model are simple and few: continuum mass and momentum conservation and intergranular stress generation governed by Coulomb friction. However, significant challenges result from the necessity of satisfying these principles when deforming avalanches interact with steep and highly variable 3D terrain. We address these challenges in four ways. (1) We formulate depthaveraged governing equations that are referenced to a rectangular Cartesian coordinate system (with z vertical) and that account explicitly for the effect of nonzero vertical accelerations on depthaveraged mass and momentum fluxes and stress states. (2) We compute fluxes of mass and momentum across vertical cell boundaries using a highresolution finite volume method and Roetype Riemann solver. Our algorithm incorporates flux difference splitting, an entropy correction for the flux, and eigenvector decomposition to embed the effects of driving and resisting forces in Riemann solutions. (3) We use a finite element method and
The surface gradient method for the treatment of source terms in the shallowwater equations
 Journal of Computational Physics
, 2001
"... A novel scheme has been developed for data reconstruction within a Godunovtype method for solving the shallowwater equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstru ..."
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Cited by 28 (0 self)
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A novel scheme has been developed for data reconstruction within a Godunovtype method for solving the shallowwater equations with source terms. In contrast to conventional data reconstruction methods based on conservative variables, the water surface level is chosen as the basis for data reconstruction. This provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a Riemann solver. The main advantages are: (1) a simple centered discretization is used for the source terms; (2) the scheme is no more complicated than the conventional method for the homogeneous terms; (3) small perturbations in the water surface elevation can be accurately predicted; and (4) the method is generally suitable for both steady and unsteady shallowwater problems. The accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems. Excellent agreement has been obtained between the numerical predictions and analytical solutions. The results indicate that the new scheme is accurate, simple, efficient, and robust. c ° 2001 Academic Press Key Words: source terms; shallowwater equations; data reconstruction; highresolution method; Godunov method; MUSCL scheme.
On the reduction of numerical dissipation in centralupwind schemes
 Commun. Comput. Phys
"... We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods t ..."
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Cited by 28 (7 self)
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We study centralupwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707–740]. Similarly to the staggered central schemes, these schemes are central Godunovtype projectionevolution methods that enjoy the advantages of high resolution, simplicity, universality, and robustness. At the same time, the centralupwind framework allows one to decrease a relatively large amount of numerical dissipation present at the staggered central schemes. In this paper, we present a modification of the onedimensional fully and semidiscrete centralupwind schemes, in which the numerical dissipation is reduced even further. The goal is achieved by a more accurate projection of the evolved quantities onto the original grid. In the semidiscrete case, the reduction of dissipation procedure leads to a new, less dissipative numerical flux. We also extend the new semidiscrete scheme to the twodimensional case via the rigorous, genuinely multidimensional derivation. The new semidiscrete schemes are tested on a number of numerical examples, where one can observe an improved resolution, especially of the contact waves. 1
An adaptive numerical scheme for highspeed reactive on overlapping grids
, 2003
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Entropy satisfying flux vector splittings and kinetic BGK models
, 2000
"... We establish forward and backward relations between entropy satisfying BGK models such as those introduced previously by the author and the first order flux vector splitting numerical methods for systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of ..."
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Cited by 25 (3 self)
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We establish forward and backward relations between entropy satisfying BGK models such as those introduced previously by the author and the first order flux vector splitting numerical methods for systems of conservation laws. Classically, to a kinetic BGK model that is compatible with some family of entropies we can associate an entropy flux vector splitting. We prove that the converse is true: any entropy flux vector splitting can be interpreted by a kinetic model, and we obtain an explicit characterization of entropy satisfying flux vector splitting schemes. We deduce a new proof of discrete entropy inequalities under a sharp CFL condition that generalizes the monotonicity criterion in the scalar case. In particular, this gives a stability condition for numerical kinetic methods with noncompact velocity support. A new interpretation of general kinetic schemes is also provided via approximate Riemann solvers. We deduce the construction of finite velocity relaxation systems for gas dyn...
Shock capturing and front tracking methods for granular avalanches
 J. Comput. Phys
, 2002
"... Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme ..."
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Cited by 25 (6 self)
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Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shockcapturing numerical scheme for the onedimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a fronttracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the fronttracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat runout region, where a shock is formed as the avalanche comes to a halt. c ○ 2002 Elsevier Science Key Words: granular avalanche; shockcapturing; nonoscillatory central scheme; free moving boundary; fronttracking. 1.