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A relaxation scheme for conservation laws with a discontinuous coefficient
 Math. Comp
, 2007
"... Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a wea ..."
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Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.
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 29 (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.
Relaxation Schemes for the Shallow Water Equations
 INT. J. NUMER. METH. FLUIDS
, 2003
"... We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKut ..."
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Cited by 9 (0 self)
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We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with RungeKutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present.
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INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI
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"... Numerical solution of the the twodimensional shallow water equations by the application of relaxation methods. ..."
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Numerical solution of the the twodimensional shallow water equations by the application of relaxation methods.
c © 2004 Birkhäuser Verlag, Basel
"... A gaskinetic scheme for shallow–water equations with source terms ..."
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"... Convergence of a continuous BGK model for initial boundaryvalue problems for conservation laws ∗ ..."
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Convergence of a continuous BGK model for initial boundaryvalue problems for conservation laws ∗