## A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes (2001)

Venue: | J. Comput. Phys |

Citations: | 26 - 5 self |

### BibTeX

@ARTICLE{Leveque01aclass,

author = {Randall J. Leveque and Marica Pelanti},

title = {A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes},

journal = {J. Comput. Phys},

year = {2001},

volume = {172},

pages = {200--1}

}

### OpenURL

### Abstract

We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235--276] 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.

### Citations

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(Show Context)
Citation Context ... in terms of these waves and speeds, for example using the wave-propagation approach of [33], which is reviewed in Section 8. One important approximate Riemann solver of this form was proposed by Roe =-=[42]-=- and has been extensively used. A special averaging of f 0 (U l ) and f 0 (U r ) is used to define a matrixsA with the property thatsA(U r \Gamma U l ) = f(U r ) \Gamma f(U l ): (1.7) The eigenvaluess... |

483 |
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Citation Context ...the transonic rarefaction case ks0 typically and the proper spreading does not occur. An entropy fix is often used to address this problem. One possibility proposed by Harten and Hyman [17] (see also =-=[32]-=-) is to replace the single wavesff ksr k in this case by a pair of waves ff k lsr k and ff k rsr k propagating at speeds s k l ! 0 ! s k r that are chosen to approximate the characteristic speeds at e... |

296 |
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(Show Context)
Citation Context ...ssuming the system is genuinely nonlinear) and then apply the first-order upwind method together with this approximate Riemann solver, the resulting method is simply Rusanov's method, as discussed in =-=[43]-=-, for example. This method is also known as the Local Lax-Friedrichs (LLF) method. If we choose d = \Deltax=\Deltat, an upper bound on all possible wave speeds provided the CFL condition is satisfied ... |

229 |
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Citation Context ...Chapman-Enskog expansion [9], [36]. The structure of this viscosity matrix can play a role in determining whether the correct entropy-satisfying solution is obtained in the limits! 0, see for example =-=[16]-=-, [37]. For a relaxation system with the coefficient matrix (7.2), we find that B(u) = \GammasA 2 + 2sAf 0 (u) \Gamma (f 0 (u)) 2 : (7.4) Note that if U lsU r thensAsf 0 (u) and the viscosity matrix v... |

219 | Non-oscillation central differencing for hyperbolic conservation laws
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(Show Context)
Citation Context ...ethod reduces to the classical Lax-Friedrichs (LxF) method. We note in passing that the LxF and LLF methods can be extended to second-order accuracy to obtain the central schemes of Nessyahu & Tadmor =-=[40]-=- and Kurganov & Tadmor [28] respectively, and connections between these methods and relaxation schemes are briefly discussed in the introduction to [28]. If D is diagonal but the diagonal elements d j... |

187 | The Relaxation Schemes for Systems of Conservation Laws in Arbitrary Space Dimensions
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(Show Context)
Citation Context ...lvers can be viewed as special cases, including the Roe solver, the Roe solver with an entropy fix, and the simple HLL and HLLE solvers. 2 We have also found that the relaxation scheme of Jin and Xin =-=[24]-=- (at least in the limit as the relaxation time vanishes) is closely related to an approximate Riemann solver of this type. Indeed the decomposition (1.8) was first suggested to us by an attempt to gen... |

158 |
On upstream differencing and Godunovtype schemes for hyperbolic conservation laws
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- 1983
(Show Context)
Citation Context ...section we will show that this can be viewed as a generalization of the HLL Riemann solver. 3 Relation to the HLL solver A simple approximate Riemann solver was discussed by Harten, Lax, and van Leer =-=[18]-=-. This HLL solver consists of approximating the Riemann solution by two waves (regardless of the dimension m of the system) with some speeds a l and a r chosen to approximate the minimum and maximum c... |

141 | New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations
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- 1997
(Show Context)
Citation Context ...cal Lax-Friedrichs (LxF) method. We note in passing that the LxF and LLF methods can be extended to second-order accuracy to obtain the central schemes of Nessyahu & Tadmor [40] and Kurganov & Tadmor =-=[28]-=- respectively, and connections between these methods and relaxation schemes are briefly discussed in the introduction to [28]. If D is diagonal but the diagonal elements d j are not equal (as in the c... |

136 | Hyperbolic conservation laws with stiff relaxation terms and entropy
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- 1994
(Show Context)
Citation Context ... as modified relaxation schemes. This may aid in the theoretical analysis of these methods (see Section 7). Relaxation schemes have recently been widely applied and studied, see for example [1], [8], =-=[9], [22], [2-=-6], [35]. Another related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connect... |

129 |
Hyperbolic conservation laws with relaxation
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- 1987
(Show Context)
Citation Context ...d j is the spectral radius of D. The insures that the characteristic speeds of the hyperbolic part of (2.1) are at least as large as the characteristic speeds of the original problem. See [10], [19], =-=[36]-=-, [38], [39], [46], [45] for some discussions of this condition and convergence properties. The relaxation scheme we describe is not exactly the same as Jin and Xin's, but is the simplest variant of i... |

79 | Wave propagation algorithms for multi-dimensional hyperbolic systems
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- 1997
(Show Context)
Citation Context ...; (1.6) which is a generalization of (1.5). Then an upwind algorithm and high-resolution variants can be defined in terms of these waves and speeds, for example using the wave-propagation approach of =-=[33]-=-, which is reviewed in Section 8. One important approximate Riemann solver of this form was proposed by Roe [42] and has been extensively used. A special averaging of f 0 (U l ) and f 0 (U r ) is used... |

74 | Convergence to equilibrium for the relaxation approximation of conservation laws
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- 1996
(Show Context)
Citation Context ... the spectral radius of D. The insures that the characteristic speeds of the hyperbolic part of (2.1) are at least as large as the characteristic speeds of the original problem. See [10], [19], [36], =-=[38]-=-, [39], [46], [45] for some discussions of this condition and convergence properties. The relaxation scheme we describe is not exactly the same as Jin and Xin's, but is the simplest variant of it. We ... |

72 | Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm
- LeVeque
- 1998
(Show Context)
Citation Context ...naccuracy in some cases, for example if the solution is nearly in steady state with f(u) xs/ and we wish to study the propagation of small disturbances on this background state (see the discussion in =-=[34]-=-, for example). Another approach is to somehow incorporate / into the solution of Riemann problems. One way to do this is to discretize the source terms as a sum of delta function singularities with s... |

71 |
Self-adjusting grid methods for one-dimensional hyperbolic conservation laws
- Harten, Hyman
- 1983
(Show Context)
Citation Context ...e ofsA, and in the transonic rarefaction case ks0 typically and the proper spreading does not occur. An entropy fix is often used to address this problem. One possibility proposed by Harten and Hyman =-=[17]-=- (see also [32]) is to replace the single wavesff ksr k in this case by a pair of waves ff k lsr k and ff k rsr k propagating at speeds s k l ! 0 ! s k r that are chosen to approximate the characteris... |

59 |
Upwind difference schemes for hyperbolic systems of conservation laws
- Osher, Solomon
- 1982
(Show Context)
Citation Context ...s in Brenier's transport-collapse method [7] or the large time step method of [31]. This also has similarities to the method developed by Engquist & Osher [14] for scalar problems and Osher & Solomon =-=[41]-=- for systems, often called the Osher solver in general. In this approach only the integral curves of the eigenvectors are used to compute an approximate Riemann solution, so that rarefaction waves and... |

56 |
Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic
- Klingenberg, Risebro
- 1995
(Show Context)
Citation Context ...blem with data U l ; U r and two different flux functions f l (u) and f r (u). When f l and f r are nonlinear, determining the exact Riemann solution for this situation may be nontrivial, e.g., [15], =-=[27]-=-, [29], [44]. We are currently investigating the possibility of using a Riemann solver of the form (1.8) for such problems and here only report some preliminary observations. One natural way to use (1... |

49 | Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics equations
- Coquel, Perthame
- 1998
(Show Context)
Citation Context ... simpler but related system of equations as an approximation. This could be useful for problems where a Roe matrix cannot be found directly. A similar idea has been proposed by by Coquel and Perthame =-=[11]-=- and implemented by In [20] for one particular system. They use the classical Roe solver for the polytropic Euler equations in order to solve real-gas problems with more complicated equations of state... |

47 |
Stable viscosity matrices for systems of conservation laws
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- 1985
(Show Context)
Citation Context ...n-Enskog expansion [9], [36]. The structure of this viscosity matrix can play a role in determining whether the correct entropy-satisfying solution is obtained in the limits! 0, see for example [16], =-=[37]-=-. For a relaxation system with the coefficient matrix (7.2), we find that B(u) = \GammasA 2 + 2sAf 0 (u) \Gamma (f 0 (u)) 2 : (7.4) Note that if U lsU r thensAsf 0 (u) and the viscosity matrix vanishe... |

46 | Natalini R, Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws
- Aregba-Driollet
- 2000
(Show Context)
Citation Context ... studied, see for example [1], [8], [9], [22], [26], [35]. Another related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilib=-=rium. See [2]-=- for one discussion of these connections. Bouchet [6] has recently presented an interpretation of kinetic schemes as approximate Riemann solvers for flux-vector splitting methods, though these take a ... |

44 | Contractive relaxation systems and the scalar multidimensional conservation law
- Katsoulakis, Tzavaras
- 1997
(Show Context)
Citation Context ...d relaxation schemes. This may aid in the theoretical analysis of these methods (see Section 7). Relaxation schemes have recently been widely applied and studied, see for example [1], [8], [9], [22], =-=[26], [35]. An-=-other related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connections. Bouche... |

41 | Construction of BGK models with a family of kinetic entropies for a given system of conservation laws
- Bouchut
- 1999
(Show Context)
Citation Context ...[38]. Another related class of numerical methods are the \kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connections. Bou=-=chut [6-=-], [7] has recently presented an interpretation of kinetic schemes as approximate Riemann solvers forsux-vector splitting methods, though these take a rather dierent form from what we introduce here. ... |

36 |
On Godunov-Type Methods for Gas Dynamics
- Einfeldt
- 1988
(Show Context)
Citation Context ...peeds a l and a r chosen to approximate the minimum and maximum characteristic speeds of the system. It is often called the HLLE solver when the specific choice of a l and a r recommended by Einfeldt =-=[13]-=- is used. The wave strengths are W 1 = Um \Gamma U l ; W 2 = U r \Gamma Um ; (3.1) where the middle state Um is chosen to preserve conservation by requiring (a r \Gamma a l )Um = a r U r \Gamma a l U ... |

36 |
Stable and entropy satisfying approximations for transonic flow calculations
- Engquist, Osher
(Show Context)
Citation Context ...by averaging an overturned compression wave as in Brenier's transport-collapse method [7] or the large time step method of [31]. This also has similarities to the method developed by Engquist & Osher =-=[14]-=- for scalar problems and Osher & Solomon [41] for systems, often called the Osher solver in general. In this approach only the integral curves of the eigenvectors are used to compute an approximate Ri... |

34 | Convergence of relaxation schemes for conservation laws
- Aregba-Driollet, Natalini
- 1996
(Show Context)
Citation Context ...nterpreted as modified relaxation schemes. This may aid in the theoretical analysis of these methods (see Section 7). Relaxation schemes have recently been widely applied and studied, see for example =-=[1], [8], [9]-=-, [22], [26], [35]. Another related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of the... |

30 |
Zero relaxation and dissipation limits for hyperbolic conservation laws
- Chen, Liu
- 1993
(Show Context)
Citation Context ...max = max j d j is the spectral radius of D. The insures that the characteristic speeds of the hyperbolic part of (2.1) are at least as large as the characteristic speeds of the original problem. See =-=[10]-=-, [19], [36], [38], [39], [46], [45] for some discussions of this condition and convergence properties. The relaxation scheme we describe is not exactly the same as Jin and Xin's, but is the simplest ... |

29 |
Riemann problems with a discontinuous flux function
- Gimse, Risebro
- 1990
(Show Context)
Citation Context ...nn problem with data U l ; U r and two different flux functions f l (u) and f r (u). When f l and f r are nonlinear, determining the exact Riemann solution for this situation may be nontrivial, e.g., =-=[15]-=-, [27], [29], [44]. We are currently investigating the possibility of using a Riemann solver of the form (1.8) for such problems and here only report some preliminary observations. One natural way to ... |

28 |
Averaged multivalued solutions for scalar conservation laws
- Brenier
- 1984
(Show Context)
Citation Context ...e approximate solution between these two waves can be viewed as an approximation to the value that would be found by averaging an overturned compression wave as in Brenier's transport-collapse method =-=[7]-=- or the large time step method of [31]. This also has similarities to the method developed by Engquist & Osher [14] for scalar problems and Osher & Solomon [41] for systems, often called the Osher sol... |

27 |
Nonoscillatory central dierencing for hyperbolic conservation laws
- Nessyahu, Tadmor
- 1990
(Show Context)
Citation Context ...ethod reduces to the classical Lax-Friedrichs (LxF) method. We note in passing that the LxF and LLF methods can be extended to second-order accuracy to obtain the central schemes of Nessyahu & Tadmor =-=[43]-=- and Kurganov & Tadmor [30] respectively, and connections between these methods and relaxation schemes are brie y discussed in the introduction to [30]. If D is diagonal but the diagonal elements d j ... |

24 |
Hyperbolic Conservation Laws with Sti Relaxation Terms and Entropy
- Chen, Levermore, et al.
- 1995
(Show Context)
Citation Context ... applications to conservation laws with discontinuous coecients and with source terms. 2 Relaxation schemes Relaxation schemes have recently been widely applied and studied, see for example [1], [9], =-=[10], [18-=-], [24], [28], [31], [38]. Another related class of numerical methods are the \kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of t... |

22 | Capturing shock reflections: an improved flux formula
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- 1996
(Show Context)
Citation Context ...rical difficulties, such as nonphysical oscillations near slowly moving shocks. The addition of more dissipation in this case is one approach to improving solutions in this case. See for example [3], =-=[12]-=-, [23], [25]. A further generalization of this solver is obtained by using vectorssr j in (5.4) that are not the eigenvectors of the Roe matrix. This may be useful for problems where a Roe average sat... |

21 | Recent mathematical results on hyperbolic relaxation problems
- Natalini
(Show Context)
Citation Context ...pectral radius of D. The insures that the characteristic speeds of the hyperbolic part of (2.1) are at least as large as the characteristic speeds of the original problem. See [10], [19], [36], [38], =-=[39]-=-, [46], [45] for some discussions of this condition and convergence properties. The relaxation scheme we describe is not exactly the same as Jin and Xin's, but is the simplest variant of it. We use a ... |

20 |
On postshock oscillations due to shock capturing schemes in unsteady flows
- Arora, Roe
- 1997
(Show Context)
Citation Context ... numerical difficulties, such as nonphysical oscillations near slowly moving shocks. The addition of more dissipation in this case is one approach to improving solutions in this case. See for example =-=[3]-=-, [12], [23], [25]. A further generalization of this solver is obtained by using vectorssr j in (5.4) that are not the eigenvectors of the Roe matrix. This may be useful for problems where a Roe avera... |

19 |
A large time step generalization of godunov’s method for systems of conservation laws
- Leveque
- 1985
(Show Context)
Citation Context ...two waves can be viewed as an approximation to the value that would be found by averaging an overturned compression wave as in Brenier's transport-collapse method [7] or the large time step method of =-=[31]-=-. This also has similarities to the method developed by Engquist & Osher [14] for scalar problems and Osher & Solomon [41] for systems, often called the Osher solver in general. In this approach only ... |

18 |
Numerical tests of evolution systems, gauge conditions, and boundary conditions for 1d colliding gravitational plane waves
- Bardeen, Buchman
(Show Context)
Citation Context ...spatially-varyingsux functions due to the metric terms. This approach is also being used in a wave-propagation algorithm for the Einstein equations in numerical relativity work by Bardeen and Buchman =-=[5]-=-. 10 Source terms Now consider a conservation law u t + f(u) x =s(10.1) with a source terms. One common approach to solving this equation is to use a fractional step method, alternating between solvin... |

18 |
On Upstream Dierencing and Godunov-Type Schemes for Hyperbolic Conservation
- Harten, Lax, et al.
- 1983
(Show Context)
Citation Context ...section we will show that this can be viewed as a generalization of the HLL Riemann solver. 3 Relation to the HLL solver A simple approximate Riemann solver was discussed by Harten, Lax, and van Leer =-=[20]-=-. This HLL solver consists of approximating the Riemann solution by two waves (regardless of the dimension m of the system) with some speeds a l and a r chosen to approximate the minimum and maximum c... |

16 | Entropy satisfying flux vector splittings and kinetic
- Bouchut
(Show Context)
Citation Context ...[35]. Another related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connections=-=. Bouchet [6]-=- has recently presented an interpretation of kinetic schemes as approximate Riemann solvers for flux-vector splitting methods, though these take a rather different form from what we have introduced ab... |

14 | Wave propagation methods for conservation laws with source terms
- LeVeque, Bale
(Show Context)
Citation Context ...ming the decomposition (1.4). Approximate Riemann solvers based on splitting the jump in f have recently been studied numerically for various applications in work with Derek Bale and James Rossmanith =-=[4]-=-. This work started directly from (9.13) and we only recently realized the connection with relaxation schemes. Preliminary results indicate that it may be a useful approach for many problems, includin... |

14 |
Quasilinear hyperbolic systems and dissipative mechanisms”, World Scientific
- Hsiao
- 1998
(Show Context)
Citation Context ...max j d j is the spectral radius of D. The insures that the characteristic speeds of the hyperbolic part of (2.1) are at least as large as the characteristic speeds of the original problem. See [10], =-=[19]-=-, [36], [38], [39], [46], [45] for some discussions of this condition and convergence properties. The relaxation scheme we describe is not exactly the same as Jin and Xin's, but is the simplest varian... |

13 | The effects of numerical viscosities. I. Slowly moving shocks
- Jin, Liu
- 1996
(Show Context)
Citation Context ...difficulties, such as nonphysical oscillations near slowly moving shocks. The addition of more dissipation in this case is one approach to improving solutions in this case. See for example [3], [12], =-=[23]-=-, [25]. A further generalization of this solver is obtained by using vectorssr j in (5.4) that are not the eigenvectors of the Roe matrix. This may be useful for problems where a Roe average satisfyin... |

13 |
The solution of nonstrictly hyperbolic conservation laws may be hard to compute
- Tveito, Winther
- 1995
(Show Context)
Citation Context ...ta U l ; U r and two different flux functions f l (u) and f r (u). When f l and f r are nonlinear, determining the exact Riemann solution for this situation may be nontrivial, e.g., [15], [27], [29], =-=[44]-=-. We are currently investigating the possibility of using a Riemann solver of the form (1.8) for such problems and here only report some preliminary observations. One natural way to use (1.8) might be... |

11 |
clawpack software. http://www.amath.washington.edu/~claw
- LeVeque
(Show Context)
Citation Context ...nite-volume method that gives high-resolution results and directly uses a wave decomposition of the form (1.2) is the wavepropagation method described in [33] and implemented in the clawpack software =-=[30]-=-. This method uses an updating formula of the form U n+1 i = U n i \Gamma \Deltat \Deltax (A + \DeltaU i\Gamma1=2 +A \Gamma \DeltaU i+1=2 ) \Gamma \Deltat \Deltax ( ~ F i+1=2 \Gamma ~ F i\Gamma1=2 ); ... |

11 |
Convergence of a relaxation scheme for hyperbolic systems of conservation laws
- Lattanzio, Serre
(Show Context)
Citation Context ...ation laws with discontinuous coecients and with source terms. 2 Relaxation schemes Relaxation schemes have recently been widely applied and studied, see for example [1], [9], [10], [18], [24], [28], =-=[31], [38-=-]. Another related class of numerical methods are the \kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connections. Bouchu... |

10 | Convergence rates for relaxation schemes approximating conservation laws
- Liu, Warnecke
(Show Context)
Citation Context ...xation schemes. This may aid in the theoretical analysis of these methods (see Section 7). Relaxation schemes have recently been widely applied and studied, see for example [1], [8], [9], [22], [26], =-=[35]. Another -=-related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these connections. Bouchet [6] ... |

10 | Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics
- Gosse, E
(Show Context)
Citation Context ...cations to conservation laws with discontinuous coecients and with source terms. 2 Relaxation schemes Relaxation schemes have recently been widely applied and studied, see for example [1], [9], [10], =-=[18], [24-=-], [28], [31], [38]. Another related class of numerical methods are the \kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these c... |

9 |
Computations of slowly moving shocks
- Karni, C̆anić
- 1997
(Show Context)
Citation Context ...ulties, such as nonphysical oscillations near slowly moving shocks. The addition of more dissipation in this case is one approach to improving solutions in this case. See for example [3], [12], [23], =-=[25]-=-. A further generalization of this solver is obtained by using vectorssr j in (5.4) that are not the eigenvectors of the Roe matrix. This may be useful for problems where a Roe average satisfying (1.7... |

8 |
Numerical evaluation of an energy relaxation method for inviscid real fluids
- In
- 1999
(Show Context)
Citation Context ... of equations as an approximation. This could be useful for problems where a Roe matrix cannot be found directly. A similar idea has been proposed by by Coquel and Perthame [11] and implemented by In =-=[20]-=- for one particular system. They use the classical Roe solver for the polytropic Euler equations in order to solve real-gas problems with more complicated equations of state. An additional energy vari... |

8 |
di#erence schemes for hyperbolic systems of conservation laws
- Osher, Solomon, et al.
- 1982
(Show Context)
Citation Context ...s in Brenier's transport-collapse method [8] or the large time step method of [34]. This also has similarities to the method developed by Engquist & Osher [15] for scalar problems and Osher & Solomon =-=[44]-=- for systems, often called the Osher solver in general. In this approach only the integral curves of the eigenvectors are used to compute an approximate Riemann solution, so that rarefaction waves and... |

7 | Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms
- Chalabi
- 1999
(Show Context)
Citation Context ...reted as modified relaxation schemes. This may aid in the theoretical analysis of these methods (see Section 7). Relaxation schemes have recently been widely applied and studied, see for example [1], =-=[8], [9], [22-=-], [26], [35]. Another related class of numerical methods are the "kinetic schemes" based on the Boltzmann equation and relaxation towards equilibrium. See [2] for one discussion of these co... |

7 |
Rankine-Hugoniot-Riemann solver considering source terms and multidimensional effects
- Jenny, Müller
- 1998
(Show Context)
Citation Context ...delta function singularities with strength proportional to \Deltax at the cell interfaces, so that the effect of the source is concentrated at these points. This approach is taken by Jenny and Muller =-=[21]-=- in their Rankine-Hugoniot Riemann solver, for example. In this case we must solve a more general Riemann problem of the form u t + f(u) x = \Psiffi(x) (10.2) where \Psi = \Deltax / i\Gamma1=2 is the ... |

7 |
Capturing shock re an improved formula
- Donat, Marquina
- 1996
(Show Context)
Citation Context ...umerical diculties, such as nonphysical oscillations near slowly moving shocks. The addition of more dissipation in this case is one approach to improving solutions in this case. See for example [3], =-=[13]-=-, [25], [27]. A further generalization of this solver is obtained by using vectors ^ r j in (5.4) that are not the eigenvectors of the Roe matrix. This may be useful for problems where a Roe average s... |