## Unconditionally Stable Explicit Schemes for the Approximation of Conservation Laws

Citations: | 2 - 1 self |

### BibTeX

@MISC{Helzel_unconditionallystable,

author = {Christiane Helzel and Gerald Warnecke},

title = {Unconditionally Stable Explicit Schemes for the Approximation of Conservation Laws},

year = {}

}

### OpenURL

### Abstract

We consider explicit schemes for homogeneous conservation laws which satisfy the geometric Courant-Friedrichs-Lewy condition in order to guarantee stability but allow a time step with CFL-number larger than one. A brief overview over existing unconditionally stable schemes for hyperbolic conservation laws is provided, although the focus is on LeVeque's large time step Godunov scheme. For this scheme we explore the question of entropy consistency for the approximation of one-dimensional scalar conservation laws with convex ux function and describe a possible way to extend the scheme to the two-dimensional case. Numerical calculations and analytical results show that an increase of accuracy can be obtained because the error introduced by the modi ed evolution step of the large time step Godunov scheme may be less important than the error due to the projection step.

### Citations

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- 1981
(Show Context)
Citation Context ...ny applications it is not necessary or possible to get an exact solution of the Riemann problem. Therefore approximative Riemann solvers were developed. One which is often used is the Roe solver, see =-=[31]-=- or any text book on numerical schemes for conservation laws [5], [26], [35]. The use of the Roe Riemann solver for an unconditionally stable extension of a Godunov-type scheme is considered in Sectio... |

516 |
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(Show Context)
Citation Context ...). Nice descriptions of the theory of hyperbolic conservation laws as well as the concepts of numerical schemes can be found in the text books of Godlewski, Raviart [4], [5], Kroner [16] and LeVeque [=-=26-=-], [28]. Here we brie y want to cover those denitions and theoretical results which will be needed for our considerations. The solution to the initial value problem (1), (2) is in general not continuo... |

512 |
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(Show Context)
Citation Context ...s a consistency result which says that if in the limit, when k and h go to zero, the scheme is convergent, then the limit function is a weak solution. This is a consequence of the Lax-Wendro Theorem [=-=17]-=-, which more generally holds for schemes which can be written in conservative form so that the numericalsux function is consistent with thesux function of the conservation law, i.e. F(u; u) = f (u). I... |

247 |
Numerical Approximation of Hyperbolic Systems of Conservation Laws
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(Show Context)
Citation Context ...ectorssr 1 (u); : : : ; r p (u). Nice descriptions of the theory of hyperbolic conservation laws as well as the concepts of numerical schemes can be found in the text books of Godlewski, Raviart [4], =-=[-=-5], Kroner [16] and LeVeque [26], [28]. Here we brie y want to cover those denitions and theoretical results which will be needed for our considerations. The solution to the initial value problem (1),... |

86 | Wave propagation algorithms for multidimensional hyperbolic systems
- LeVeque
- 1997
(Show Context)
Citation Context ...nov scheme is implemented in the software packagesCLAWPACK [18], which was used for our calculations. It may be used with any Riemann solver. There the wave propagation algorithm developed by LeVeque =-=[27]-=- is used to update the solution u at every time step. Highresolution can be obtained by considering piecewise linear reconstruction. This requires the use of limiter functions in order to avoid spurio... |

84 |
Numerical Schemes for Conservation Laws
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- 1997
(Show Context)
Citation Context ...); : : : ; r p (u). Nice descriptions of the theory of hyperbolic conservation laws as well as the concepts of numerical schemes can be found in the text books of Godlewski, Raviart [4], [5], Kroner [=-=16-=-] and LeVeque [26], [28]. Here we brie y want to cover those denitions and theoretical results which will be needed for our considerations. The solution to the initial value problem (1), (2) is in gen... |

78 |
Self-adjusting grid methods for one-dimensional hyperbolic conservation laws
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(Show Context)
Citation Context ...hand side will be in uenced. In the case of a transonic rarefaction wave both cell averages should be changed. An entropy-x has to be used to overcome this problem, see for instance Harten and Hyman [=-=8-=-]. For the LTS scheme the entropy-x has to be extended. In order to explain this, we want to consider the approximation of a rarefaction wave for a scalar conservation law, for instance the Burgers eq... |

66 |
A dierence scheme for numerical computation of the discontinuous solutions of equations of dynamics
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(Show Context)
Citation Context ...asic tool for computationalsuid dynamics of compressiblesows and multidimenensional schemes for hyperbolic conservation laws in general. The idea of solving Riemann problems was introduced by Godunov =-=[6]-=- and was developed considerably in the last two decades. The available number of variants of this numerical approach is quite large and there are many special techniques with corresponding theoretical... |

46 |
Shock waves and reaction-diusion equations
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(Show Context)
Citation Context ... conservation laws and systems in one space dimension a number of mathematical criteria such as the Lax condition, various entropy inequalities or the viscosity method are in use, see [4], [5], [26], =-=[34]-=-. For a given numerical scheme we do not only have to know whether the sequence of approximate solutions is convergent. We also have to consider whether the limit solution is actually the physically r... |

38 |
Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics i: one dimensional flow
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- 1995
(Show Context)
Citation Context ... periods that have to be considered, see Iben and Tadmor [14]. Other numerical problems arise if very dierent scales are modeled by the equations, for instance in low Mach numbersow calculations. In [=-=15]-=-, Klein uses the unconditionally stable Godunov-type scheme of LeVeque, which is reviewed in this paper, to calculate the in uence of the fast acoustic waves in such a system. 2 Christiane Helzel and ... |

25 |
A method of fractional steps for scalar conservation laws without the CFL condition
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(Show Context)
Citation Context ...point in the mesh cell instead of the cell average over the solution which is use by Godunov type schemes. An unconditionally stable scheme based on Front tracking was developed by Holden and Risebro =-=[13]-=-. The good performance of this scheme for the Euler equations is demonstrated in [12]. Billett and Toro [1] consider the weighted averagesux (WAF) scheme which allows time steps with Courant numbers u... |

23 |
On upstream dierencing and Godunov-type schemes for hyperbolic conservation laws
- Harten, Lax, et al.
- 1983
(Show Context)
Citation Context ...iane Helzel and Gerald Warnecke the physical relevant solution of the problem considered. For the Godunov scheme we get a positive answer to the latter question from the Harten-Lax Theorem, see [10], =-=[11]-=-. There the discrete form of the entropy inequality, a special form of the entropy condition, is used to pick out the physical relevant weak solution. The concept of the Godunov scheme is implemented ... |

19 |
A large time step generalization of godunovâ€™s method for systems of conservation laws
- Leveque
- 1985
(Show Context)
Citation Context ...airer [7] and Satofuka [32]. They were also used insow calculations, see Satofuka [33]. 4 The large time step Godunov scheme of LeVeque Now we want to consider the LTS Godunov scheme of LeVeque [20], =-=[21-=-], [22], [25] applied to one dimensional systems of conservation laws. As we have seen in Section 2.2 the classical concept of the Godunov scheme, where the solution u is dened as the combination of t... |

16 |
On WAF-type schemes for multidimensional hyperbolic conservation laws
- Billett, Toro
- 1997
(Show Context)
Citation Context ...n unconditionally stable scheme based on Front tracking was developed by Holden and Risebro [13]. The good performance of this scheme for the Euler equations is demonstrated in [12]. Billett and Toro =-=[1]-=- consider the weighted averagesux (WAF) scheme which allows time steps with Courant numbers up to two, see [36]. Finally, we note that unconditionally stable explicit methods, based on rational Runge-... |

16 | Random choice solution of hyperbolic systems
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- 1976
(Show Context)
Citation Context ...ks, rarefaction waves or contact discontinuities, see Smoller [34]. For some conservation laws, for instance the Euler equations of gas dynamics, the Riemann problem can be solved exactly. See Chorin =-=[2]-=- for a scheme where the exact Riemann solution is approximated. For many applications it is not necessary or possible to get an exact solution of the Riemann problem. Therefore approximative Riemann s... |

12 |
Large time step shock-capturing techniques for scalar conservation laws
- LeVeque
- 1982
(Show Context)
Citation Context ... is satised if the space stencil of the scheme is automatically increased with a larger time step. A generalization of the Godunov scheme which allows larger time steps was developed by LeVeque, see [=-=20]-=-, [22]. In the following we will refer to this scheme as the LTS Godunov-type scheme. Here the interaction of waves corresponding to neighboring Riemann problems is approximated as a linear superposit... |

11 | Error bounds for the methods of Glimm, Godunov and LeVeque
- Lucier
- 1985
(Show Context)
Citation Context ...ps with Courant number larger then one at least if the solution is smooth. Numerical results conrm this. Further evidence for the advantage of using larger time steps is the error estimate of Lucier [=-=29]-=- for a shock-capturing scheme for scalar conservation laws of LeVeque [20], which is similar to the LTS Godunov-type scheme considered here. 5 Entropy consistency In this section we consider the speci... |

9 |
Uber die partiellen Dierenzengleichungen der mathematischen Physik, Mathematische Annalen 100
- Courant, Lewy
- 1928
(Show Context)
Citation Context ...he cell interface between the mesh cells i 1 and i. This is a special and, as we see below, more restrictive than necessary form of the Courant-Friedrich-Lewy (CFL) condition, which was introduced in =-=[3]-=-. In general the CFL condition says, see [3], that a necessary condition for a numerical scheme to be stable and therefore convergent is that its numerical domain of dependence contains the true domai... |

9 | Nonlinear conservation laws and finite volume methods, in: Computational methods for astrophysical fluid flow, (Eds) LeVeque
- LeVeque
- 1998
(Show Context)
Citation Context ...e descriptions of the theory of hyperbolic conservation laws as well as the concepts of numerical schemes can be found in the text books of Godlewski, Raviart [4], [5], Kroner [16] and LeVeque [26], [=-=28-=-]. Here we brie y want to cover those denitions and theoretical results which will be needed for our considerations. The solution to the initial value problem (1), (2) is in general not continuous, i.... |

8 | An unconditionally stable method for the Euler equations
- Holden, Lie, et al.
- 1999
(Show Context)
Citation Context ...Godunov type schemes. An unconditionally stable scheme based on Front tracking was developed by Holden and Risebro [13]. The good performance of this scheme for the Euler equations is demonstrated in =-=[12]-=-. Billett and Toro [1] consider the weighted averagesux (WAF) scheme which allows time steps with Courant numbers up to two, see [36]. Finally, we note that unconditionally stable explicit methods, ba... |

5 |
On dierence approximations and entropy conditions for shocks
- Harten, Hyman, et al.
- 1976
(Show Context)
Citation Context ...easing then the solution of the scalar conservation law at a later time consists of shocks which may cross each other. The entropy consistency for monotone schemes was proved by Harten, Hyman and Lax =-=[9]-=-. Hence, it follows, that the LTS Godunov scheme leads to an entropy consistent approximation as long as the initial values are monotone. This means especially that the linear superposition of arbitra... |

5 |
Riemann solvers and numerical methods for dynamics
- Toro
- 1997
(Show Context)
Citation Context ...the Riemann problem. Therefore approximative Riemann solvers were developed. One which is often used is the Roe solver, see [31] or any text book on numerical schemes for conservation laws [5], [26], =-=[35]-=-. The use of the Roe Riemann solver for an unconditionally stable extension of a Godunov-type scheme is considered in Section 4.1. As long as the waves coming from neighboring Riemann problems do not ... |

3 |
clawpack software. available from http://www.amath.washington.edu/~ claw
- LeVeque
(Show Context)
Citation Context ...entropy inequality, a special form of the entropy condition, is used to pick out the physical relevant weak solution. The concept of the Godunov scheme is implemented in the software packagesCLAWPACK =-=[18]-=-, which was used for our calculations. It may be used with any Riemann solver. There the wave propagation algorithm developed by LeVeque [27] is used to update the solution u at every time step. Highr... |

3 |
On the analysis of volume methods for evolutionary problems
- Morton
- 1998
(Show Context)
Citation Context ...s. Numerical approximation always leads to the introduction of dissipation, which is especially strong at shocks. The numerical propagation of waves may also be distorted by a phase error, see Morton =-=[30]-=- for a recent discussion. Even one-dimensional problems may provide a challenge to modern highresolution schemes, when numerical results for long time dynamics are required. Examples of such practical... |

3 |
Billett,S.J.: A unified Riemann-problem-based extension of the Warming-beam and Lax-Wendroff schemes
- Toro
(Show Context)
Citation Context ...formance of this scheme for the Euler equations is demonstrated in [12]. Billett and Toro [1] consider the weighted averagesux (WAF) scheme which allows time steps with Courant numbers up to two, see =-=[36]-=-. Finally, we note that unconditionally stable explicit methods, based on rational Runge-Kutta methods, have been considered for parabolic equations in the past, see Hairer [7] and Satofuka [32]. They... |

2 |
Unconditionally stable explicit methods for parabolic equations
- Hairer
- 1980
(Show Context)
Citation Context ...numbers up to two, see [36]. Finally, we note that unconditionally stable explicit methods, based on rational Runge-Kutta methods, have been considered for parabolic equations in the past, see Hairer =-=[7]-=- and Satofuka [32]. They were also used insow calculations, see Satofuka [33]. 4 The large time step Godunov scheme of LeVeque Now we want to consider the LTS Godunov scheme of LeVeque [20], [21], [22... |

2 |
Hyperbolic conservation laws and numerical methods. Von Karman Institute of Fluid Dynamics Lecture Series
- LeVeque
- 1990
(Show Context)
Citation Context ...d Satofuka [32]. They were also used insow calculations, see Satofuka [33]. 4 The large time step Godunov scheme of LeVeque Now we want to consider the LTS Godunov scheme of LeVeque [20], [21], [22], =-=[25-=-] applied to one dimensional systems of conservation laws. As we have seen in Section 2.2 the classical concept of the Godunov scheme, where the solution u is dened as the combination of the solution ... |

2 |
Large time step generalizations of Glimm's scheme for systems of conservation laws
- Wang
- 1988
(Show Context)
Citation Context ...roblems is approximated as a linear superposition. We will review this approach in the next section. A generalization of the Glimm scheme to an unconditionally stable scheme was considered by Wang in =-=[37-=-]. In dierence to the LTS Godunov-type Title Suppressed Due to Excessive Length 9 scheme the piecewise constant initial values for the next time step are obtained by taking the exact or approximative ... |

2 |
On entropy consistency of large time step schemes II. approximate Riemann solvers
- Wang, Warnecke
(Show Context)
Citation Context ...be dicult due to the fact that the scheme is not monotone. Some preliminary results concerning the entropy consistency of the LTS Godunov scheme and the LTS Glimm scheme where shown in Wang, Warnecke =-=[38]-=-, [39]. 4.1 The solution of the Riemann problem A necessary condition in order to get an entropy stable approximation is that the solution u n j (x; t) of each single Riemann problem is entropy consis... |

1 |
A random choice dierence scheme for hyperbolic conservation laws
- Harten, Lax
- 1981
(Show Context)
Citation Context ...Christiane Helzel and Gerald Warnecke the physical relevant solution of the problem considered. For the Godunov scheme we get a positive answer to the latter question from the Harten-Lax Theorem, see =-=[10]-=-, [11]. There the discrete form of the entropy inequality, a special form of the entropy condition, is used to pick out the physical relevant weak solution. The concept of the Godunov scheme is implem... |

1 |
Tadmor: High resolution schemes for conservation laws in the simulation of injection systems
- Iben, E
- 2000
(Show Context)
Citation Context .... Due to the accumulation of the numerical dissipation and phase error, currently available schemes produce useless results for the longer time periods that have to be considered, see Iben and Tadmor =-=[14-=-]. Other numerical problems arise if very dierent scales are modeled by the equations, for instance in low Mach numbersow calculations. In [15], Klein uses the unconditionally stable Godunov-type sche... |

1 |
CLAWPACK User Notes
- LeVeque
(Show Context)
Citation Context ...nts, see [27]. The CLAWPACK program considers very general problems, including conservation laws with a capacity function or conservation laws with source terms in one, two, and three dimensions, see =-=[19]-=-, [27] or [28] for details. The simplest way to extend a one dimensional method to a multidimensional scheme is to use a dimension splitting approach. For a two-dimensional problem one could for insta... |

1 |
Some preliminary results using a large time step generalization of the Godunov method
- LeVeque
- 1985
(Show Context)
Citation Context ...d the most right going wave are chosen in such a way that the jump in the initial values is equally distributed to all of these waves. A similar approximation of rarefaction waves was used in LeVeque =-=[2-=-3]. 4.2 A geometric interpretation of the entropy-x In order to approximate the correct cell average for a rarefaction wave, we do not always need to use several waves. In Figure 4 dierent possibiliti... |

1 |
Second order accuracy of Brenier's time-discrete method for nonlinear systems of conservation laws
- LeVeque
- 1988
(Show Context)
Citation Context ...systems is given. For smooth solutions the error made during one time step due to the linear superposition is of the order O(k 3 ) which gives a contribution to the global error of order O(k 2 ), see =-=[24]-=- for a proof. The projection error introduced in every time step is O(h 2 ) which gives a contribution of O(h 2 =k) to the global error. Hence the global error of the LTS Godunov scheme is c 1 k 2 + c... |

1 |
A new explicit method for the numerical solution of parabolic dierential equations. Numerical properties and methodologies
- Satofuka
- 1981
(Show Context)
Citation Context ..., see [36]. Finally, we note that unconditionally stable explicit methods, based on rational Runge-Kutta methods, have been considered for parabolic equations in the past, see Hairer [7] and Satofuka =-=[32]-=-. They were also used insow calculations, see Satofuka [33]. 4 The large time step Godunov scheme of LeVeque Now we want to consider the LTS Godunov scheme of LeVeque [20], [21], [22], [25] applied to... |

1 |
Numerical solution of dynamic equations using rational Runge-Kutta methods
- Satofuka
- 1986
(Show Context)
Citation Context ...plicit methods, based on rational Runge-Kutta methods, have been considered for parabolic equations in the past, see Hairer [7] and Satofuka [32]. They were also used insow calculations, see Satofuka =-=[33]-=-. 4 The large time step Godunov scheme of LeVeque Now we want to consider the LTS Godunov scheme of LeVeque [20], [21], [22], [25] applied to one dimensional systems of conservation laws. As we have s... |