## Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems (1998)

Venue: | SIAM J. Numer. Anal |

Citations: | 52 - 7 self |

### BibTeX

@ARTICLE{Berger98adaptivemesh,

author = {Marsha J. Berger and Randall and J. Leveque},

title = {Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems},

journal = {SIAM J. Numer. Anal},

year = {1998},

volume = {35},

pages = {2298--2316}

}

### OpenURL

### Abstract

Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ high-resolution wave-propagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.

### Citations

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Citation Context ... heterogeneous medium, can be found in [14] and the documentation with [11]. The wave-propagation algorithm is extended to nonlinear systems of conservation laws using a Roe approximate Riemann solver=-=[16]-=-, which linearizes the problem at each cell interface in such a way that the wave-propagation approach is guaranteed to be conservative. The general wavepropagation algorithm is written symbolically a... |

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Citation Context ...ed in various ways, (e.g. by introducing slope information), and "slope-limiters" or "fluxlimiters " are then used to give good resolution of discontinuities without spurious oscil=-=lations. See, e.g., [12] for a gen-=-eral discussion of such methods. The wave-propagation algorithms are based on using the waves directly to update cell values, including second order corrections with "wave limiters". For con... |

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Citation Context ...e solution, and for time-dependent problems, the region of refinement must move adaptively with the interesting structure. An effective adaptive mesh refinement (AMR) strategy has been developed [1], =-=[2]-=-, [3], [4], [5], [6] that uses Cartesian grids, with refinement in both space and time, over rectangular patches. The refinement is by an arbitrary even integer ratio (typically 4), and further recurs... |

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Citation Context ... conservation laws. A crucial ingredient is the manner in which fluxes at grid-refinement interfaces are coordinated to insure that the method is globally conservative. This is described in detail in =-=[5]-=- and reviewed in Section 2.1. In this paper we show how the wave-propagation algorithm can be used in conjunction with this AMR strategy. This algorithm is written in a more general form that does not... |

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Citation Context ...ed to the "Strang splitting" in which one advances first by a half time step on one equation, then by a full time step on the other equation, and ends with a half time step on the first equa=-=tion again[17]-=-, [18]. Formally the Strang splitting can give second order accuracy in situations where only first order accuracy is achieved with our choice, but in practical problems where methods of this nature a... |

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Citation Context ...\Gamma k h ( ~ F i+1 \Gamma ~ F i ) = q i \Gamma k h (F i+1 \Gamma F i ) where F i = uq i\Gamma1 + ~ F i : This is the numerical flux for a standard flux-limiter method on the advection equation (see =-=[19]-=-, for example). In particular, if no limiter is used and ~ W i j W i , this reduces to the Lax-Wendroff method. The more general form (9) is not in flux-differencing form, but works just as effectivel... |

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Citation Context ...mula (7) would then become q n+1 i = q i \Gamma k h (u + i\Gamma1=2 (q i \Gamma q i\Gamma1 ) + u \Gamma i+1=2 (q i+1 \Gamma q i )): (8) This has advantages in two-dimensional incompressible flow (see =-=[13]-=-), but for illustration here we will use the formulation (7), which is based on the cell-centered velocites (6) rather than edge values. High-resolution second-order corrections are easily introduced,... |

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Citation Context ...and using the resulting wave structure to update the solution in the grid cell to each side. This is, of course, the basis for a host of methods for conservation laws, dating back to Godunov’s method =-=[8]-=-. The Riemann problem consists of the original conservation law together with piecewise constant initial data defined by the two neighboring cell values. For a wide class of conservation laws, the Rie... |

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Citation Context ... interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available. 1 Introduction The multi-dimensional wave-propagation algorithm described in =-=[14] is a &quo-=-t;high-resolution" method that is second order accurate on smooth solutions while maintaining sharp discontinuities through the use of slope-limiters. While based on ideas developed for hyperboli... |

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Citation Context ...oid refining too many cells, but this must be balanced with the competing desire to create relatively few separate patches, and to minimize computational overhead on the boundaries of fine grids. See =-=[1]-=- for more details on the refinement and clustering algorithms. See Figure 6 for an example of refinement on logically rectangular patches. An alternative would be to use a quadtree data structure (e.g... |

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Citation Context ...the "Strang splitting" in which one advances first by a half time step on one equation, then by a full time step on the other equation, and ends with a half time step on the first equation a=-=gain[17], [18]-=-. Formally the Strang splitting can give second order accuracy in situations where only first order accuracy is achieved with our choice, but in practical problems where methods of this nature are use... |

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Citation Context ...ution, and for time-dependent problems, the region of refinement must move adaptively with the interesting structure. An effective adaptive mesh refinement (AMR) strategy has been developed [1], [2], =-=[3]-=-, [4], [5], [6] that uses Cartesian grids, with refinement in both space and time, over rectangular patches. The refinement is by an arbitrary even integer ratio (typically 4), and further recursive r... |

27 | A wave propagation method for three-dimensional hyperbolic conservation laws
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Citation Context ...l for some applications. Adaptive mesh refinement is even more crucial in solving three-dimensional problems than in two space dimensions. Recently clawpack has been extended to three dimensions [9], =-=[10]-=- and work is underway to extend the amrclaw version. The underlying AMR code for the Euler equations has previously been successfully extended to three dimensions (e.g., [1], [20]) and the ideas prese... |

14 |
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Citation Context ...d in a general software package (in fortran) called clawpack (Conservation LAWs PACKage, a holdover from earlier versions that applied only to conservation laws). This package is available from netlib=-=[11]-=-. This software has recently been combined with Berger's implementation of AMR, incorporating the extensions described in this paper. This is now freely available as the amrclaw package[6]. All of the... |

10 |
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Citation Context ...r more details on the refinement and clustering algorithms. See Figure 6 for an example of refinement on logically rectangular patches. An alternative would be to use a quadtree data structure (e.g., =-=[7]-=-, [15]) in which only the flagged cells are refined, but the storage overhead of these data structures, typically 30 to 50 words per cell, usually exceeds the storage overhead associated with the bloc... |

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4 | Three-dimensional Euler computations using clawpack
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(Show Context)
Citation Context ...rucial for some applications. Adaptive mesh refinement is even more crucial in solving three-dimensional problems than in two space dimensions. Recently clawpack has been extended to three dimensions =-=[9]-=-, [10] and work is underway to extend the amrclaw version. The underlying AMR code for the Euler equations has previously been successfully extended to three dimensions (e.g., [1], [20]) and the ideas... |

2 |
amrclaw software. A test version is available at http://www.amath.washington.edu/~rjl/amrclaw
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(Show Context)
Citation Context ...from netlib[11]. This software has recently been combined with Berger's implementation of AMR, incorporating the extensions described in this paper. This is now freely available as the amrclaw package=-=[6]-=-. All of the numerical results presented in this paper were obtained with amrclaw, and several such examples are included with the package. Details on the use of this package can be found in the on-li... |

2 |
Some Aspects of the Computational Dynamics of Colliding Flows
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(Show Context)
Citation Context ...hree dimensions [9], [10] and work is underway to extend the amrclaw version. The underlying AMR code for the Euler equations has previously been successfully extended to three dimensions (e.g., [1], =-=[20]-=-) and the ideas presented in this paper carry over easily as well. (a) -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 title (b) -0.5-0.4-0.3-0.2-0.1 0 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag ... |

2 |
clawpack software, available online from http://www.amath.washington. edu/˜claw
- LeVeque
(Show Context)
Citation Context ... in a general software package (in fortran) called clawpack (Conservation LAWs PACKage, a holdover from earlier versions that applied only to conservation laws). This package is available from netlib =-=[11]-=-. This software has recently been combined with Berger’s implementation of AMR, incorporating the extensions described in this paper. This is now freely available as the amrclaw package [6]. All of th... |