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## TIME- DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS

### Citations

559 |
Adaptive mesh refinement for hyperbolic partial differential equations
- Berger, Oliger
- 1984
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Citation Context ...been the subject of a great deal of recent attention (cf. Babuska et al. [8, 9) and Thompson [29]) and are generally capable of introducing finer- meshes in regions where greater resolution is needed =-=[1, 2, 3, 6, 14, 15, 22, 26]-=-, moving meshes in order to follow isolated dynamic phenomena [1, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific regions of the problem domain [17, 21]. The utility of such adapti... |

519 | Local adaptive mesh refinement for shock hydrodynamics - Berger, Colella - 1989 |

250 |
The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks
- Woodward, Colella
- 1984
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Citation Context ...inement Me#tod 65 We solve a problem where a Mach 10 shock in air (-y = 1.4) moves down a channel containing a wedge with a half-angle of thirty degrees. This problem was used by Woodward and Collela =-=[30]-=- to compare several finite difference schemes on uniform grids. Like them, we orient a rectangular computational domain, -0.3 _s x -s 3.4, 0 _5 y _5 1, so that the top edge of the wedge is on the bott... |

116 | Adaptive zoning for singular problems in two dimensions - Brackbill, Sal&man - 1982 |

80 | The Effect of Viscosity in Hypervelocity Impact Cratering. - MacCormack - 1969 |

38 |
A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations,
- Adjerid, Flaherty
- 1986
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Citation Context ...been the subject of a great deal of recent attention (cf. Babuska et al. [8, 9) and Thompson [29]) and are generally capable of introducing finer- meshes in regions where greater resolution is needed =-=[1, 2, 3, 6, 14, 15, 22, 26]-=-, moving meshes in order to follow isolated dynamic phenomena [1, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific regions of the problem domain [17, 21]. The utility of such adapti... |

30 |
On conservation at grid interfaces.
- Berger
- 1987
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Citation Context ... well as coarse grid data for the interpolation. The order and nature of the interpolation also needs further investigation, and we are studying methods that, for example, conserve fluxes (cf. Berger =-=[12]-=- or Rai [27]). 3. COMPUTATIONAL EXAMPLES In order to demonstrate the capabilities of the adaptive procedure described in Section 2, we applied it to three hyperbolic systems. We used a two-step MacCor... |

30 | Numerical solution of the atmospheric diffusion equation for chemically reactive flows. - McRae, Goodin, et al. - 1982 |

25 | On the Stability of Mesh Equidistribution Strategies for Time-Dependent Partial Differential Equations - Coyle, Flaherty, et al. - 1986 |

24 |
R.: The approximation theory for the p-version of the finite element method
- Dorr
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Citation Context ... needed [1, 2, 3, 6, 14, 15, 22, 26], moving meshes in order to follow isolated dynamic phenomena [1, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific regions of the problem domain =-=[17, 21]-=-. The utility of such adaptive techniques is greatly enhanced when they are capable of providing an estimate of the accuracy of the computed solution. Local error estimates are often used as refinemen... |

19 |
A Survey of Dynamically-Adaptive Grids in the Numerical Solution of Partial Differential Equations
- Thompson
- 1985
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Citation Context ...es that evolve with the solution offer a robust, reliable, and efficient alternative. Such techniques have been the subject of a great deal of recent attention (cf. Babuska et al. [8, 9) and Thompson =-=[29]-=-) and are generally capable of introducing finer- meshes in regions where greater resolution is needed [1, 2, 3, 6, 14, 15, 22, 26], moving meshes in order to follow isolated dynamic phenomena [1, 2, ... |

15 |
A two dimensional mesh moving technique for time dependent partial differential equations
- Arney, Flaherty
- 1986
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Citation Context ...ds greater reliability and efficiency to these methods. The three components of our adaptive algorithm are described in Section 2; however, frequent references are made to our previous investigations =-=[5, 6]-=-. A computer code based on the adaptive algorithm of Section 2 has been combined with a MacCormack finite difference scheme and an error indicator based on Richardson extrapolation. It has been used t... |

14 | An Adaptive Finite Element Method for Initial-Boundary Value Problems for Partial Differential Equations - Davis, Flaherty - 1982 |

14 | Generalized Coordinate Forms of Governing Fluid Equations and Associated Geometrically Induced Errors - Hindman - 1982 |

13 | Non-stationary Oblique Shock-Wave A Reflections - Ben-Dor, Glass - 1978 |

11 |
A Local Refinement Finite Element Method for Two-Dimensional Parabolic Systems
- Adjerid, Flaherty
- 1986
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Citation Context ...been the subject of a great deal of recent attention (cf. Babuska et al. [8, 9) and Thompson [29]) and are generally capable of introducing finer- meshes in regions where greater resolution is needed =-=[1, 2, 3, 6, 14, 15, 22, 26]-=-, moving meshes in order to follow isolated dynamic phenomena [1, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific regions of the problem domain [17, 21]. The utility of such adapti... |

11 |
An adaptive local mesh refinement method for time dependent partial differential equations
- Arney, Flaherty
- 1989
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11 | Adaptive Computational Methods for Partial Differential Equations, - Babuska, Chandra, et al. - 1984 |

10 |
A Moving Mesh Finite Element Method with Local Refinement for Parabolic Partial Differential Equations
- Adjerid, Flaherty
- 1986
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Citation Context |

10 | Domains and Boundaries of Non-stationary Oblique ShockGlass - Ben-Dor |

9 | Grid Evolution in Time Asymptotic Problems - Rai, Anderson - 1981 |

7 |
TVD Finite Difference Schemes and Artificial Viscosity
- Davis
- 1984
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Citation Context ...at it is usually not noticed in computations. The MacCormack finite difference scheme needs artificial viscosity to "capture" shocks without excessive oscillations. We used a model developed by Davis =-=[19]-=-, which is total variation, diminishing in one space dimension. This computational strategy is not competitive with the higher-order methods considered by Woodward and Collela [301 and Berger and Coll... |

6 |
Computing with high-resolution upwind schemes for hyperbolic equations
- CHAKRAVARTHY, OSHER
- 1990
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Citation Context ... needed [1, 2, 3, 6, 14, 15, 22, 26], moving meshes in order to follow isolated dynamic phenomena [1, 2, 5, 20, 22, 26, 281, or changing the order of methods in specific regions of the problem domain =-=[17, 21]-=-. The utility of such adaptive techniques is greatly enhanced when they are capable of providing an estimate of the accuracy of the computed solution. Local error estimates are often used as refinemen... |

6 |
Adaptive finite element methods for the analysis of inviscid compressible flow: Part I. Fast refinement/unrefinement and moving mesh methods for unstructured meshes. Computer Methods in Applied Mechanics and Engineering 59
- Oden, Strouboulis, et al.
- 1986
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5 |
Local uniform mesh refinement with moving grids,
- Gropp
- 1987
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4 |
Adaptive Refinement Methods for Non-Linear Parabolic Partial Differential Equations," Chap. 19 in Accuracy Estimates and Adaptive Refinements in Finite Element Computations
- Bieterman, Flaherty, et al.
- 1986
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Citation Context ...tation [ 13, 17, 30]. The present Richardson's extrapolation-based error indicator is expensive, and we are seeking ways of replacing it by techniques using p-refinement. Such methods have been shown =-=[1, 2, 3, 15, 21]-=- to have an excellent cost performance ratio when used in conjunction with finite element methods. An appropriate error indicator or estimator can be used to control a differential refinement algorith... |

3 |
An Adaptive Method With Mesh Moving and Local Mesh Refinement for Time-Dependent Partial Differential Equations
- Arney, Flaherty
- 1987
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Citation Context ...rectangular clusters and are moved toward the center of the nearest error cluster by a procedure similar to the one described in Section 2.1. Additional details are presented by Arney and Flaherty in =-=[7]-=-. A new base mesh can be generated whenever the existing one becomes severely distorted. Since this new mesh is created at a specific time, rather than by mesh motion, we refer to this process as stat... |

3 |
M.M.: Patched-Grid Calculations with the Euler and Navier-Stokes Equations: Theory and Application, NASA TM-88228
- Rai
- 1986
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Citation Context ...rse grid data for the interpolation. The order and nature of the interpolation also needs further investigation, and we are studying methods that, for example, conserve fluxes (cf. Berger [12] or Rai =-=[27]-=-). 3. COMPUTATIONAL EXAMPLES In order to demonstrate the capabilities of the adaptive procedure described in Section 2, we applied it to three hyperbolic systems. We used a two-step MacCormack finite ... |

2 | A posteriori error estimation of adaptive finite difference schemes for hyperbolic systems - ARNEY, BISWAS, et al. - 1987 |

1 | Accuracy Estimates and Adaptive Refinements in Finite Element Computations - GAGO, R, et al. |