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Construction and Application of an AMR Algorithm for Distributed Memory Computers
 PROC. OF CHICAGO WORKSHOP ON ADAPTIVE MESH REFINEMENT METHODS
, 2003
"... ... In this paper, a localitypreserving domain decomposition is proposed that partitions the entire AMR hierarchy from the base level on. It is shown that the approach reduces the communication costs and simplifies the implementation. Emphasis is put on the effective parallelization of the flux cor ..."
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Cited by 25 (11 self)
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... In this paper, a localitypreserving domain decomposition is proposed that partitions the entire AMR hierarchy from the base level on. It is shown that the approach reduces the communication costs and simplifies the implementation. Emphasis is put on the effective parallelization of the flux correction procedure at coarsefine boundaries, which is indispensable for conservative finite volume schemes. An easily reproducible standard benchmark and a highly resolved parallel AMR simulation of a diffrracting hydrogenoxygen detonation demonstrate the proposed strategy in practice.
A Wave Propagation Method for Three Dimensional Hyperbolic Problems
, 1996
"... A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. ..."
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Cited by 23 (5 self)
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A class of wave propagation algorithms for threedimensional conservation laws is developed. This unsplit nite volume method is based on solving onedimensional Riemann problems at the cell interfaces and applying fluxlimiter functions to suppress oscillations arising from second derivative terms. Waves emanating from the Riemann problem are further split by solving Riemann problems in the transverse direction to model crossderivative terms. Due to proper upwinding, the method is stable for Courant numbers up to one. Several examples using the Euler equations are included.
A Cartesian Grid FiniteVolume Method for the AdvectionDiffusion Equation in Irregular Geometries
 J. COMPUT. PHYS
, 1999
"... We present a fully conservative, highresolution, finite volume algorithm for advectiondiffusion equations in irregular geometries. The algorithm uses a Cartesian grid in which some cells are cut by the embedded boundary. A novel feature is the use of a "capacity function" to model the fact that so ..."
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Cited by 9 (2 self)
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We present a fully conservative, highresolution, finite volume algorithm for advectiondiffusion equations in irregular geometries. The algorithm uses a Cartesian grid in which some cells are cut by the embedded boundary. A novel feature is the use of a "capacity function" to model the fact that some cells are only partially available to the fluid. The advection portion then uses the explicit wavepropagation methods implemented in clawpack, and is stable for Courant numbers up to 1. Diffusion is modelled with an implicit finite volume algorithm. Results are shown for several geometries. Convergence if verified and the 1norm order of accuracy is found to between 1.2 and 2 depending on the geometry and Peclet number. Software is available on the web.
An Adaptive Finite Difference Method for Traveltime and Amplitude
 GEOPHYSICS
, 1999
"... The eikonal equation with point source is difficult to solve with high order accuracy because of the singularity of the solution at the source. All the formally high order schemes turn out to be first order accurate without special treatment of this singularity. Adaptive upwind finite difference met ..."
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Cited by 9 (5 self)
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The eikonal equation with point source is difficult to solve with high order accuracy because of the singularity of the solution at the source. All the formally high order schemes turn out to be first order accurate without special treatment of this singularity. Adaptive upwind finite difference methods based on high order ENO (Essentially NonOscillatory) RungeKutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than fixed grid methods. Reliable auxiliary quantities, such as takeoff angle and geometrical spreading factor, are byproducts.
Adaptive Mesh Refinement for conservative systems: multidimensional efficiency evaluation
"... Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro and magnetohydrodynamic simulations, AMR is used in combination with several shockcapturi ..."
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Cited by 8 (2 self)
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Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro and magnetohydrodynamic simulations, AMR is used in combination with several shockcapturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multidimensional calculations and only 1.5  8 % overhead is observed. For AMR calculations of multidimensional magnetohydrodynamic problems, several strategies for controlling the r B = 0 constraint are examined. Three source term approaches suitable for cellcentered B representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing LaxFriedrichs method on the coarser levels. This leveldependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.
An Adaptive FiniteDifference Method for Traveltimes and Amplitudes
 GEOPHYSICS
, 1999
"... The point source traveltime field has an upwind singularity at the source point. Consequently, all formally highorder finitedifference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finitedifference methods based on highorder Weighted Essentially NonO ..."
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Cited by 8 (1 self)
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The point source traveltime field has an upwind singularity at the source point. Consequently, all formally highorder finitedifference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finitedifference methods based on highorder Weighted Essentially NonOscillatory (WENO) RungeKutta difference schemes for the paraxial eikonal equation overcome this difficulty. The method controls error by automatic grid refinement and coarsening based on an a posteriori error estimation. It achieves prescribed accuracy at far lower cost than does the fixedgrid method. Moreover, the achieved high accuracy of traveltimes yields reliable estimates of auxiliary quantities such as takeoff angles and geometrical spreading factors.
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 8 (3 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
Finite Volume Methods and Adaptive Refinement for Tsunami Propagation and Inundation
, 2006
"... ..."
Nonlinear Conservation Laws and Finite Volume Methods for Astrophysical Fluid Flow
 Computational Methods for Astrophysical Fluid Flow, 27th SaasFee Advanced Course Lecture Notes
, 1998
"... Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Cited by 7 (0 self)
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Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans