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73
An unconditionally stable MacCormack method
, 2006
"... The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other nonreversible partial differential equations, it is useful for ..."
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Cited by 64 (16 self)
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The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other nonreversible partial differential equations, it is useful for often troublesome advection terms. The error estimate is used to correct the data before advection raising the method to second order accuracy, even though each individual step is only first order accurate. In this paper, we rewrite the MacCormack method to illustrate that it estimates the error in the same exact fashion as BFECC. The difference is that the MacCormack method uses this error estimate to correct the already computed forward advected data. Thus, it does not require the third advection step in BFECC reducing the cost of the method while still obtaining second order accuracy in space and time. Recent work replaced each of the three BFECC advection steps with a simple first order accurate unconditionally stable semiLagrangian method yielding a second order accurate unconditionally stable BFECC scheme. We use a similar approach to create a second order accurate unconditionally stable MacCormack method.
An accurate adaptive solver for surfacetensiondriven interfacial flows
 Journal of Computational Physics
, 2009
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Hierarchical RLE level set: A compact and versatile deformable surface representation
, 2006
"... This article introduces the Hierarchical RunLength Encoded (HRLE) Level Set data structure. This novel data structure combines the best features of the DTGrid (of Nielsen and Museth [2004]) and the RLE Sparse Level Set (of Houston et al. [2004]) to provide both optimal efficiency and extreme vers ..."
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Cited by 49 (9 self)
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This article introduces the Hierarchical RunLength Encoded (HRLE) Level Set data structure. This novel data structure combines the best features of the DTGrid (of Nielsen and Museth [2004]) and the RLE Sparse Level Set (of Houston et al. [2004]) to provide both optimal efficiency and extreme versatility. In brief, the HRLE level set employs an RLE in a dimensionally recursive fashion. The RLE scheme allows the compact storage of sequential nonnarrowband regions while the dimensionally recursive encoding along each axis efficiently compacts nonnarrowband planes and volumes. Consequently, this new structure can store and process level sets with effective voxel resolutions exceeding 500030003000 (45 billion voxels) on commodity PCs with only 1 GB of memory. This article, besides introducing the HRLE level set data structure and its efficient core algorithms, also describes numerous applications that have benefited from our use of this structure: our unified implicit object representation, efficient and robust mesh to level set conversion, rapid ray tracing, level set metamorphosis, collision detection, and fully sparse fluid simulation (including RLE vector and matrix representations.) Our comparisons of the popular octree level set and Peng level set structures to the HRLE level set indicate that the latter is superior in both narrowband sequential access speed and overall memory usage
Delaunay Deformable Models: Topologyadaptive Meshes Based on the Restricted Delaunay triangulation
, 2006
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An unconditionally stable fully conservative semilagrangian method
 J. Comput. Phys
"... SemiLagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFLbased time step restriction of the smallest cell can ..."
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Cited by 19 (10 self)
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SemiLagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFLbased time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semiLagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semiLagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressiblestyle pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had. 1.
Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids
"... When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentri ..."
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Cited by 17 (2 self)
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When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentric pressures. Second, meshes must align with potentially complex moving surfaces and boundaries, necessitating continuous remeshing. Unfortunately, sacrificing mesh quality in favour of speed yields inaccurate velocities and simulation artifacts. We describe how to eliminate the boundarymatching constraint by adapting recent embedded boundary techniques to tetrahedra, so that neither air nor solid boundaries need to align with mesh geometry. This enables the use of high quality, arbitrarily graded, nonconforming Delaunay meshes, which are simpler and faster to generate. Temporal coherence can also be exploited by reusing meshes over adjacent timesteps to further reduce meshing costs. Lastly, our free surface boundary condition eliminates the spurious currents that previous methods exhibited for slow or static scenarios. We provide several examples demonstrating that our efficient tetrahedral embedded boundary method can substantially increase the flexibility and accuracy of adaptive Eulerian fluid simulation.
TopologyAdaptive Mesh Deformation for Surface Evolution, Morphing, and MultiView Reconstruction
, 2009
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Adaptive characteristicsbased matching for compressible multifluid dynamics
 J. COMPUT. PHYS
, 2005
"... This paper presents an evolutionary step in sharp capturing of shocked, high Acoustic Impedance Mismatch (AIM) interfaces in an adaptive mesh refinement (AMR) environment. The central theme which guides the present development addresses the need to optimize between the algorithmic complexities in a ..."
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Cited by 14 (4 self)
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This paper presents an evolutionary step in sharp capturing of shocked, high Acoustic Impedance Mismatch (AIM) interfaces in an adaptive mesh refinement (AMR) environment. The central theme which guides the present development addresses the need to optimize between the algorithmic complexities in advanced front capturing and front tracking methods developed recently for high AIM interfaces with the simplicity requirements imposed by the AMR multilevel dynamic solutions implementation. The paper shows that we have achieved this objective by means of relaxing the strict conservative treatment of AMR prolongation/restriction operators in the interfacial region and by using a NaturalNeighborInterpolation (NNI) algorithm to eliminate the need for ghost cell extrapolation into the other fluid in a characteristicsbased matching (CBM) scheme. The later is based on a twofluid Riemann solver, which brings the accuracy and robustness of fronttracking approach into the fast local level set frontcapturing implementation of the CBM method. A broad set of test prob
Adaptive finite volume methods for distributed nonsmooth parameter identification
, 2007
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Advecting normal vectors: a new method for calculating interface normals and curvatures when modeling twophase flows
 J. Comput. Phys
"... In simulating twophase flows, interface normal vectors and curvatures are needed for modeling surface tension. In the traditional approach, these quantities are calculated from the spatial derivatives of a scalar function (e.g. the volumeoffluid or the level set function) at any instant in time. ..."
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Cited by 12 (2 self)
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In simulating twophase flows, interface normal vectors and curvatures are needed for modeling surface tension. In the traditional approach, these quantities are calculated from the spatial derivatives of a scalar function (e.g. the volumeoffluid or the level set function) at any instant in time. The orders of accuracy of normals and curvatures calculated from these functions are studied. A new method for calculating these quantities is then presented, where the interface unit normals are advected along with whatever function represents the interface, and curvatures are calculated directly from these advected normals. To illustrate this new approach, the volumeoffluid method is used to represent the interface, and the advected normals are used for interface reconstruction. The accuracy and performance of the new method are demonstrated via test cases with prescribed velocity fields. The results are compared with those of traditional approaches.