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Gerris: A TreeBased Adaptive Solver For The Incompressible Euler Equations In Complex Geometries
 J. Comp. Phys
, 2003
"... An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Sec ..."
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Cited by 100 (16 self)
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An adaptive mesh projection method for the timedependent incompressible Euler equations is presented. The domain is spatially discretised using quad/octrees and a multilevel Poisson solver is used to obtain the pressure. Complex solid boundaries are represented using a volumeoffluid approach. Secondorder convergence in space and time is demonstrated on regular, statically and dynamically refined grids. The quad/octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features. The source code of the method implementation is freely available.
A second order Coupled Level Set and VolumeofFluid Method for . . .
, 2002
"... We present a coupled Level Set / VolumeofFluid (CLSVOF) method for computing growth and collapse of vapor bubbles. The liquid is assumed incompressible and the vapor is assumed to have constant pressure in space. Second order algorithms are used for nding "mass conserving" extension vel ..."
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Cited by 66 (6 self)
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We present a coupled Level Set / VolumeofFluid (CLSVOF) method for computing growth and collapse of vapor bubbles. The liquid is assumed incompressible and the vapor is assumed to have constant pressure in space. Second order algorithms are used for nding "mass conserving" extension velocities, for discretizing the local interfacial curvature and also for the discretization of the cell centered projection step. Convergence studies are given that demonstrate this second order accuracy. Examples are provided that apply to cavitating bubbles.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
"... ..."
An unsplit, cellcentered Godunov method for ideal MHD
 JOURNAL OF COMPUTATIONAL PHYSICS 203 (2005) 422–448
, 2005
"... We present a secondorder Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergencefree condition of the ..."
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Cited by 22 (3 self)
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We present a secondorder Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergencefree condition of the magnetic fields, we apply a discrete projection to the intermediate values of the field at cell faces, and apply a filter to the primary dependent variables at the end of each time step. We test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions. The test suite includes rotated planar linear waves, MHD shock tube problems, lowbeta flux tubes, and a magnetized rotor problem. For all of these cases, we observe that the algorithm is secondorder accurate for smooth solutions, converges to the correct weak solution for problems involving shocks, and exhibits no evidence of instability or loss of accuracy due to the possible presence of nonsolenoidal fields.
Chombo software package for AMR applications design document
, 2003
"... Disclaimer This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employ ..."
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Cited by 22 (6 self)
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Disclaimer This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States
SIMULATING THE FLUID DYNAMICS OF NATURAL AND PROSTHETIC HEART VALVES USING THE IMMERSED BOUNDARY METHOD
, 2009
"... The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluidstructure interaction. In the present work, we describe the application of the immersed boundary method to the simulation of the fluid dynamics of heart valves, including a ..."
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Cited by 19 (6 self)
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The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluidstructure interaction. In the present work, we describe the application of the immersed boundary method to the simulation of the fluid dynamics of heart valves, including a model of a natural aortic valve and a model of a chorded prosthetic mitral valve. Each valve is mounted in a semirigid flow chamber. In the case of the mitral valve, the flow chamber is a circular pipe, and in the case of the aortic valve, the flow chamber is a model of the aortic root. The model valves and flow chambers are immersed in a viscous incompressible fluid, and realistic fluid boundary conditions are prescribed at the upstream and downstream ends of the chambers. To connect the immersed boundary models to the boundaries of the fluid domain, we introduce a novel modification of the standard immersed boundary scheme. In particular, near the outer boundaries of the fluid domain, we modify the construction of the regularized delta function which mediates fluidstructure coupling in the immersed boundary method, whereas in the interior of the fluid domain, we employ a standard fourpoint delta function which is frequently used with the immersed boundary method. The standard delta
Performance and scaling of locallystructured grid methods for partial differential equations
 Journal of Physics: Conference Series
"... Abstract. In this paper, we discuss some of the issues in obtaining high performance for blockstructured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and sc ..."
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Cited by 18 (3 self)
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Abstract. In this paper, we discuss some of the issues in obtaining high performance for blockstructured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach. Introduction A broad range of applied PDE problems exhibit multiscale behavior, i.e. variation in the solution over scales that are much smaller than the global large scales in the problem. Examples include flame fronts arising in the burning of hydrocarbon fuels and nuclear burning in supernovae; in geophysical problems, ocean currents, effects of localized features in orography or bathymetry, and tropical cyclones; and in plasma physics, a variety of small scale effects due to nonlinear instabilities and localized kinetic effects. In all of these problems, the fundamental mathematical description is given in terms of various combinations of PDE of classical type (elliptic, parabolic, hyperbolic). To effectively compute solutions to such problems, we need simulation capabilities with the following features.
Adaptive Mesh Refinement in Titanium
 In: Proceedings of the 19th International Parallel and Distributed Processing Symposium (IPDPS
, 2005
"... In this paper, we evaluate Titanium’s usability as a highlevel parallel programming language through a case study, where we implement a subset of Chombo’s functionality in Titanium. Chombo is a software package applying the Adaptive Mesh Refinement methodology to numerical Partial Differential Equa ..."
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Cited by 14 (3 self)
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In this paper, we evaluate Titanium’s usability as a highlevel parallel programming language through a case study, where we implement a subset of Chombo’s functionality in Titanium. Chombo is a software package applying the Adaptive Mesh Refinement methodology to numerical Partial Differential Equations at the production level. In Chombo, the library approach is used to parallel programming (C++ and Fortran, with MPI), whereas Titanium is a Java dialect designed for highperformance scientific computing. The performance of our implementation is studied and compared with that of Chombo in solving Poisson’s equation based on two grid configurations from a real application. Also provided are the counts of lines of code from both sides.
On the Volume Conservation of the Immersed Boundary Method
, 2012
"... Abstract. The immersed boundary (IB) method is an approach to problems of fluidstructure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the f ..."
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Cited by 12 (4 self)
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Abstract. The immersed boundary (IB) method is an approach to problems of fluidstructure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volumeconservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finitedifference discretizations. In this paper, we use quasistatic and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volumeconservation properties of a formally secondorder accurate IB method. We consider both collocated and staggeredgrid discretization methods. For the tests considered herein, the staggeredgrid IB scheme generally yields at least a modest improvement in volume conservation when compared to cellcentered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggeredgrid IB method are more than an order of magnitude smaller than those of
Block Structured Adaptive Mesh and Time Refinement for Hybrid, Hyperbolic + Nbody Systems
, 2007
"... We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the num ..."
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Cited by 11 (1 self)
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We present a new numerical algorithm for the solution of coupled collisional and collisionless systems, based on the block structured adaptive mesh and time refinement strategy (AMR). We describe the issues associated with the discretization of the system equations and the synchronization of the numerical solution on the hierarchy of grid levels. We implement a code based on a higher order, conservative and directionally unsplit Godunov’s method for hydrodynamics; a symmetric, time centered modified symplectic scheme for collisionless component; and a multilevel, multigrid relaxation algorithm for the elliptic equation coupling the two components. Numerical results that illustrate the accuracy of the code and the relative merit of various implemented schemes are also presented.