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A limiter for PPM that preserves accuracy at smooth extrema
 J. Comput. Phys
"... We present a new limiter for the PPM method of Colella and Woodward [4] that preserves accuracy at smooth extrema. It is based on constraining the interpolated values at extrema (and only at extrema) using nonlinear combinations of various difference approximations of the second derivatives. Otherwi ..."
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We present a new limiter for the PPM method of Colella and Woodward [4] that preserves accuracy at smooth extrema. It is based on constraining the interpolated values at extrema (and only at extrema) using nonlinear combinations of various difference approximations of the second derivatives. Otherwise, we use a standard geometric limiter to preserve monotonicity away from extrema. This leads to a method that has the same accuracy for smooth initial data as the underlying PPM method without limiting, while providing sharp, nonoscillatory representations of discontinuities. Key words: Upwind methods, PPM, limiters. 1
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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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.
W.S.: Scalability Challenges for Massively Parallel AMR Applications
 In: Proceedings of the 23rd IEEE International Parallel and Distributed Processing Symposium
, 2009
"... PDE solvers using Adaptive Mesh Refinement on block structured grids are some of the most challenging applications to adapt to massively parallel computing environments. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to thousands of processors on the Cray X ..."
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PDE solvers using Adaptive Mesh Refinement on block structured grids are some of the most challenging applications to adapt to massively parallel computing environments. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to thousands of processors on the Cray XT4. The optimization process also uncovered OSrelated performance variations that were not explained by conventional OS interference benchmarks. Ultimately the variability was traced back to complex interactions between the application, system software, and the memory hierarchy. Once identified, software modifications to control the variability improved performance by 20 % and decreased the variation in computation time across processors by a factor of 3. These newly identified sources of variation will impact many applications and suggest new benchmarks for OSservices be developed. 1
An embedded boundary method for the NavierStokes equations on a timedependent domain
 Comm. App. Math. Comp. Sci
"... mathematical sciences publishersCOMM. APP. MATH. AND COMP. SCI. ..."
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mathematical sciences publishersCOMM. APP. MATH. AND COMP. SCI.
Petascale BlockStructured AMR Applications Without Distributed Metadata,” EuroPar 2011
 Parallel Processing17th International Conference, EuroPar 2011, August 29  September 2, 2011, Proceedings, Part II. Lecture Notes in Computer Science 6853 Springer 2011
, 2011
"... Abstract. Adaptive mesh refinement (AMR) applications to solve partial differential equations (PDE) are very challenging to scale efficiently to the petascale regime. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to petascale on the Cray XT5. We describe a ..."
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Abstract. Adaptive mesh refinement (AMR) applications to solve partial differential equations (PDE) are very challenging to scale efficiently to the petascale regime. We describe optimizations to the Chombo AMR framework that enable it to scale efficiently to petascale on the Cray XT5. We describe an example of a hyperbolic solver (inviscid gas dynamics) and an matrixfree geometric multigrid elliptic solver. Both show good weak scaling to 131K processors without any threadlevel or SIMD vector parallelism. This paper describes the algorithms used to compress the Chombo metadata and the optimizations of the Chombo infrastructure that are necessary for this scaling result. That we are able to achieve petascale performance without distribution of the metadata is a significant advance which allows for much simpler and faster AMR codes. 1
A SecondOrder Accurate Conservative FrontTracking Method in One Dimension
 SIAM J. Sci. Comput
, 2010
"... Abstract. This paper presents a conservative fronttracking method for shocks and contact discontinuities that is secondorder accurate. It is based on a volumeoffluid method that treats the moving front with concepts similar to those of an embeddedboundary method. Special care is taken in the ce ..."
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Abstract. This paper presents a conservative fronttracking method for shocks and contact discontinuities that is secondorder accurate. It is based on a volumeoffluid method that treats the moving front with concepts similar to those of an embeddedboundary method. Special care is taken in the centering of the data to ensure the right order of accuracy at the front, and a redistribution step guarantees conservation. A suite of test problems, for tracking both shocks and contact discontinuities, is presented that confirms that the method is secondorder accurate.
A THREEDIMENSIONAL, UNSPLIT GODUNOV METHOD FOR SCALAR CONSERVATION LAWS ∗
"... Abstract. Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transpo ..."
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Abstract. Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transport in Darcy models of saturated subsurface flow. An important characteristic of these problems is that the velocity field is not known analytically. Instead, an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization. In this paper, we present a customized threedimensional finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles. The method is an extension of an algorithm by Bell, Dawson, and Shubin (BDS), which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions. The original BDS algorithm is a variant of unsplit, higherorder Godunov methods based on construction of a limited bilinear profile within each computational cell. Here we present a threedimensional extension of the original BDS algorithm that is based on a limited trilinear profile within each cell. We compare this new method to several other unsplit approaches, including piecewise linear methods, piecewise parabolic methods, and wave propagation schemes. Key words. Godunov method, scalar conservation law, linear advection AMS subject classifications. 3504, 35L65 DOI. 10.1137/100809520
MINIMAL ROTATIONALLY INVARIANT BASES FOR HYPERELASTICITY
, 2004
"... Rotationally invariant polynomial bases of the hyperelastic strain energy function are rederived using methods of group theory, invariant theory, and computational algebra. A set of minimal basis functions is given for each of the 11 Laue groups, with a complete set of rewriting syzygies. The ideal ..."
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Rotationally invariant polynomial bases of the hyperelastic strain energy function are rederived using methods of group theory, invariant theory, and computational algebra. A set of minimal basis functions is given for each of the 11 Laue groups, with a complete set of rewriting syzygies. The ideal generated from this minimal basis agrees with the classic work of Smith and Rivlin [Trans. Amer. Math. Soc., 88 (1958), pp. 175–193]. However, the structure of the invariant algebra described here calls for fewer terms, beginning with the fourth degree in strain, for most groups.
An Unsplit Godunov Method for Ideal MHD via Constrained Transport
, 2005
"... We describe a single step, secondorder accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, the corner transport upwind (CTU) method of Colella for multidimensional integration, and the constrained transport (CT) algori ..."
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We describe a single step, secondorder accurate Godunov scheme for ideal MHD based on combining the piecewise parabolic method (PPM) for performing spatial reconstruction, the corner transport upwind (CTU) method of Colella for multidimensional integration, and the constrained transport (CT) algorithm for preserving the divergencefree constraint on the magnetic field. We adopt the most compact form of CT, which requires the field be represented by areaaverages at cell faces. We demonstrate that the fluxes of the areaaveraged field used by CT can be made consistent with the fluxes of the volumeaveraged field returned by a Riemann solver if they obey certain simple relationships. We use these relationships to derive new algorithms for constructing the CT fluxes at grid cell corners which reduce exactly to the equivalent onedimensional solver for planeparallel, gridaligned flow. We show that the PPM reconstruction algorithm must include multidimensional terms for MHD, and we describe a number of important extensions that must be made to CTU in order for it to be used for MHD with CT. We present the results of a variety of test problems to demonstrate the method is accurate and robust. Key words: PACS: