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GLIMM'S METHOD FOR GAS DYNAMICS

Description: We investigate Glimm's method, a method for constructing approximate solutions to systems of hyperbolic conservation laws in one space variable by sampling explicit wave solutions. It is extended to several space variables by operator splitting. We consider two functional problems. 1) We propose a highly accurate form of the sampling procedure, in one space variable, based on the van der Corput sampling sequence. We test the improved sampling procedure numerically in the case of inviscid compressible flow in one space dimension and find that it gives high resolution results both in the smooth parts of the solution, as well as the discontinuities. 2) We investigate the operator splitting procedure by means of which the multidimensional method is constructed. An 0(1) error stemming from the use of this procedure near shocks oblique to the spatial grid is analyzed numerically in the case of the equations for inviscid compressible flow in two space dimensions. We present a hybrid method which eliminates this error, consisting of Glimm's method, used in continuous parts of the flow, and the nonlinear Godunov's method, used in regions where large pressure jumps are generated. The resulting method is seen to be a substantial improvement over either of the component methods for multidimensional calculations.
Date: July 1, 1980
Creator: Colella, Phillip
Partner: UNT Libraries Government Documents Department

Final Report: Computational Fluid Dynamics and Combustion Dynamics, February 15, 1995 - February 14, 1998

Description: The HPCC Grand Challenge Project on Computational Fluid Dynamics and Combustion Dynamics focuses on the development of advanced numerical methodologies for modeling realistic engineering problems in combustion and other areas of fluid dynamics. The project was a collaboration between two DOE Laboratories (LBNL and LANL) and two universities (University of California, Berkeley, and New York University). In this document, we report on the work done under the UC Berkeley portion of the grant.
Date: September 22, 1998
Creator: Colella, Phillip
Partner: UNT Libraries Government Documents Department

A high-order finite-volume method for hyperbolic conservation laws on locally-refined grids

Description: We present a fourth-order accurate finite-volume method for solving time-dependent hyperbolic systems of conservation laws on Cartesian grids with multiple levels of refinement. The underlying method is a generalization of that in [5] to nonlinear systems, and is based on using fourth-order accurate quadratures for computing fluxes on faces, combined with fourth-order accurate Runge?Kutta discretization in time. To interpolate boundary conditions at refinement boundaries, we interpolate in time in a manner consistent with the individual stages of the Runge-Kutta method, and interpolate in space by solving a least-squares problem over a neighborhood of each target cell for the coefficients of a cubic polynomial. The method also uses a variation on the extremum-preserving limiter in [8], as well as slope flattening and a fourth-order accurate artificial viscosity for strong shocks. We show that the resulting method is fourth-order accurate for smooth solutions, and is robust in the presence of complex combinations of shocks and smooth flows.
Date: January 28, 2011
Creator: McCorquodale, Peter & Colella, Phillip
Partner: UNT Libraries Government Documents Department

A Filtering Method For Gravitationally Stratified Flows

Description: Gravity waves arise in gravitationally stratified compressible flows at low Mach and Froude numbers. These waves can have a negligible influence on the overall dynamics of the fluid but, for numerical methods where the acoustic waves are treated implicitly, they impose a significant restriction on the time step. A way to alleviate this restriction is to filter out the modes corresponding to the fastest gravity waves so that a larger time step can be used. This paper presents a filtering strategy of the fully compressible equations based on normal mode analysis that is used throughout the simulation to compute the fast dynamics and that is able to damp only fast gravity modes.
Date: April 25, 2005
Creator: Gatti-Bono, Caroline & Colella, Phillip
Partner: UNT Libraries Government Documents Department

Adaptive mesh refinement in titanium

Description: In this paper, we evaluate Titanium's usability as a high-level 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 high-performance 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.
Date: January 21, 2005
Creator: Colella, Phillip & Wen, Tong
Partner: UNT Libraries Government Documents Department

A stable and convergent scheme for viscoelastic flow in contraction channels

Description: We present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels. In our approach we split the viscoelastic terms of the Oldroyd-B constitutive equation using Duhamel's formula and discretize the resulting PDEs using a semi-implicit finite difference method based on a Lax-Wendroff method for hyperbolic terms. In particular, we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well-posed. A projection method is used to impose the incompressibility constraint. We are able to compute the full range of elastic flows in an abrupt contraction channel--from the viscous limit to the elastic limit--in a stable and convergent manner for elastic Mach numbers less than one. We demonstrate the method for unsteady Oldroyd-B and Maxwell fluids in planar contraction channels.
Date: February 15, 2004
Creator: Trebotich, David; Colella, Phillip & Miller, Gregory
Partner: UNT Libraries Government Documents Department

A cartesian grid embedded boundary method for the heat equationand poisson's equation in three dimensions

Description: We present an algorithm for solving Poisson's equation and the heat equation on irregular domains in three dimensions. Our work uses the Cartesian grid embedded boundary algorithm for 2D problems of Johansen and Colella (1998, J. Comput. Phys. 147(2):60-85) and extends work of McCorquodale, Colella and Johansen (2001, J. Comput. Phys. 173(2):60-85). Our method is based on a finite-volume discretization of the operator, on the control volumes formed by intersecting the Cartesian grid cells with the domain, combined with a second-order accurate discretization of the fluxes. The resulting method provides uniformly second-order accurate solutions and gradients and is amenable to geometric multigrid solvers.
Date: November 2, 2004
Creator: Schwartz, Peter; Barad, Michael; Colella, Phillip & Ligocki, Terry
Partner: UNT Libraries Government Documents Department

An Anelastic Allspeed Projection Method for GravitationallyStratified Flows

Description: This paper looks at gravitationally-stratified atmospheric flows at low Mach and Froude numbers and proposes a new algorithm to solve the compressible Euler equations, in which the asymptotic limits are recovered numerically and the boundary conditions for block-structured local refinement methods are well-posed. The model is non-hydrostatic and the numerical algorithm uses a splitting to separate the fast acoustic dynamics from the slower anelastic dynamics. The acoustic waves are treated implicitly while the anelastic dynamics is treated semi-implicitly and an embedded-boundary method is used to represent mountain ranges. We present an example that verifies our asymptotic analysis and a set of results that compares very well with the classical gravity wave results presented by Durran.
Date: February 24, 2005
Creator: Gatti-Bono, Caroline & Colella, Phillip
Partner: UNT Libraries Government Documents Department

An Incompressible Navier-Stokes with Particles Algorithm andParallel Implementation

Description: We present a variation of an adaptive projection method forcomputing solutions to the incompressible Navier-Stokes equations withsuspended particles. To compute the divergence-free component of themomentum forcing due to the particle drag, we employ an approach whichexploits the locality and smoothness of the Laplacian of the projectionoperator applied to the discretized particle drag force. We presentconvergence and performance results to demonstrate the effectiveness ofthis approach.
Date: November 28, 2006
Creator: Martin, Daniel F.; Colella, Phillip & Keen, Noel D.
Partner: UNT Libraries Government Documents Department

A Local Corrections Algorithm for Solving Poisson's Equation inThree Dimensions

Description: We present a second-order accurate algorithm for solving thefree-space Poisson's equation on a locally-refined nested grid hierarchyin three dimensions. Our approach is based on linear superposition oflocal convolutions of localized charge distributions, with the nonlocalcoupling represented on coarser grids. There presentation of the nonlocalcoupling on the local solutions is based on Anderson's Method of LocalCorrections and does not require iteration between different resolutions.A distributed-memory parallel implementation of this method is observedto have a computational cost per grid point less than three times that ofa standard FFT-based method on a uniform grid of the same resolution, andscales well up to 1024 processors.
Date: October 30, 2006
Creator: McCorquodale, Peter; Colella, Phillip; Balls, Gregory T. & Baden,Scott B.
Partner: UNT Libraries Government Documents Department

Performance and scaling of locally-structured grid methods forpartial differential equations

Description: In this paper, we discuss some of the issues in obtaining high performance for block-structured 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.
Date: July 19, 2007
Creator: Colella, Phillip; Bell, John; Keen, Noel; Ligocki, Terry; Lijewski, Michael & Van Straalen, Brian
Partner: UNT Libraries Government Documents Department

An unsplit, cell-centered Godunov method for ideal MHD

Description: We present a second-order Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. vol. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergence-free 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, low-beta flux tubes, and a magnetized rotor problem. For all of these cases, we observe that the algorithm is second-order 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 non-solenoidal fields.
Date: August 29, 2003
Creator: Crockett, Robert K.; Colella, Phillip; Fisher, Robert T.; Klein, Richard I. & McKee, Christopher F.
Partner: UNT Libraries Government Documents Department

A fourth order accurate adaptive mesh refinement method forpoisson's equation

Description: We present a block-structured adaptive mesh refinement (AMR) method for computing solutions to Poisson's equation in two and three dimensions. It is based on a conservative, finite-volume formulation of the classical Mehrstellen methods. This is combined with finite volume AMR discretizations to obtain a method that is fourth-order accurate in solution error, and with easily verifiable solvability conditions for Neumann and periodic boundary conditions.
Date: August 20, 2004
Creator: Barad, Michael & Colella, Phillip
Partner: UNT Libraries Government Documents Department

A node-centered local refinement algorithm for poisson's equation in complex geometries

Description: This paper presents a method for solving Poisson's equation with Dirichlet boundary conditions on an irregular bounded three-dimensional region. The method uses a nodal-point discretization and adaptive mesh refinement (AMR) on Cartesian grids, and the AMR multigrid solver of Almgren. The discrete Laplacian operator at internal boundaries comes from either linear or quadratic (Shortley-Weller) extrapolation, and the two methods are compared. It is shown that either way, solution error is second order in the mesh spacing. Error in the gradient of the solution is first order with linear extrapolation, but second order with Shortley-Weller. Examples are given with comparison with the exact solution. The method is also applied to a heavy-ion fusion accelerator problem, showing the advantage of adaptivity.
Date: May 4, 2004
Creator: McCorquodale, Peter; Colella, Phillip; Grote, David P. & Vay, Jean-Luc
Partner: UNT Libraries Government Documents Department

A cartesian grid embedded boundary method for hyperbolic conservation laws

Description: We present a second-order Godunov algorithm to solve time-dependent hyperbolic systems of conservation laws on irregular domains. Our approach is based on a formally consistent discretization of the conservation laws on a finite-volume grid obtained from intersecting the domain with a Cartesian grid. We address the small-cell stability problem associated with such methods by hybridizing our conservative discretization with a stable, nonconservative discretization at irregular control volumes, and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation. The resulting method is second-order accurate in L{sup 1} for smooth problems, and is robust in the presence of large-amplitude discontinuities intersecting the irregular boundary.
Date: October 3, 2004
Creator: Colella, Phillip; Graves, Daniel T.; Keen, Benjamin J. & Modiano, David
Partner: UNT Libraries Government Documents Department

A Cartesian grid embedded boundary method for the heat equation on irregular domains

Description: We present an algorithm for solving the heat equation on irregular time-dependent domains. It is based on the Cartesian grid embedded boundary algorithm of Johansen and Colella (J. Comput. Phys. 147(2):60--85) for discretizing Poisson's equation, combined with a second-order accurate discretization of the time derivative. This leads to a method that is second-order accurate in space and time. For the case where the boundary is moving, we convert the moving-boundary problem to a sequence of fixed-boundary problems, combined with an extrapolation procedure to initialize values that are uncovered as the boundary moves. We find that, in the moving boundary case, the use of Crank--Nicolson time discretization is unstable, requiring us to use the L{sub 0}-stable implicit Runge--Kutta method of Twizell, Gumel, and Arigu.
Date: March 14, 2001
Creator: McCorquodale, Peter; Colella, Phillip & Johansen, Hans
Partner: UNT Libraries Government Documents Department

Numerical Computation of Diffusion on a Surface

Description: We present a numerical method for computing diffusive transport on a surface derived from image data. Our underlying discretization method uses a Cartesian grid embedded boundary method for computing the volume transport in region consisting of all points a small distance from the surface. We obtain a representation of this region from image data using a front propagation computation based on level set methods for solving the Hamilton-Jacobi and eikonal equations. We demonstrate that the method is second-order accurate in space and time, and is capable of computing solutions on complex surface geometries obtained from image data of cells.
Date: February 24, 2005
Creator: Schwartz, Peter; Adalsteinsson, David; Colella, Phillip; Arkin, Adam Paul & Onsum, Matthew
Partner: UNT Libraries Government Documents Department