12 results on '"M. R. Dorr"'
Search Results
2. High-order, finite-volume methods in mapped coordinates
- Author
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Phillip Colella, Jeffrey Hittinger, Daniel F. Martin, and M. R. Dorr
- Subjects
Numerical Analysis ,Partial differential equation ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Order of accuracy ,Computer Science Applications ,law.invention ,Quadrature (mathematics) ,Computational Mathematics ,law ,Modeling and Simulation ,Cartesian coordinate system ,Coordinate space ,Hyperbolic partial differential equation ,Mathematics - Abstract
We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge-Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.
- Published
- 2011
3. A numerical algorithm for the solution of a phase-field model of polycrystalline materials
- Author
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M. Wickett, M. R. Dorr, Jean-Luc Fattebert, James Belak, and Patrice E. A. Turchi
- Subjects
Backward differentiation formula ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Preconditioner ,Applied Mathematics ,Linear system ,MathematicsofComputing_NUMERICALANALYSIS ,Newton's method in optimization ,Mathematics::Numerical Analysis ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Modeling and Simulation ,Temporal discretization ,Quaternion ,Algorithm ,Mathematics - Abstract
We describe an algorithm for the numerical solution of a phase-field model (PFM) of microstructure evolution in polycrystalline materials. The PFM system of equations includes a local order parameter, a quaternion representation of local orientation and a species composition parameter. The algorithm is based on the implicit integration of a semidiscretization of the PFM system using a backward difference formula (BDF) temporal discretization combined with a Newton-Krylov algorithm to solve the nonlinear system at each time step. The BDF algorithm is combined with a coordinate-projection method to maintain quaternion unit length, which is related to an important solution invariant. A key element of the Newton-Krylov algorithm is the selection of a preconditioner to accelerate the convergence of the Generalized Minimum Residual algorithm used to solve the Jacobian linear system in each Newton step. Results are presented for the application of the algorithm to 2D and 3D examples.
- Published
- 2010
4. Progress Report for 'High-Resolution Methods for Phase Space Problems in Complex Geometry'
- Author
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P. McCorquodale, Jeffrey Hittinger, M. R. Dorr, Peter Schwartz, and Phillip Colella
- Subjects
Complex geometry ,Computer science ,Phase space ,Calculus ,Applied mathematics ,High resolution - Published
- 2015
5. Simulating time-dependent energy transfer between crossed laser beams in an expanding plasma
- Author
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Jeffrey Hittinger, E. A. Williams, M. R. Dorr, and Richard Berger
- Subjects
Physics ,Numerical Analysis ,Beam diameter ,Physics and Astronomy (miscellaneous) ,Differential equation ,Velocity gradient ,business.industry ,Applied Mathematics ,Paraxial approximation ,Plasma ,Mechanics ,Refraction ,Computer Science Applications ,Computational Mathematics ,Nonlinear system ,Optics ,Modeling and Simulation ,business ,Beam (structure) - Abstract
A coupled mode system is derived to investigate a three-wave parametric instability leading to energy transfer between co-propagating laser beams crossing in a plasma flow. The model includes beams of finite width refracting in a prescribed transverse plasma flow with spatial and temporal gradients in velocity and density. The resulting paraxial light equations are discretized spatially with a Crank-Nicholson-type scheme, and these algebraic constraints are nonlinearly coupled with ordinary differential equations in time that describe the ion acoustic response. The entire nonlinear differential-algebraic system is solved using an adaptive, backward-differencing method coupled with Newton's method. A numerical study is conducted in two dimensions that compares the intensity gain of the fully time-dependent coupled mode system with the gain computed under the further assumption of a strongly damped ion acoustic response. The results demonstrate a time-dependent gain suppression when the beam diameter is commensurate with the velocity gradient scale length. The gain suppression is shown to depend on time-dependent beam refraction and is interpreted as a time-dependent frequency shift.
- Published
- 2005
6. Simulation of Laser Plasma Filamentation Using Adaptive Mesh Refinement
- Author
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F.Xabier Garaizar, M. R. Dorr, and Jeffrey Hittinger
- Subjects
Physics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Hierarchy (mathematics) ,Adaptive mesh refinement ,Applied Mathematics ,Paraxial approximation ,Godunov's scheme ,Plasma ,Laser ,Grid ,Topology ,Computer Science Applications ,law.invention ,Computational Mathematics ,Classical mechanics ,Filamentation ,Physics::Plasma Physics ,law ,Modeling and Simulation - Abstract
We investigate the use of adaptive mesh refinement in the simulation of laser plasma filamentation. A numerical algorithm is constructed to solve model equations consisting of a fluid approximation of a quasineutral plasma combined with a paraxial light propagation model. The algorithm involves high-resolution plasma and light model discretizations on a block-structured, locally refined grid hierarchy, which is dynamically modified during the time integration to follow evolving fine-scale solution features. Comparisons of the efficiency of this approach to that of uniform grid calculations are presented.
- Published
- 2002
7. Numerical Solution of Plasma Fluid Equations Using Locally Refined Grids
- Author
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Daniel D. Wake, M. R. Dorr, and Phillip Colella
- Subjects
Physics ,Numerical Analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Discretization ,Adaptive mesh refinement ,Differential equation ,Applied Mathematics ,Numerical analysis ,Grid ,Computer Science Applications ,Euler equations ,Computational Mathematics ,symbols.namesake ,Modeling and Simulation ,symbols ,Applied mathematics ,Poisson's equation ,Algorithm - Abstract
This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasmas under consideration are typical of those used for the processing of semiconductors. The governing equations consist of a drift?diffusion model of the electrons, together with an energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electric field, collisional, and source/sink terms. A discretization previously developed for a uniform spatial grid is generalized to enable local grid refinement. This extension involves the time integration of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. This approach represents an advancement of methodologies developed for neutral flows using block-structured adaptive mesh refinement (AMR) to include the significant additional effect of the electrostatic forces that couple the ion and electron fluid components. Numerical results that assess the accuracy and efficiency of the method and illustrate the importance of using adequate resolution are also presented.
- Published
- 1999
8. A Conservative Finite Difference Method for the Numerical Solution of Plasma Fluid Equations
- Author
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Daniel D. Wake, M. R. Dorr, and Phillip Colella
- Subjects
Physics ,Numerical Analysis ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Finite difference method ,Backward Euler method ,Computer Science Applications ,Regular grid ,Euler equations ,Computational Mathematics ,Boundary layer ,symbols.namesake ,Classical mechanics ,Modeling and Simulation ,symbols ,Boundary value problem - Abstract
This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level.
- Published
- 1999
9. A domain decomposition preconditioner with reduced rank interdomain coupling
- Author
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M. R. Dorr
- Subjects
Numerical Analysis ,Discretization ,Rank (linear algebra) ,Preconditioner ,Applied Mathematics ,Linear system ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Domain decomposition methods ,Finite element method ,Computational Mathematics ,Conjugate gradient method ,Applied mathematics ,Conjugate residual method ,Mathematics - Abstract
A preconditioner for the conjugate gradient solution of the linear system arising from a standard finite element discretization of a self-adjoint, second-order elliptic problem is proposed. To achieve an efficient implementation on multiprocessors, the preconditioner employs a domain decomposition strategy designed to substantially reduce the amount of interdomain (and therefore interprocessor) communication required to invert the preconditioner against the residual vector in each conjugate gradient iteration. A spectral equivalence result and some numerical experiments are presented.
- Published
- 1991
10. High-order finite-volume adaptive methods on locally rectangular grids
- Author
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Daniel F. Martin, P. McCorquodale, Phillip Colella, M. R. Dorr, and Jeffrey Hittinger
- Subjects
History ,Finite volume method ,Adaptive mesh refinement ,Geometry ,Polytropic process ,Computer Science Applications ,Education ,Regular grid ,law.invention ,law ,Applied mathematics ,Cartesian coordinate system ,High order ,Mathematics - Abstract
We are developing a new class of finite-volume methods on locally-refined and mapped grids, which are at least fourth-order accurate in regions where the solution is smooth. This paper discusses the implementation of such methods for time-dependent problems on both Cartesian and mapped grids with adaptive mesh refinement. We show 2D results with the Berger-Colella shock-ramp problem in Cartesian coordinates, and fourth-order accuracy of the solution of a Gaussian pulse problem in a polytropic gas in mapped coordinates.
- Published
- 2009
11. Calculation of electromagnetic scattering by a perfect conductor
- Author
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M. R. Dorr, A. K. Aziz, and R. B. Kellogg
- Subjects
Numerical Analysis ,Partial differential equation ,Physics and Astronomy (miscellaneous) ,Scattering ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,Boundary (topology) ,Computer Science Applications ,Computational Mathematics ,symbols.namesake ,Maxwell's equations ,Modeling and Simulation ,symbols ,T-matrix method ,Boundary value problem ,Perfect conductor ,Mathematics - Abstract
A new method is presented for calculating the scattering of an arbitrary electromagnetic wave by a bounded, perfectly conducting body of general shape. The strategy is to replace the corresponding exterior boundary-value problem by an approximate problem on the boundary of the scattering body. This involves the introduction of a certain bilinear form and non-local boundary operator, together with the use of a special class of known solutions of the reduced Maxwell's equations satisfying the Sommerfeld radiation boundary conditions at infinity. Two computer programs implementing this method are described and numerical results showing the successful application of this method to some model problems are presented.
- Published
- 1982
12. A New Approximation Method for the Helmholtz Equation in an Exterior Domain
- Author
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R. B. Kellogg, A. K. Aziz, and M. R. Dorr
- Subjects
Numerical Analysis ,Computational Mathematics ,Helmholtz equation ,Applied Mathematics ,Mathematical analysis ,Nonlocal boundary ,Bilinear form ,Approximate solution ,Linear subspace ,Domain (mathematical analysis) ,Density property ,Mathematics - Abstract
In this paper we consider the numerical solution of the Helmholtz equation in an exterior domain in $\mathbb{R}^3 $. We introduce a new variational formulation of the problem involving a nonlocal boundary condition. We show that the discrete bilinear form associated with our variational formulation satisfies the “inf-sup” condition, and we give a quasi-optimal error estimate for the approximate solution. Moreover, we show a certain density property of our subspaces which plays a crucial role in the justification of the method.
- Published
- 1982
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