Papers by Thomas Hagstrom
Applied Computational Electromagnetics Society, 2021
We describe the construction, analysis, and implementation of arbitrary-order local radiation bou... more We describe the construction, analysis, and implementation of arbitrary-order local radiation boundary condition sequences for Maxwell’s equations. In particular we use the complete radiation boundary conditions which implicitly apply uniformly accurate exponentially convergent rational approximants to the exact radiation boundary conditions. Numerical experiments for waveguide and free space problems using high- order discontinuous Galerkin spatial discretizations are presented.
Journal of Scientific Computing, 2019
We develop dissipative, energy-stable difference methods for linear first-order hyperbolic system... more We develop dissipative, energy-stable difference methods for linear first-order hyperbolic systems by applying an upwind, discontinuous Galerkin construction of derivative matrices to a space of discontinuous piecewise polynomials on a structured mesh. The space is spanned by translates of a function spanning multiple cells, yielding a class of implicit difference formulas of arbitrary order. We examine the properties of the method, including the scaling of the derivative operator with method order, and demonstrate its accuracy for problems in one and two space dimensions.
International Journal for Numerical Methods in Engineering, 2019
SummaryWe consider wave propagation in a coupled fluid‐solid region separated by a static but pos... more SummaryWe consider wave propagation in a coupled fluid‐solid region separated by a static but possibly curved interface. The wave propagation is modeled by the acoustic wave equation in terms of a velocity potential in the fluid, and the elastic wave equation for the displacement in the solid. At the fluid solid interface, we impose suitable interface conditions to couple the two equations. We use a recently developed energy‐based discontinuous Galerkin method to discretize the governing equations in space. Both energy conserving and upwind numerical fluxes are derived to impose the interface conditions. The highlights of the developed scheme include provable energy stability and high order accuracy. We present numerical experiments to illustrate the accuracy property and robustness of the developed scheme.
SIAM Journal on Scientific Computing, 2018
Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are pre... more Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are presented. Both methods use (m + 1) d degrees of freedom per node for the displacement in d-dimensions; the dissipative and conservative methods achieve orders of accuracy (2m − 1) and 2m, respectively. Stability and error analyses as well as implementation strategies for accelerators are also given.
We review time-domain formulations of radiation boundary conditions for Maxwell's equations, focu... more We review time-domain formulations of radiation boundary conditions for Maxwell's equations, focusing on methods which can deliver arbitrary accuracy at acceptable computational cost. Examples include fast evaluations of nonlocal conditions on symmetric and general boundaries, methods based on identifying and evaluating equivalent sources, and local approximations such as the perfectly matched layer and sequences of local boundary conditions. Complexity estimates are derived to assess work and storage requirements as a function of wavelength and simulation time.
Communications in Computational Physics, 2017
Hermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and stag... more Hermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.
Journal of Mathematical Fluid Mechanics, 2003
In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stok... more In this paper we consider the Cauchy problem for incompressible flows governed by the Navier-Stokes or MHD equations. We give a new proof for the time decay of the spatial L 2 norm of the solution, under the assumption that the solution of the heat equation with the same initial data decays. By first showing decay of the first derivatives of the solution, we avoid some technical difficulties of earlier proofs based on Fourier splitting.
On High-Order Radiation Boundary Conditions
Computational Wave Propagation, 1997
Boundary conditions and the simulation of low Mach number flows
The problem of accurately computing low Mach number flows, with the specific intent of studying t... more The problem of accurately computing low Mach number flows, with the specific intent of studying the interaction of sound waves with incompressible flow structures, such as concentrations of vorticity is considered. This is a multiple time (and/or space) scales problem, leading to various difficulties in the design of numerical methods. Concentration is on one of these difficulties - the development of boundary conditions at artificial boundaries which allow sound waves and vortices to radiate to the far field. Nonlinear model equations are derived based on assumptions about the scaling of the variables. Then these are linearized about a uniform flow and exact boundary conditions are systematically derived using transform methods. Finally, useful approximations to the exact conditions which are valid for small Mach number and small viscosity are computed.
Journal of Computational Physics, 2014
We derive new, explicit representations for the solution to the scalar wave equation in the exter... more We derive new, explicit representations for the solution to the scalar wave equation in the exterior of a sphere, subject to either Dirichlet or Robin boundary conditions. Our formula leads to a stable and high-order numerical scheme that permits the evaluation of the solution at an arbitrary target, without the use of a spatial grid and without numerical dispersion error. In the process, we correct some errors in the analytic literature concerning the asymptotic behavior of the logarithmic derivative of the spherical modified Hankel function. We illustrate the performance of the method with several numerical examples.
Mathematics of Computation, 1996
We present some relations that allow the efficient approximate inversion of linear differential o... more We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple 3-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e., matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of trun...
Journal of Computational Physics, 2015
In this paper, we extend the frequency domain Lorenz-Mie-Debye formalism for the Maxwell equation... more In this paper, we extend the frequency domain Lorenz-Mie-Debye formalism for the Maxwell equations to the time domain. In particular, we show that the problem of scattering from a perfectly conducting sphere can be reduced to the solution of two scalar wave equations-one with Dirichlet boundary conditions and the other with Robin boundary conditions. An explicit, stable, and high-order numerical scheme is then developed, based on our earlier treatment of the scalar case. This new representation may provide some insight into transient electromagnetic phenomena, and can also serve as a reference solution for general purpose time-domain software packages.
Advanced Modeling and Simulation in Engineering Sciences, 2015
Background Recently the Double Absorbing Boundary (DAB) method was introduced as a new approach f... more Background Recently the Double Absorbing Boundary (DAB) method was introduced as a new approach for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. Methods The DAB method is based on truncating the unbounded domain to produce a finite computational domain, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and within the layer, and participate in the numerical scheme. In previous studies DAB was developed for acoustic waves which are solutions to the scalar wave equation. Here the approach is extended to time-dependent elastic waves in homogeneous and layered media. The equations are written...
Computers & Fluids, 2014
We consider the application of a multidomain, sparse, and modal spectral-tau method to the helica... more We consider the application of a multidomain, sparse, and modal spectral-tau method to the helically reduced Navier Stokes equations describing pipe flow. This work (i) formulates the corresponding modal approximations, (ii) describes improved boundary conditions for the helically reduced equations, and (iii) constructs iterative solutions of the corresponding elliptic problem that arises in the reduction. Regarding (iii), we also present and test a method for preconditioning the matching conditions between subdomains, a method based on statistical sampling and the interpolative decomposition. Although the following application is only discussed in our concluding section, a partial motivation for this work has been our ongoing development of similar spectral methods for the construction binary neutron star spacetimes.
Regularization Strategies for Hyperplane Classifiers: Application to Cancer Classification with Gene Expression Data
Journal of Bioinformatics and Computational Biology, 2007
Linear discrimination, from the point of view of numerical linear algebra, can be treated as solv... more Linear discrimination, from the point of view of numerical linear algebra, can be treated as solving an ill-posed system of linear equations. In order to generate a solution that is robust in the presence of noise, these problems require regularization. Here, we examine the ill-posedness involved in the linear discrimination of cancer gene expression data with respect to outcome and tumor subclasses. We show that a filter factor representation, based upon Singular Value Decomposition, yields insight into the numerical ill-posedness of the hyperplane-based separation when applied to gene expression data. We also show that this representation yields useful diagnostic tools for guiding the selection of classifier parameters, thus leading to improved performance.
Complete Radiation Boundary Conditions for Convective Waves
Communications in Computational Physics, 2012
Local approximate radiation boundary conditions of optimal efficiency for the convective wave equ... more Local approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4].
A General Perfectly Matched Layer Model for Hyperbolic-Parabolic Systems
SIAM Journal on Scientific Computing, 2009
SIAM Journal on Applied Mathematics, 2006
Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust ... more Since its introduction the Perfectly Matched Layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limted to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is well-posed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell's equations, the linearized Euler equations, as well as arbitrary 2 × 2 systems in (2 + 1) dimensions.
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Papers by Thomas Hagstrom