Daniel R. Reynolds

Daniel R. Reynolds
Southern Methodist University | SMU · Department of Mathematics

PhD Computational and Applied Mathematics

About

70
Publications
9,898
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1,497
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Introduction
Daniel R. Reynolds currently works at the Department of Mathematics at Southern Methodist University. Daniel does research in Computational and Applied Mathematics, Computational Physics and Parallel Computing. One of Daniel's current projects is 'Scalable numerical methods for weather prediction and for simulation of the fate of chemical constituents in the atmosphere'.
Additional affiliations
May 2014 - present
Southern Methodist University
Position
  • Professor (Associate)
May 2014 - present
Southern Methodist University
Position
  • Professor (Associate)
Description
  • I teach courses in applied mathematics, primarily those with a strong computational component, including courses in scientific computing, numerical analysis, linear algebra, numerical ordinary differential equations, and parallel scientific computing.
August 2008 - May 2014
Southern Methodist University
Position
  • Professor (Assistant)
Education
August 1998 - May 2003
Rice University
Field of study
  • Computational and Applied Mathematics
August 1994 - May 1998
Southwestern University
Field of study
  • Mathematics

Publications

Publications (70)
Article
In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing max...
Preprint
Full-text available
We describe the ARKODE library of one-step time integration methods for initial-value problems (IVPs). In addition to providing standard explicit and diagonally-implicit Runge--Kutta methods, ARKODE also supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge--Kutta methods and m...
Preprint
Full-text available
Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stabi...
Article
As part of the Exascale Computing Project (ECP), a recent focus of development efforts for the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been to enable GPU-accelerated time integration in scientific applications at extreme scales. This effort has resulted in several new GPU-enabled implementations of core SUNDIAL...
Preprint
Full-text available
In this paper we present a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. Following from recent work on multirate exponential Runge--Kutta (MERK) methods, that construct multirate schemes by approximating the action of matrix $\varphi$-functions within explic...
Article
We study the parallelization of a flexible order Cartesian treecode algorithm for evaluating electrostatic potentials of charged particle systems in which N particles are located on the molecular surfaces of biomolecules such as proteins. When the well-separated condition is satisfied, the treecode algorithm uses a far-field Taylor expansion to com...
Preprint
Full-text available
As part of the Exascale Computing Project (ECP), a recent focus of development efforts for the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been to enable GPU-accelerated time integration in scientific applications at extreme scales. This effort has resulted in several new GPU-enabled implementations of core SUNDIAL...
Preprint
Full-text available
In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing max...
Preprint
Full-text available
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this flexibility at the slow tim...
Article
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriat...
Article
Full-text available
Abstract The nonhydrostatic High‐Order Method Modeling Environment (HOMME‐NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the time step size that can be used with standard explicit time integration methods. Resolving acoustic waves is unnecessary for acc...
Preprint
Full-text available
In this report we document performance test results on a SUNDIALS-based multiphysics demonstration application. We aim to assess the large-scale parallel performance of new capabilities that have been added to the SUNDIALS suite of time integrators and nonlinear solvers in recent years under funding from both the Exascale Computing Project (ECP) an...
Preprint
Full-text available
The nonhydrostatic High Order Method Modeling Environment (HOMME-NH) atmospheric dynamical core supports acoustic waves that propagate significantly faster than the advective wind speed, thus greatly limiting the timestep size that can be used with standard explicit time-integration methods. Resolving acoustic waves is unnecessary for accurate clim...
Preprint
Full-text available
This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriat...
Article
Full-text available
In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods for the shallow water equations on the sphere. This stiff, nonlinear model provides a ‘testing ground’ for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We th...
Preprint
Full-text available
This work focuses on the construction of a new class of fourth-order accurate methods for multirate time evolution of systems of ordinary differential equations. We base our work on the Recursive Flux Splitting Multirate version of the Multirate Infinitesimal Step methods, and use recent theoretical developments for Generalized Additive Runge-Kutta...
Article
The spectral element method (SEM) is a mimetic finite element method with several properties that make it a desirable choice for numerical modeling. Although the linear dispersion properties of this method have been analyzed extensively for the case of the 1D inviscid advection equation, practical implementations of the SEM frequently employ hyperd...
Preprint
Full-text available
In this paper, we investigate the use of higher-order exponential Rosenbrock time integration methods on the shallow water equations on the sphere. This stiff, nonlinear model provides a testing ground for accurate and stable time integration methods in weather modeling, serving as the focus for exploration of novel methods for many years. We there...
Article
Full-text available
The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable l...
Article
Full-text available
The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work we investigate various implicit-explicit (IMEX) additive Runge-Kutta (ARK) methods for evolving acoustic waves implicitly to enable la...
Article
Multiscale computational models strive to produce accurate and efficient numerical simulations of systems involving interactions across multiple spatial and temporal scales that typically differ by several orders of magnitude. Some such models utilize a hybrid continuum-atomistic approach combining continuum approximations with first-principles-bas...
Article
Full-text available
In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may he...
Article
Full-text available
We describe an extension of the Enzo code to enable fully-coupled radiation hydrodynamical simulation of inhomogeneous reionization in large $\sim (100 Mpc)^3$ cosmological volumes with thousands to millions of point sources. We solve all dynamical, radiative transfer, thermal, and ionization processes self-consistently on the same mesh, as opposed...
Article
Full-text available
We use a fully coupled cosmological simulation including dark matter dynamics, multispecies hydrodynamics, nonequilibrium chemical ionization, flux-limited diffusion radiation transport, and a parameterized model of star formation and feedback (thermal and radiative) to investigate the epoch of hydrogen reionization in detail. In this paper, the fi...
Article
Full-text available
In this first of several application papers, we investigate the mechanics of reionization from stellar sources in high-z galaxies, the utility of various clumping factors on estimating the recombination time in the IGM, and the photon budget required to achieve reionization. We test the accuracy of the static and time-dependent models of Madau et a...
Article
Full-text available
This paper describes the open-source code Enzo, which uses block-structured adaptive mesh refinement to provide high spatial and temporal resolution for modeling astrophysical fluid flows. The code is Cartesian, can be run in 1, 2, and 3 dimensions, and supports a wide variety of physics including hydrodynamics, ideal and non-ideal magnetohydrodyna...
Article
Full-text available
We describe an extension of the {\em Enzo} code to enable the direct numerical simulation of inhomogeneous reionization in large cosmological volumes. By direct we mean all dynamical, radiative, and chemical properties are solved self-consistently on the same mesh, as opposed to a postprocessing approach which coarse-grains the radiative transfer....
Article
Full-text available
We consider multiphysics applications from algorithmic and architectural perspectives, where ‘‘algorithmic’’ includes both mathematical analysis and computational complexity, and ‘‘architectural’’ includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a...
Article
We examine the epoch of hydrogen reionization using a new numerical method that allows us to self-consistently couple all the relevant physical processes (gas dynamics, dark matter dynamics, self-gravity, star formation/feedback, radiative transfer, ionization, recombination, heating and cooling) and evolve the system of coupled equations on the sa...
Conference Paper
The light from early galaxies had a dramatic impact on the gasses filling the universe. This video highlights the spatial structure of the light's effect, by comparing two simulations: one with a self-consistent radiation field (radiative), and one without (non-radiative), each with a very high dynamic range. Looking at the simulations side-by-side...
Conference Paper
The light from early galaxies had a dramatic impact on the gasses filling the universe. This video highlights the spatial structure of the light's effect, by comparing two simulations: one with a self-consistent radiation field (radiative), and one without (non-radiative), each with a very high dynamic range. Ionization fraction is the amount of th...
Conference Paper
The submitted visualization represents work performed by the Enzo PRAC team lead by Brian O'Shea on the Blue Waters Early Science system. A relatively small test calculation was performed followed by several much larger AMR cosmological runs. The overall scientific goal was to understand how galaxies in the early Universe (the first billion years o...
Article
Full-text available
Single-fluid resistive magnetohydrodynamics (MHD) is a fluid description of fusion plasmas which is often used to investigate macroscopic instabilities in tokamaks. In MHD modeling of tokamaks, it is often desirable to compute MHD phenomena to resistive time scales or a combination of resistive-Alfvén time scales, which can render explicit time ste...
Book
Full-text available
We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a comm...
Article
We apply the automatic differentiation tool OpenAD toward constructing a preconditioner for fully implicit simulations of mapped grid visco-resistive magnetohydrodynamics (MHD), used in modeling tokamak fusion devices. Our simulation framework employs a fully implicit formulation in time, and a mapped finite volume spatial discretization. We solve...
Conference Paper
This simulation uses a flux-limited diffusion solver to explore the radiation hydrodynamics of early galaxies, in particular, the ionizing radiation created by Population III stars. At the time of this rendering, the simulation has evolved to a redshift of 3.5. The simulation volume is 11.2 comoving megaparsecs, and has a uniform grid of 1024³ cell...
Article
Full-text available
We introduce an operator-based scheme for preconditioning stiff components encountered in implicit methods for hyperbolic systems of PDEs posed on regular grids. The method is based on a directional splitting of the implicit operator, followed by a characteristic decomposition of the resulting directional parts. This approach allows for the solutio...
Article
Full-text available
We describe an extension of the cosmological hydrodynamics code ENZO to include the self-consistent transport of ionizing radiation modeled in the flux-limited diffusion approximation. A novel feature of our algorithm is a coupled implicit solution of radiation transport, ionization kinetics, and gas photoheating, making the timestepping for this p...
Article
The development of radiation hydrodynamical methods that are able to follow gas dynamics and radiative transfer self-consistently is key to the solution of many problems in numerical astrophysics. Such fluid flows are highly complex, rarely allowing even for approximate analytical solutions against which numerical codes can be tested. An alternativ...
Patent
Full-text available
A method and apparatus wherein phase changes in a material can dampen vibrational energy, dampen noise and facilitate heat transfer. One embodiment includes a method for damping vibrational energy in a body. The method comprises attaching a material to the body, wherein the material comprises a substrate, a shape memory alloy layer, and a plurality...
Article
We consider a PDE system comprising compressible hydrodynamics, flux-limited diffusion radiation transport and chemical ionization kinetics in a cosmologically-expanding universe. Under an operator-split framework, the cosmological hydrodynamics equations are solved through the piecewise parabolic method, as implemented in the Enzo community hydrod...
Article
This paper describes an implicit approach and nonlinear solver for solution of radiation-hydrodynamic problems in the context of supernovae and proto-neutron star cooling. The robust approach applies Newton-Krylov methods and overcomes the difficulties of discontinuous limiters in the discretized equations and scaling of the equations over wide ran...
Article
Full-text available
In this paper we describe our massively parallel version of Enzo, a multiphysics, parallel, AMR application for simulating cosmological structure formation developed at UCSD and Columbia. We describe its physics, numerical algorithms, implementation, and performance on current terascale platforms. We also discuss our future plans and some of the ch...
Article
We present a mathematical model describing the thermodynamic behavior of shape memory alloy wires, as well as a computational technique to solve the resulting system of partial differential equations. The model consists of continuum-level conservation equations based on a new construction of the Helmholtz free energy potential. This construction al...
Article
We present a nonlinearly implicit, conservative numerical method for integration of the single-fluid resistive MHD equations. The method uses a high-order spatial discretization that preserves the solenoidal property of the magnetic field. The fully coupled PDE system is solved implicitly in time, providing for increased interaction between physica...
Article
Full-text available
Computational scientists are grappling with increasingly complex, multi-rate applications that couple such physical phenomena as fluid dynamics, electromagnetics, radiation transport, chemical and nuclear reactions, and wave and material propagation in inhomogeneous media. Parallel computers with large storage capacities are paving the way for high...
Article
Computational MHD of tokamak plasmas poses severe challenges due to its wide range of spatio-temporal scales, strong anisotropy and nonlinearity. Additionally, resistive MHD experiences increased ill-conditioning as the spatial meshes are refined to resolve diffusive layers. We present a fully implicit Jacobian-Free Newton-Krylov method for resisti...
Article
Full-text available
Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provide an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic model and...
Article
We consider a class of alloys and ceramics with equilibria described by non-attainable infima of non-quasiconvex variational integrals. Such situations frequently arise when atomic lattice structure plays an important role at the mesoscopic continuum level.We prove that standard variational approaches associated with gradient based relaxation of no...
Article
We provide an example of a stochastic approach to relaxation of the variational integrals with non-attainable infima in one dimension. We provide an approximation for the coefficients of the Laplace transformation of the probability density function. This approaximation yields the relaxing microstructures.
Article
We show that fast, localized heating and cooling of a Shape Memory material can provide a very effective means of damping vibrational energy. We model the thermally induced pseudo-elastic behavior of a NiTi Shape Memory wire using the variant of Landau-Devonshire potential. We assume that the wire consists of martensitic NiTi single crystal. Dynami...
Article
We propose a model for the control of high-frequency oscillations in shape memory alloy wires. We introduce a notion of generalized solution for a generalized control and in this context prove a local exact controllability result effectively corresponding to an approximate controllability result for the nonconvex pseudoelastic material system.
Article
Full-text available
Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provides an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic descript...
Article
We propose a subgrid projection method which is suitable for computing microstructures describing non-attainable infima of variational integrals. We document that a descent method in combination with the proposed method yields relaxing sequences which converge in weak-* topology as well as in the sense of approximate Young measures. We show that a...
Conference Paper
We introduce a variational principle suitable for the computational modeling of crystalline materials. We consider a class of materials that are described by non-quasiconvex variational integrals. We are further focused on equlibria of such materials that have non-attainment structure, i.e., Dirichlet boundary conditions prohibit these variational...
Article
Full-text available
Our group was formed as a collection of graduate students and one post-doc from the Compu-tational and Applied Math Department at Rice University. For nearly all the problems, we had at least two distinct approaches that gave similar results. Our solutions are given in Table 1 to a level we are confident of accuracy. For the problems marked with a...
Article
Combination of object-oriented programming with automatic differentiation techniques facilitates the solution of data fitting, control, and design problems driven by explicit time stepping schemes for initial-boundary value problems. The C++ class fdtd takes a complete specification of a single step, along with some associated code, and assembles f...
Article
Full-text available
The C++ class fdtd uses automatic differentiation techniques to implement an abstract time stepping scheme in an object-oriented fashion, making it possible to use the resulting simulator to solve inverse or control problems. The class takes a complete specification of a single step of the scheme, and assembles from it a complete simulator, along w...
Article
Full-text available
We present a mathematical model describing the thermodynamic behavior of shape memory alloy wires, as well as a computational technique to solve the resulting system of partial differential equations. The model consists of conservation equations based on a new Helmholtz free energy potential. The computational technique introduces a viscosity-based...
Article
Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provides an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic model an...

Projects

Project (1)
Project
The objectives of this project is to develop numerical methods for the simulation of Earth's atmosphere on polyhedral meshes with the help of exponential time integration schemes. The project is divided into three stages: The investigation of metric properties of the terrain following coordinates on ellipsoid and the derivation of covariant equations of meteorology Numerical solution of the three-dimensional advection--diffusion--reaction equation on an ellipsoidal planet with mountains. Discretization of the Euler Equations on an ellipsoidal planet with mountains