Communications in Applied Mathematics and Computational Science Articles (Project Euclid)
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The latest articles from Communications in Applied Mathematics and Computational Science on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 19 Oct 2017 13:04 EDTThu, 19 Oct 2017 13:04 EDThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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A single-stage flux-corrected transport algorithm for high-order finite-volume methods
https://projecteuclid.org/euclid.camcos/1508432637
<strong>Christopher Chaplin</strong>, <strong>Phillip Colella</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 12, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. We modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the preconstraint on the high-order fluxes. We compute the high-order fluxes via a method-of-lines approach with fourth-order Runge–Kutta as the time integrator. For computing low-order fluxes, we select the corner-transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered-difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high-wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.
</p>projecteuclid.org/euclid.camcos/1508432637_20171019130406Thu, 19 Oct 2017 13:04 EDTAchieving algorithmic resilience for temporal integration through spectral deferred corrections
https://projecteuclid.org/euclid.camcos/1508432638
<strong>Ray Grout</strong>, <strong>Hemanth Kolla</strong>, <strong>Michael Minion</strong>, <strong>John Bell</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 12, Number 1, 25--50.</p><p><strong>Abstract:</strong><br/>
Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. We demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier–Stokes equations for combustion research.
</p>projecteuclid.org/euclid.camcos/1508432638_20171019130406Thu, 19 Oct 2017 13:04 EDTA fourth-order Cartesian grid embedded boundary method for Poisson's equation
https://projecteuclid.org/euclid.camcos/1508432639
<strong>Dharshi Devendran</strong>, <strong>Daniel Graves</strong>, <strong>Hans Johansen</strong>, <strong>Terry Ligocki</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 12, Number 1, 51--79.</p><p><strong>Abstract:</strong><br/>
In this paper, we present a fourth-order algorithm to solve Poisson’s equation in two and three dimensions. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We use a weighted least squares algorithm to solve for our stencils. We use convergence tests to demonstrate accuracy and we show the eigenvalues of the operator to demonstrate stability. We compare accuracy and performance with an established second-order algorithm. We also discuss in depth strategies for retaining higher-order accuracy in the presence of nonsmooth geometries.
</p>projecteuclid.org/euclid.camcos/1508432639_20171019130406Thu, 19 Oct 2017 13:04 EDTA central-upwind geometry-preserving method for hyperbolic conservation laws on the sphere
https://projecteuclid.org/euclid.camcos/1508432640
<strong>Abdelaziz Beljadid</strong>, <strong>Philippe LeFloch</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 12, Number 1, 81--107.</p><p><strong>Abstract:</strong><br/>
We introduce a second-order, central-upwind finite volume method for the discretization of nonlinear hyperbolic conservation laws on the two-dimensional sphere. The semidiscrete version of the proposed method is based on a technique of local propagation speeds, and the method is free of any Riemann solver. The main advantages of our scheme are its high resolution of discontinuous solutions, its low numerical dissipation, and its simplicity of implementation. We do not use any splitting approach, which is often applied to upwind schemes in order to simplify the resolution of Riemann problems. The semidiscrete form of our scheme is strongly built upon the analytical properties of nonlinear conservation laws and the geometry of the sphere. The curved geometry is treated here in an analytical way so that the semidiscrete form of the proposed scheme is consistent with a geometric compatibility property. Furthermore, the time evolution is carried out by using a total-variation diminishing Runge–Kutta method. A rich family of (discontinuous) stationary solutions is available for the conservation laws under consideration when the flux is nonlinear and foliated (in a suitable sense). We present a series of numerical tests, encompassing various nontrivial steady state solutions and therefore providing a good validation of the accuracy and efficiency of the proposed central-upwind finite volume scheme. Our numerical tests confirm that the scheme is stable and succeeds in accurately capturing discontinuous steady state solutions to conservation laws posed on the sphere.
</p>projecteuclid.org/euclid.camcos/1508432640_20171019130406Thu, 19 Oct 2017 13:04 EDTTime-parallel gravitational collapse simulation
https://projecteuclid.org/euclid.camcos/1508432641
<strong>Andreas Kreienbuehl</strong>, <strong>Pietro Benedusi</strong>, <strong>Daniel Ruprecht</strong>, <strong>Rolf Krause</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 12, Number 1, 109--128.</p><p><strong>Abstract:</strong><br/>
This article demonstrates the applicability of the parallel-in-time method Parareal to the numerical solution of the Einstein gravity equations for the spherical collapse of a massless scalar field. To account for the shrinking of the spatial domain in time, a tailored load balancing scheme is proposed and compared to load balancing based on number of time steps alone. The performance of Parareal is studied for both the subcritical and black hole case; our experiments show that Parareal generates substantial speedup and, in the supercritical regime, can reproduce Choptuik’s black hole mass scaling law.
</p>projecteuclid.org/euclid.camcos/1508432641_20171019130406Thu, 19 Oct 2017 13:04 EDTAdaptively weighted least squares finite element methods for partial differential equations with singularitieshttps://projecteuclid.org/euclid.camcos/1522202435<strong>Brian Hayhurst</strong>, <strong>Mason Keller</strong>, <strong>Chris Rai</strong>, <strong>Xidian Sun</strong>, <strong>Chad R. Westphal</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 1, 1--25.</p><p><strong>Abstract:</strong><br/>
The overall effectiveness of finite element methods may be limited by solutions that lack smoothness on a relatively small subset of the domain. In particular, standard least squares finite element methods applied to problems with singular solutions may exhibit slow convergence or, in some cases, may fail to converge. By enhancing the norm used in the least squares functional with weight functions chosen according to a coarse-scale approximation, it is possible to recover near-optimal convergence rates without relying on exotic finite element spaces or specialized meshing strategies. In this paper we describe an adaptive algorithm where appropriate weight functions are generated from a coarse-scale approximate solution. Several numerical tests, both linear and nonlinear, illustrate the robustness of the adaptively weighted approach compared with the analogous standard [math] least squares finite element approach.
</p>projecteuclid.org/euclid.camcos/1522202435_20180327220043Tue, 27 Mar 2018 22:00 EDTOn the convergence of iterative solvers for polygonal discontinuous Galerkin discretizationshttps://projecteuclid.org/euclid.camcos/1522202438<strong>Will Pazner</strong>, <strong>Per-Olof Persson</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 1, 27--51.</p><p><strong>Abstract:</strong><br/>
We study the convergence of iterative linear solvers for discontinuous Galerkin discretizations of systems of hyperbolic conservation laws with polygonal mesh elements compared with traditional triangular elements. We solve the semidiscrete system of equations by means of an implicit time discretization method, using iterative solvers such as the block Jacobi method and GMRES. We perform a von Neumann analysis to analytically study the convergence of the block Jacobi method for the two-dimensional advection equation on four classes of regular meshes: hexagonal, square, equilateral-triangular, and right-triangular. We find that hexagonal and square meshes give rise to smaller eigenvalues, and thus result in faster convergence of Jacobi’s method. We perform numerical experiments with variable velocity fields, irregular, unstructured meshes, and the Euler equations of gas dynamics to confirm and extend these results. We additionally study the effect of polygonal meshes on the performance of block ILU(0) and Jacobi preconditioners for the GMRES method.
</p>projecteuclid.org/euclid.camcos/1522202438_20180327220043Tue, 27 Mar 2018 22:00 EDTTheoretically optimal inexact spectral deferred correction methodshttps://projecteuclid.org/euclid.camcos/1522202439<strong>Martin Weiser</strong>, <strong>Sunayana Ghosh</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 1, 53--86.</p><p><strong>Abstract:</strong><br/>
In several initial value problems with particularly expensive right-hand side evaluation or implicit step computation, there is a tradeoff between accuracy and computational effort. We consider inexact spectral deferred correction (SDC) methods for solving such initial value problems. SDC methods are interpreted as fixed-point iterations and, due to their corrective iterative nature, allow one to exploit the accuracy-work tradeoff for a reduction of the total computational effort. First we derive error models bounding the total error in terms of the evaluation errors. Then we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal local tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. The properties of optimal local tolerances and the predicted efficiency gain compared to simpler heuristics, and reasonable practical performance, are illustrated with simple numerical examples.
</p>projecteuclid.org/euclid.camcos/1522202439_20180327220043Tue, 27 Mar 2018 22:00 EDTA third order finite volume WENO scheme for Maxwell's equations on tetrahedral mesheshttps://projecteuclid.org/euclid.camcos/1522202440<strong>Marina Kotovshchikova</strong>, <strong>Dmitry K. Firsov</strong>, <strong>Shiu Hong Lui</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 1, 87--106.</p><p><strong>Abstract:</strong><br/>
A third order type II WENO finite volume scheme for tetrahedral unstructured meshes is applied to the numerical solution of Maxwell’s equations. Stability and accuracy of the scheme are severely affected by mesh distortions, domain geometries, and material inhomogeneities. The accuracy of the scheme is enhanced by a clever choice of a small parameter in the WENO weights. Also, hybridization with a polynomial scheme is proposed to eliminate unnecessary and costly WENO reconstructions in regions where the solution is smooth. The proposed implementation is applied to several test problems to demonstrate the accuracy and efficiency, as well as usefulness of the scheme to problems with singularities.
</p>projecteuclid.org/euclid.camcos/1522202440_20180327220043Tue, 27 Mar 2018 22:00 EDTOn a scalable nonparametric denoising of time series signalshttps://projecteuclid.org/euclid.camcos/1522202441<strong>Lukáš Pospíšil</strong>, <strong>Patrick Gagliardini</strong>, <strong>William Sawyer</strong>, <strong>Illia Horenko</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 1, 107--138.</p><p><strong>Abstract:</strong><br/>
Denoising and filtering of time series signals is a problem emerging in many areas of computational science. Here we demonstrate how the nonparametric computational methodology of the finite element method of time series analysis with [math] regularization can be extended for denoising of very long and noisy time series signals. The main computational bottleneck is the inner quadratic programming problem. Analyzing the solvability and utilizing the problem structure, we suggest an adapted version of the spectral projected gradient method (SPG-QP) to resolve the problem. This approach increases the granularity of parallelization, making the proposed methodology highly suitable for graphics processing unit (GPU) computing. We demonstrate the scalability of our open-source implementation based on PETSc for the Piz Daint supercomputer of the Swiss Supercomputing Centre (CSCS) by solving large-scale data denoising problems and comparing their computational scaling and performance to the performance of the standard denoising methods.
</p>projecteuclid.org/euclid.camcos/1522202441_20180327220043Tue, 27 Mar 2018 22:00 EDTA numerical study of the extended Kohn–Sham ground states of atomshttps://projecteuclid.org/euclid.camcos/1530842698<strong>Eric Cancès</strong>, <strong>Nahia Mourad</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 139--188.</p><p><strong>Abstract:</strong><br/>
In this article, we consider the extended Kohn–Sham model for atoms subjected to cylindrically symmetric external potentials. The variational approximation of the model and the construction of appropriate discretization spaces are detailed together with the algorithm to solve the discretized Kohn–Sham equations used in our code. Using this code, we compute the occupied and unoccupied energy levels of all the atoms of the first four rows of the periodic table for the reduced Hartree–Fock (rHF) and the extended Kohn–Sham X [math] models. These results allow us to test numerically the assumptions on the negative spectra of atomic rHF and Kohn–Sham Hamiltonians used in our previous theoretical works on density functional perturbation theory and pseudopotentials. Interestingly, we observe accidental degeneracies between s and d shells or between p and d shells at the Fermi level of some atoms. We also consider the case of an atom subjected to a uniform electric field. For various magnitudes of the electric field, we compute the response of the density of the carbon atom confined in a large ball with Dirichlet boundary conditions, and we check that, in the limit of small electric fields, the results agree with the ones obtained with first-order density functional perturbation theory.
</p>projecteuclid.org/euclid.camcos/1530842698_20180705220500Thu, 05 Jul 2018 22:05 EDTAn equation-by-equation method for solving the multidimensional moment constrained maximum entropy problemhttps://projecteuclid.org/euclid.camcos/1530842699<strong>Wenrui Hao</strong>, <strong>John Harlim</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 189--214.</p><p><strong>Abstract:</strong><br/>
An equation-by-equation (EBE) method is proposed to solve a system of nonlinear equations arising from the moment constrained maximum entropy problem of multidimensional variables. The design of the EBE method combines ideas from homotopy continuation and Newton’s iterative methods. Theoretically, we establish the local convergence under appropriate conditions and show that the proposed method, geometrically, finds the solution by searching along the surface corresponding to one component of the nonlinear problem. We will demonstrate the robustness of the method on various numerical examples, including (1) a six-moment one-dimensional entropy problem with an explicit solution that contains components of order [math] – [math] in magnitude, (2) four-moment multidimensional entropy problems with explicit solutions where the resulting systems to be solved range from [math] – [math] equations, and (3) four- to eight-moment of a two-dimensional entropy problem, whose solutions correspond to the densities of the two leading EOFs of the wind stress-driven large-scale oceanic model. In this case, we find that the EBE method is more accurate compared to the classical Newton’s method, the Matlab generic solver, and the previously developed BFGS-based method, which was also tested on this problem. The fourth example is four-moment constrained of up to five-dimensional entropy problems whose solutions correspond to multidimensional densities of the components of the solutions of the Kuramoto–Sivashinsky equation. For the higher-dimensional cases of this example, the EBE method is superior because it automatically selects a subset of the prescribed moment constraints from which the maximum entropy solution can be estimated within the desired tolerance. This selection feature is particularly important since the moment constrained maximum entropy problems do not necessarily have solutions in general.
</p>projecteuclid.org/euclid.camcos/1530842699_20180705220500Thu, 05 Jul 2018 22:05 EDTSymmetrized importance samplers for stochastic differential equationshttps://projecteuclid.org/euclid.camcos/1530842700<strong>Andrew Leach</strong>, <strong>Kevin K. Lin</strong>, <strong>Matthias Morzfeld</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 215--241.</p><p><strong>Abstract:</strong><br/>
We study a class of importance sampling methods for stochastic differential equations (SDEs). A small noise analysis is performed, and the results suggest that a simple symmetrization procedure can significantly improve the performance of our importance sampling schemes when the noise is not too large. We demonstrate that this is indeed the case for a number of linear and nonlinear examples. Potential applications, e.g., data assimilation, are discussed.
</p>projecteuclid.org/euclid.camcos/1530842700_20180705220500Thu, 05 Jul 2018 22:05 EDTEfficient high-order discontinuous Galerkin computations of low Mach number flowshttps://projecteuclid.org/euclid.camcos/1538013628<strong>Jonas Zeifang</strong>, <strong>Klaus Kaiser</strong>, <strong>Andrea Beck</strong>, <strong>Jochen Schütz</strong>, <strong>Claus-Dieter Munz</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 243--270.</p><p><strong>Abstract:</strong><br/>
We consider the efficient approximation of low Mach number flows by a high-order scheme, coupling a discontinuous Galerkin (DG) discretization in space with an implicit/explicit (IMEX) discretization in time. The splitting into linear implicit and nonlinear explicit parts relies heavily on the incompressible solution. The method has been originally developed for a singularly perturbed ODE and applied to the isentropic Euler equations. Here, we improve, extend, and investigate the so-called RS-IMEX splitting method. The resulting scheme can cope with a broader range of Mach numbers without running into roundoff errors, it is extended to realistic physical boundary conditions, and it is shown to be highly efficient in comparison to more standard solution techniques.
</p>projecteuclid.org/euclid.camcos/1538013628_20180926220052Wed, 26 Sep 2018 22:00 EDTA numerical study of the relativistic Burgers and Euler equations on a Schwarzschild black hole exteriorhttps://projecteuclid.org/euclid.camcos/1538013632<strong>Philippe G. LeFloch</strong>, <strong>Shuyang Xiang</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 271--301.</p><p><strong>Abstract:</strong><br/>
We study the dynamical behavior of compressible fluids evolving on the outer domain of communication of a Schwarzschild background. For both the relativistic Burgers equation and the relativistic Euler system, assuming spherical symmetry we introduce numerical methods that take the Schwarzschild geometry and, specifically, the steady state solutions into account. The schemes we propose preserve the family of steady state solutions and enable us to study the nonlinear stability of fluid equilibria and the behavior of solutions near the black hole horizon. We state and numerically demonstrate several properties about the late-time behavior of perturbed steady states.
</p>projecteuclid.org/euclid.camcos/1538013632_20180926220052Wed, 26 Sep 2018 22:00 EDTA semi-implicit multiscale scheme for shallow water flows at low Froude numberhttps://projecteuclid.org/euclid.camcos/1538013633<strong>Stefan Vater</strong>, <strong>Rupert Klein</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 13, Number 2, 303--336.</p><p><strong>Abstract:</strong><br/>
A new large time step semi-implicit multiscale method is presented for the solution of low Froude number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite volume discretization is implemented in the framework of second-order Godunov-type methods for conservation laws. The basic properties of the method are validated by numerical tests. This development is a further step in the construction of asymptotically adaptive numerical methods for the computation of large-scale atmospheric flows.
</p>projecteuclid.org/euclid.camcos/1538013633_20180926220052Wed, 26 Sep 2018 22:00 EDT