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 EDTComputation of volume potentials on structured grids with the method of local correctionshttps://projecteuclid.org/euclid.camcos/1564020018<strong>Chris Kavouklis</strong>, <strong>Phillip Colella</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 1, 1--32.</p><p><strong>Abstract:</strong><br/>
We present a new version of the method of local corrections (MLC) of McCorquodale, Colella, Balls, and Baden (2007), a multilevel, low-communication, noniterative domain decomposition algorithm for the numerical solution of the free space Poisson’s equation in three dimensions on locally structured grids. In this method, the field is computed as a linear superposition of local fields induced by charges on rectangular patches of size [math] mesh points, with the global coupling represented by a coarse-grid solution using a right-hand side computed from the local solutions. In the present method, the local convolutions are further decomposed into a short-range contribution computed by convolution with the discrete Green’s function for a [math] -th-order accurate finite difference approximation to the Laplacian with the full right-hand side on the patch, combined with a longer-range component that is the field induced by the terms up to order [math] of the Legendre expansion of the charge over the patch. This leads to a method with a solution error that has an asymptotic bound of [math] , where [math] is the mesh spacing and [math] is the max norm of the charge times a rapidly decaying function of the radius of the support of the local solutions scaled by [math] . The bound [math] is essentially the error of the global potential computed on the coarsest grid in the hierarchy. Thus, we have eliminated the low-order accuracy of the original method (which corresponds to [math] in the present method) for smooth solutions, while keeping the computational cost per patch nearly the same as that of the original method. Specifically, in addition to the local solves of the original method we only have to compute and communicate the expansion coefficients of local expansions (that is, for instance, 20 scalars per patch for [math] ). Several numerical examples are presented to illustrate the new method and demonstrate its convergence properties.
</p>projecteuclid.org/euclid.camcos/1564020018_20190724220033Wed, 24 Jul 2019 22:00 EDTOn the convergence of spectral deferred correction methodshttps://projecteuclid.org/euclid.camcos/1564020020<strong>Mathew F. Causley</strong>, <strong>David C. Seal</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 1, 33--64.</p><p><strong>Abstract:</strong><br/>
In this work we analyze the convergence properties of the spectral deferred correction (SDC) method originally proposed by Dutt et al. (BIT 40 (2000), no. 2, 241–266). The framework for this high-order ordinary differential equation (ODE) solver is typically described as a low-order approximation (such as forward or backward Euler) lifted to higher-order accuracy by applying the same low-order method to an error equation and then adding in the resulting defect to correct the solution. Our focus is not on solving the error equation to increase the order of accuracy, but on rewriting the solver as an iterative Picard integral equation solver. In doing so, our chief finding is that it is not the low-order solver that picks up the order of accuracy with each correction, but it is the underlying quadrature rule of the right-hand-side function that is solely responsible for picking up additional orders of accuracy. Our proofs point to a total of three sources of errors that SDC methods carry: the error at the current time point, the error from the previous iterate, and the numerical integration error that comes from the total number of quadrature nodes used for integration. The second of these two sources of errors is what separates SDC methods from Picard integral equation methods; our findings indicate that as long as the difference between the current and previous iterates always gets multiplied by at least a constant multiple of the time step size, then high-order accuracy can be found even if the underlying ODE “solver” is inconsistent. From this vantage, we solidify the prospects of extending spectral deferred correction methods to a larger class of solvers, of which we present some examples.
</p>projecteuclid.org/euclid.camcos/1564020020_20190724220033Wed, 24 Jul 2019 22:00 EDTA theoretical study of aqueous humor secretion based on a continuum model coupling electrochemical and fluid-dynamical transmembrane mechanismshttps://projecteuclid.org/euclid.camcos/1564020021<strong>Lorenzo Sala</strong>, <strong>Aurelio Giancarlo Mauri</strong>, <strong>Riccardo Sacco</strong>, <strong>Dario Messenio</strong>, <strong>Giovanna Guidoboni</strong>, <strong>Alon Harris</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 1, 65--103.</p><p><strong>Abstract:</strong><br/>
Intraocular pressure, resulting from the balance of aqueous humor (AH) production and drainage, is the only approved treatable risk factor in glaucoma. AH production is determined by the concurrent function of ion pumps and aquaporins in the ciliary processes, but their individual contribution is difficult to characterize experimentally. In this work, we propose a novel unified modeling and computational framework for the finite element simulation of the role of the main ion pumps and exchangers involved in AH secretion, namely, the sodium-potassium pump, the calcium-sodium exchanger, the chloride-bicarbonate exchanger, and the sodium-proton exchanger. The theoretical model is developed at the cellular scale and is based on the coupling between electrochemical and fluid-dynamical transmembrane mechanisms characterized by a novel description of the electric pressure exerted by the ions on the intrapore fluid that includes electrochemical and osmotic corrections. Considering a realistic geometry of the ion pumps, the proposed model is demonstrated to correctly predict their functionality as a function of (1) the permanent electric charge density over the pore surface, (2) the osmotic gradient coefficient, and (3) the stoichiometric ratio between the ion pump currents enforced at the inlet and outlet sections of the pore. In particular, theoretical predictions of the transepithelial membrane potential for each simulated pump/exchanger allow us to perform a first significant model comparison with experimental data for monkeys. This is a significant step for future multidisciplinary studies on the action of molecules on AH production.
</p>projecteuclid.org/euclid.camcos/1564020021_20190724220033Wed, 24 Jul 2019 22:00 EDTAn adaptive local discrete convolution method for the numerical solution of Maxwell's equationshttps://projecteuclid.org/euclid.camcos/1564020022<strong>Boris Lo</strong>, <strong>Phillip Colella</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 1, 105--119.</p><p><strong>Abstract:</strong><br/>
We present a numerical method for solving the free-space Maxwell’s equations in three dimensions using compact convolution kernels on a rectangular grid. We first rewrite Maxwell’s equations as a system of wave equations with auxiliary variables and discretize its solution from the method of spherical means. The algorithm has been extended to be used on a locally refined nested hierarchy of rectangular grids.
</p>projecteuclid.org/euclid.camcos/1564020022_20190724220033Wed, 24 Jul 2019 22:00 EDTSimple second-order finite differences for elliptic PDEs with discontinuous coefficients and interfaceshttps://projecteuclid.org/euclid.camcos/1584669735<strong>Chung-Nan Tzou</strong>, <strong>Samuel N. Stechmann</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 2, 121--147.</p><p><strong>Abstract:</strong><br/>
In multiphase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise due to changes in material properties at an immersed interface or embedded boundary, which may have an irregular shape. Consequently, the solution and its gradient can be discontinuous, and numerical methods can be difficult to design. Here a new method is presented and analyzed, using a simple formulation of one-dimensional finite differences on a Cartesian grid, allowing for a relatively easy setup for one-, two-, or three-dimensional problems. The derivation is relatively simple and mainly involves centered finite difference formulas, with less reliance on the Taylor series expansions of typical immersed interface method derivations. The method preserves a sharp interface with discontinuous solutions, obtained from a small number of iterations (approximately five) of solving a symmetric linear system with updates to the right-hand side. Second-order accuracy is rigorously proven in one spatial dimension and demonstrated through numerical examples in two and three spatial dimensions. The method is tested here on the variable-coefficient Poisson equation, and it could be extended for use on time-dependent problems of heat transfer, fluid dynamics, or other applications.
</p>projecteuclid.org/euclid.camcos/1584669735_20200319220227Thu, 19 Mar 2020 22:02 EDT2D force constraints in the method of regularized Stokesletshttps://projecteuclid.org/euclid.camcos/1584669736<strong>Ondrej Maxian</strong>, <strong>Wanda Strychalski</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 2, 149--174.</p><p><strong>Abstract:</strong><br/>
For many biological systems that involve elastic structures immersed in fluid, small length scales mean that inertial effects are also small, and the fluid obeys the Stokes equations. One way to solve the model equations representing such systems is through the Stokeslet, the fundamental solution to the Stokes equations, and its regularized counterpart, which treats the singularity of the velocity at points where force is applied. In two dimensions, an additional complication arises from Stokes’ paradox, whereby the velocity from the Stokeslet is unbounded at infinity when the net hydrodynamic force within the domain is nonzero, invalidating any solutions that use the free space Stokeslet. A straightforward computationally inexpensive method is presented for obtaining valid solutions to the Stokes equations for net nonzero forcing. The approach is based on modifying the boundary conditions of the Stokes equations to impose a mean zero velocity condition on a large curve that surrounds the domain of interest. The corresponding Green’s function is derived and used as a fundamental solution in the case of net nonzero forcing. The numerical method is applied to models of cellular motility and blebbing, both of which involve tether forces that are not required to integrate to zero.
</p>projecteuclid.org/euclid.camcos/1584669736_20200319220227Thu, 19 Mar 2020 22:02 EDTPotential field formulation based on decomposition of the electric field for a nonlinear induction hardening modelhttps://projecteuclid.org/euclid.camcos/1584669737<strong>Tong Kang</strong>, <strong>Ran Wang</strong>, <strong>Huai Zhang</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 2, 175--205.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate a mathematical model of induction heating including eddy current equations coupled with a nonlinear heat equation. A nonlinear law between the magnetic field and the magnetic induction field in the workpiece is assumed. Meanwhile the electric conductivity is temperature dependent. We present a potential field formulation (the [math] - [math] method) based on decomposition of the electric field for the electromagnetic part. Using the theory of monotone operator and Rothe’s method, we prove the existence of a weak solution to the coupled nonlinear system in the conducting domain. Finally, we solve it by means of the [math] - [math] finite element method and show some numerical simulation results.
</p>projecteuclid.org/euclid.camcos/1584669737_20200319220227Thu, 19 Mar 2020 22:02 EDTComputing the quasipotential for highly dissipative and chaotic SDEs an application to stochastic Lorenz'63https://projecteuclid.org/euclid.camcos/1584669738<strong>Maria Cameron</strong>, <strong>Shuo Yang</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 2, 207--246.</p><p><strong>Abstract:</strong><br/>
The study of noise-driven transitions occurring rarely on the time scale of systems modeled by SDEs is of crucial importance for understanding such phenomena as genetic switches in living organisms and magnetization switches of the Earth. For a gradient SDE, the predictions for transition times and paths between its metastable states are done using the potential function. For a nongradient SDE, one needs to decompose its forcing into a gradient of the so-called quasipotential and a rotational component, which cannot be done analytically in general.
We propose a methodology for computing the quasipotential for highly dissipative and chaotic systems built on the example of Lorenz’63 with an added stochastic term. It is based on the ordered line integral method, a Dijkstra-like quasipotential solver, and combines 3D computations in whole regions, a dimensional reduction technique, and 2D computations on radial meshes on manifolds or their unions. Our collection of source codes is available on M. Cameron’s web page and on GitHub.
</p>projecteuclid.org/euclid.camcos/1584669738_20200319220227Thu, 19 Mar 2020 22:02 EDTEfficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methodshttps://projecteuclid.org/euclid.camcos/1584669739<strong>Robert I. Saye</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 14, Number 2, 247--283.</p><p><strong>Abstract:</strong><br/>
With an emphasis on achieving ideal multigrid solver performance, this paper explores the design of local discontinuous Galerkin schemes for multiphase elliptic interface problems. In particular, for cases exhibiting coefficient discontinuities several orders in magnitude, the role of viscosity-weighted numerical fluxes on interfacial mesh faces is examined: findings support a known strategy of harmonic weighting, but also show that further improvements can be made via a stronger kind of biasing, denoted herein as viscosity-upwinded weighting. Applying this strategy, multigrid performance is assessed for a variety of elliptic interface problems in 1D, 2D, and 3D, across 16 orders of viscosity ratio. These include constant- and variable-coefficient problems, multiphase checkerboard patterns, implicitly defined interfaces, and 3D problems with intricate geometry. With the exception of a challenging case involving a lattice of vanishingly small droplets, in all demonstrated examples the condition number of the multigrid V-cycle preconditioned system has unit order magnitude, independent of the mesh size [math] .
</p>projecteuclid.org/euclid.camcos/1584669739_20200319220227Thu, 19 Mar 2020 22:02 EDTInvestigation of finite-volume methods to capture shocks and turbulence spectra in compressible flowshttps://projecteuclid.org/euclid.camcos/1593072123<strong>Emmanuel Motheau</strong>, <strong>John Wakefield</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 15, Number 1, 1--36.</p><p><strong>Abstract:</strong><br/>
The aim of the present paper is to provide a comparison between several finite-volume methods of different numerical accuracy: the second-order Godunov method with PPM interpolation and the high-order finite-volume WENO method. The results show that while on a smooth problem the high-order method performs better than the second-order one, when the solution contains a shock all the methods collapse to first-order accuracy. In the context of the decay of compressible homogeneous isotropic turbulence with shocklets, the actual overall order of accuracy of the methods reduces to second-order, despite the use of fifth-order reconstruction schemes at cell interfaces. Most important, results in terms of turbulent spectra are similar regardless of the numerical methods employed, except that the PPM method fails to provide an accurate representation in the high-frequency range of the spectra. It is found that this specific issue comes from the slope-limiting procedure and a novel hybrid PPM/WENO method is developed that has the ability to capture the turbulent spectra with the accuracy of a high-order method, but at the cost of the second-order Godunov method. Overall, it is shown that virtually the same physical solution can be obtained much faster by refining a simulation with the second-order method and carefully chosen numerical procedures, rather than running a coarse high-order simulation. Our results demonstrate the importance of evaluating the accuracy of a numerical method in terms of its actual spectral dissipation and dispersion properties on mixed smooth/shock cases, rather than by the theoretical formal order of convergence rate.
</p>projecteuclid.org/euclid.camcos/1593072123_20200625040211Thu, 25 Jun 2020 04:02 EDTA stochastic version of Stein variational gradient descent for efficient samplinghttps://projecteuclid.org/euclid.camcos/1593072124<strong>Lei Li</strong>, <strong>Yingzhou Li</strong>, <strong>Jian-Guo Liu</strong>, <strong>Zibu Liu</strong>, <strong>Jianfeng Lu</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 15, Number 1, 37--63.</p><p><strong>Abstract:</strong><br/>
We propose in this work RBM-SVGD, a stochastic version of the Stein variational gradient descent (SVGD) method for efficiently sampling from a given probability measure, which is thus useful for Bayesian inference. The method is to apply the random batch method (RBM) for interacting particle systems proposed by Jin et al. to the interacting particle systems in SVGD. While keeping the behaviors of SVGD, it reduces the computational cost, especially when the interacting kernel has long range. We prove that the one marginal distribution of the particles generated by this method converges to the one marginal of the interacting particle systems under Wasserstein-2 distance on fixed time interval [math] . Numerical examples verify the efficiency of this new version of SVGD.
</p>projecteuclid.org/euclid.camcos/1593072124_20200625040211Thu, 25 Jun 2020 04:02 EDTA third-order multirate Runge–Kutta scheme for finite volume solution of 3D time-dependent Maxwell's equationshttps://projecteuclid.org/euclid.camcos/1593072126<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 15, Number 1, 65--87.</p><p><strong>Abstract:</strong><br/>
A third-order multirate time-stepping based on an SSP Runge–Kutta method is applied to solve the three-dimensional Maxwell’s equations on unstructured tetrahedral meshes. This allows for an evolution of the solution on fine and coarse meshes with time steps satisfying a local stability condition to improve the computational efficiency of numerical simulations. Two multirate strategies with flexible time-step ratios are compared for accuracy and efficiency. Numerical experiments with a third-order finite volume discretization are presented to validate the theory. Our results of electromagnetic simulations demonstrate that 1D analysis is also valid for linear conservation laws in 3D. In one of the methods, significant speedup in 3D simulations is achieved without sacrificing third-order accuracy.
</p>projecteuclid.org/euclid.camcos/1593072126_20200625040211Thu, 25 Jun 2020 04:02 EDTFast optical absorption spectra calculations for periodic solid state systemshttps://projecteuclid.org/euclid.camcos/1593072127<strong>Felix Henneke</strong>, <strong>Lin Lin</strong>, <strong>Christian Vorwerk</strong>, <strong>Claudia Draxl</strong>, <strong>Rupert Klein</strong>, <strong>Chao Yang</strong>. <p><strong>Source: </strong>Communications in Applied Mathematics and Computational Science, Volume 15, Number 1, 89--113.</p><p><strong>Abstract:</strong><br/>
We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe–Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe–Salpeter Hamiltonian can be reduced to nearly optimal as [math] with respect to the number of samples in the Brillouin zone [math] for the first time. In addition, we show that the cost for applying the Bethe–Salpeter Hamiltonian to a vector scales as [math] . Therefore, the optical absorption spectrum, as well as selected excitation energies, can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the [math] and [math] scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian, respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with [math] , it takes [math] hours to assemble the Bethe–Salpeter Hamiltonian and [math] hours to fully diagonalize the Hamiltonian using [math] cores when the brute force approach is used. The new method takes less than [math] minutes to set up the Hamiltonian and [math] minutes to compute the absorption spectrum on a single core.
</p>projecteuclid.org/euclid.camcos/1593072127_20200625040211Thu, 25 Jun 2020 04:02 EDT