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Communications in Applied Mathematics and Computational Science
Communications in Applied Mathematics and Computational Science
Pospíšil, Lukáš; Gagliardini, Patrick; Sawyer, William; Horenko, Illia
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...
Kotovshchikova, Marina; Firsov, Dmitry K.; Lui, Shiu Hong
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...
Weiser, Martin; Ghosh, Sunayana
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...
Pazner, Will; Persson, Per-Olof
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...
Hayhurst, Brian; Keller, Mason; Rai, Chris; Sun, Xidian; Westphal, Chad R.
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...
Stinis, Panagiotis
We present a mesh refinement algorithm for detecting singularities of time-dependent partial differential equations. The algorithm is inspired by renormalization constructions used in statistical mechanics to evaluate the properties of a system near a critical point, that is, a phase transition. The main idea behind the algorithm is to treat the occurrence of singularities of time-dependent partial differential equations as phase transitions.
¶ The algorithm assumes the knowledge of an accurate reduced model. In particular, we need only assume that we know the functional form of the reduced model, that is, the terms appearing in the reduced model, but not necessarily...
Ambrose, David; Wilkening, Jon
We classify all bifurcations from traveling waves to nontrivial time-periodic solutions of the Benjamin–Ono equation that are predicted by linearization. We use a spectrally accurate numerical continuation method to study several paths of nontrivial solutions beyond the realm of linear theory. These paths are found to either reconnect with a different traveling wave or to blow up. In the latter case, as the bifurcation parameter approaches a critical value, the amplitude of the initial condition grows without bound and the period approaches zero. We then prove a theorem that gives the mapping from one bifurcation to its counterpart on the...
Barenblatt, Grigory
A modified model of turbulent shear flow of a suspension of small heavy particles in a fluid is presented. The modification is based on the assumption that in the flow there are two sorts of particles. For the particles of the first sort the velocity of free fall [math] is larger than the characteristic velocity fluctuation, for the particles of the second sort the velocity of free fall [math] is less than the characteristic velocity of fluctuation.
Sekora, Michael; Stone, James
A higher-order Godunov method for the radiation subsystem of radiation hydrodynamics is presented. A key ingredient of the method is the direct coupling of stiff source term effects to the hyperbolic structure of the system of conservation laws; it is composed of a predictor step that is based on Duhamel’s principle and a corrector step that is based on Picard iteration. The method is second-order accurate in both time and space, unsplit, asymptotically preserving, and uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit....
Mori, Yoichiro; Peskin, Charles
We present a numerical method for solving the system of equations of a model of cellular electrical activity that takes into account both geometrical effects and ionic concentration dynamics. A challenge in constructing a numerical scheme for this model is that its equations are stiff: There is a time scale associated with “diffusion” of the membrane potential that is much faster than the time scale associated with the physical diffusion of ions. We use an implicit discretization in time and a finite volume discretization in space. We present convergence studies of the numerical method for cylindrical and two-dimensional geometries for...
Nonaka, Andrew; Trebotich, David; Miller, Gregory; Graves, Daniel; Colella, Phillip
We present a conservative finite difference method designed to capture elastic wave propagation in viscoelastic fluids in two dimensions. We model the incompressible Navier–Stokes equations with an extra viscoelastic stress described by the Oldroyd-B constitutive equations. The equations are cast into a hybrid conservation form which is amenable to the use of a second-order Godunov method for the hyperbolic part of the equations, including a new exact Riemann solver. A numerical stress splitting technique provides a well-posed discretization for the entire range of Newtonian and elastic fluids. Incompressibility is enforced through a projection method and a partitioning of variables that...
Christlieb, Andrew; Ong, Benjamin; Qiu, Jing-Mei
A class of novel deferred correction methods, integral deferred correction (IDC) methods, is studied. This class of methods is an extension of ideas introduced by Dutt, Greengard and Rokhlin on spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs). The novel nature of this class of defect correction methods is that the correction of the defect is carried out using an accurate integral form of the residual instead of the more familiar differential form. As a family of methods, these schemes are capable of matching the efficiency of popular high-order RK methods.
¶ The smoothness of the error vector...
Cai, Xiao-Chuan; Liu, Si; Zou, Jun
We study an overlapping domain decomposition method for solving the coupled nonlinear system of equations arising from the discretization of inverse elliptic problems. Most algorithms for solving inverse problems take advantage of the fact that the optimality system has a natural splitting into three components: the state equation for the constraints, the adjoint equation for the Lagrange multipliers, and the equation for the parameter to be identified. Such algorithms often involve interiterations between the three separate solvers, and the intercomponent iteration is sequential. Several fully coupled or so-called one-shot approaches exist, and the main challenges in these approaches are that...
Aspden, Andrew; Nikiforakis, Nikos; Dalziel, Stuart; Bell, John
Implicit LES methods are numerical methods that capture the energy-containing and inertial ranges of turbulent flows, while relying on their own intrinsic dissipation to act as a subgrid model. We present a scheme-dependent Kolmogorov scaling analysis of the solutions produced by such methods. From this analysis we can define an effective Reynolds number for implicit LES simulations of inviscid flow. The approach can also be used to define an effective Reynolds number for under-resolved viscous simulations. Simulations of maintained homogeneous isotropic turbulence and the Taylor–Green vortex are presented to support this proposal and highlight similarities and differences with real-world viscous...
Beale, J. Thomas; Chopp, David; LeVeque, Randall; Li, Zhilin
A recent paper by Vaughan, Smith, and Chopp [Comm. App. Math. & Comp. Sci. 1 (2006), 207–228] reported numerical results for three examples using the immersed interface method (IIM) and the extended finite element method (X-FEM). The results presented for the IIM showed first-order accuracy for the solution and inaccurate values of the normal derivative at the interface. This was due to an error in the implementation. The purpose of this note is to present correct results using the IIM for the same examples used in that paper, which demonstrate the expected second-order accuracy in the maximum norm over all grid points. Results now...
Chorin, Alexandre
A sampling method for spin systems is presented. The spin lattice is written as the union of a nested sequence of sublattices, all but the last with conditionally independent spins, which are sampled in succession using their marginals. The marginals are computed concurrently by a fast algorithm; errors in the evaluation of the marginals are offset by weights. There are no Markov chains and each sample is independent of the previous ones; the cost of a sample is proportional to the number of spins (but the number of samples needed for good statistics may grow with array size). The examples...
Mokhov, Sergiy; Zeldovich, Boris
The one-dimensional wave equation describing propagation and reflection of waves in a layered medium is transformed into an exact first-order system for the amplitudes of coupled counter-propagating waves. Any choice of such amplitudes, out of continuous multitude of them, allows one to get an accurate numerical solution of the reflection problem. We discuss relative advantages of particular choices of amplitude.
¶ We also introduce the notion of reflection strength [math] of a plane wave by a nonabsorbing layer, which is related to the reflection intensity [math] by [math] . We show that the total reflection strength by a sequence of elements...
Tu, Xuemin; Li, Jing
The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. A preconditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partially subassembled interface problem is solved. A convergence rate estimate for the GMRES iteration is established for the cases where the advection is not strong, under the condition that the mesh size is small enough. The estimate deteriorates with a decrease of the viscosity and for fixed viscosity...
Dal Pont, Stefano; Dimnet, Eric
This paper presents a model for the description of instantaneous collisions and a computational method for the simulation of multiparticle systems’ evolution. The description of the behavior of a collection of discrete bodies is based on the consideration that the global system is deformable even if particles are rigid. Making use of the principle of virtual work, the equations describing the regular (that is, smooth) as well as the discontinuous (that is, the collisions) evolutions of the motion system are obtained. For an instantaneous collision involving several rigid particles, the existence and the uniqueness of the solution as well as...
Cogan, Nicholas
We consider the coupled motion of a passive interface separating two immiscible fluids of different viscosities. There are several applications where the velocity of the two fluids is needed everywhere within the domain. Examples include the transport of bacteria and diffusing substances within a biofilm matrix and the transport of cations throughout the mucociliary and periciliary layer in the lung lining. In this investigation, we use a hybrid approach which employs the boundary integral method to determine the interface velocity and the method of regularized stokeslets to determine the velocity elsewhere in the domain.
¶ Our approach capitalizes on the strengths...