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Communications in Applied Mathematics and Computational Science
Communications in Applied Mathematics and Computational Science
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...
Mathelin, Lionel; Le Maître, Olivier
We present an a posteriori error estimation for the numerical solution of a stochastic variational problem arising in the context of parametric uncertainties. The discretization of the stochastic variational problem uses standard finite elements in space and piecewise continuous orthogonal polynomials in the stochastic domain. The a posteriori methodology is derived by measuring the error as the functional difference between the continuous and discrete solutions. This functional difference is approximated using the discrete solution of the primal stochastic problem and two discrete adjoint solutions (on two imbricated spaces) of the associated dual stochastic problem. The dual problem being linear, the...
McCorquodale, Peter; Colella, Phillip; Balls, Gregory; Baden, Scott
We present a second-order accurate algorithm for solving the free-space Poisson’s equation on a locally-refined nested grid hierarchy in three dimensions. Our approach is based on linear superposition of local convolutions of localized charge distributions, with the nonlocal coupling represented on coarser grids. The representation of the nonlocal coupling on the local solutions is based on Anderson’s Method of Local Corrections and does not require iteration between different resolutions. A distributed-memory parallel implementation of this method is observed to have a computational cost per grid point less than three times that of a standard FFT-based method on a uniform grid...
Smith, Bryan; Vaughan, Benjamin; Chopp, David
In this paper, we use the eXtended Finite Element Method, with customized enrichment functions determined by asymptotic analysis, to study boundary layer behavior in elliptic equations with discontinuous coefficients. In particular, we look at equations where the coefficients are discontinuous across a boundary internal to the domain. We also show how to implement this method for Dirichlet conditions at an interface. The method requires neither the mesh to conform to the internal boundary, nor the mesh to have additional refinement near the interface, making this an ideal method for moving interface type problems. We then apply this method to equations...
Layton, Anita; Minion, Michael
High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. This study assesses the efficiency of SIPIDC methods that instead use standard semi-implicit methods with orders two through four to compute the provisional solution. Numerical results...
Vaughan, Benjamin; Smith, Bryan; Chopp, David
We compare the Immersed Interface Method (IIM) with the Extended Finite Element Method (X-FEM) for elliptic equations with singular sources and discontinuous coefficients. The IIM has been compared favorably with a number of other competing methods. These methods are of particular interest because they allow for the solution of elliptic equations with internal boundaries on nonconforming meshes. In the context of moving interface problems, the emphasis in this paper is placed on accuracy of solutions and their normal derivatives on the interface. These methods are briefly described and the results for benchmark problems are compared.