Acoustic scattering cross sections of smart furtive obstacles are studied and discussed. A smart furtive obstacle is an obstacle that, when hit by an incoming field, avoids detection through the use of a pressure current acting on its boundary. A highly parallelizable algorithm for computing the acoustic scattering cross section of smart obstacles is developed. As a case study, this algorithm is applied to the (acoustic) scattering cross section of a “smart” (furtive) simplified version of the NASA space shuttle when hit by incoming time-harmonic plane waves, the wavelengths of which are small compared to the characteristic dimensions of the shuttle. The solution to this numerically challenging scattering problem requires the solution of systems of linear equations with many unknowns and equations. Due to the sparsity of these systems of equations, they can be stored and solved using affordable computing resources. A cross section analysis of the simplified NASA space shuttle highlights three findings: i) the smart furtive obstacle reduces the magnitude of its cross section compared to the cross section of a corresponding “passive” obstacle; ii) several wave propagation directions fail to satisfactorily respond to the smart strategy of the obstacle; iii) satisfactory furtive effects along all directions may only be obtained by using a pressure current of considerable magnitude. Numerical experiments and virtual reality applications can be found at the website: http://www.ceri.uniromal.it/ceri/zirilli/w7.

The dynamics of the Liesegang type pattern formation is investigated in a centrally symmetric two-dimensional setup. According to the observations in real experiments, the qualitative change of the dynamics is exhibited for slightly different initial conditions. Two kinds of chemical mechanisms are studied; in both cases the pattern formation is described using a phase separation model including the Cahn-Hilliard equations. For the numerical simulations we make use of an adaptive grid PDE method, which successfully deals with the computationally critical cases such as steep gradients in the concentration distribution and investigation of long time behavior. The numerical simulations show a good agreement with the real experiments.

The present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Wellbalancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

In low speed flow computations, compressible finite-volume solvers are known to a) fail to converge in acceptable time and b) reach unphysical solutions. These problems are known to be cured by A) preconditioning on the time-derivative term, and B) control of numerical dissipation, respectively. There have been several methods of A) and B) proposed separately. However, it is unclear which combination is the most accurate, robust, and efficient for low speed flows. We carried out a comparative study of several well-known or recently-developed low-dissipation Euler fluxes coupled with a preconditioned LU-SGS (Lower-Upper Symmetric Gauss-Seidel) implicit time integration scheme to compute steady flows. Through a series of numerical experiments, accurate, efficient, and robust methods are suggested for low speed flow computations.

To finite-difference model elastic wave propagation in a combined structure with solid, fluid and porous subregions, a set of modified Biot’s equations are used, which can be reduced to the governing equations in solids, fluids as well as fluid-saturated porous media. Based on the modified Biot’s equations, the field quantities are finite-difference discretized into unified forms in the whole structure, including those on any interface between the solid, fluid and porous subregions. For the discrete equations on interfaces, however, the harmonic mean of shear modulus and the arithmetic mean of the other parameters on both sides of the interfaces are used. These parameter averaging equations are validated by deriving from the continuity conditions on the interfaces. As an example of using the parameter averaging technique, a 2-D finite-difference scheme with a velocity-stress staggered grid in cylindrical coordinates is implemented to simulate the acoustic logs in porous formations. The finite-difference simulations of the acoustic logging in a homogeneous formation agree well with those obtained by the analytical method. The acoustic logs with mud cakes clinging to the borehole well are simulated for investigating the effect of mud cake on the acoustic logs. The acoustic logs with a varying radius borehole embedded in a horizontally stratified formation are also simulated by using the proposed finite-difference scheme.

Two fundamental facts of the modern wave turbulence theory are 1) existence of power energy spectra in k-space, and 2) existence of “gaps” in this spectra corresponding to the resonance clustering. Accordingly, three wave turbulent regimes are singled out: kinetic, described by wave kinetic equations and power energy spectra; discrete, characterized by resonance clustering; and mesoscopic, where both types of wave field time evolution coexist. In this review paper we present the results on integrable dynamics of resonance clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a novel method based on the notion of dynamical invariant we show that some of the frequently met clusters are integrable in quadratures for arbitrary initial conditions and some others-only for particular initial conditions. We also identify chaotic behaviour in some cases. Physical implications of the results obtained are discussed.

When gas flows through corrugated pipes, pressure waves interacting with vortex shedding can produce distinct tonal noise and structural vibration. Based on established observations, a model is proposed which couples an acoustic pipe and self-excited oscillations with vortex shedding over the corrugation cavities. In the model, the acoustic response of the corrugated pipe is simulated by connecting the lossless medium moving with a constant velocity with a source based on a discrete distribution of van der Pol oscillators arranged along the pipe. Our time accurate solutions exhibit dynamic behavior consistent with that experimentally observed, including the lock-in frequency of vortex shedding, standing waves and the onset fluid velocity capable of generating the lock-in.

The multi-mode approximation is presented to compute the interior wave function of Schrödinger equation. This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered. The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method. The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al. [Phys. Rev. E., 64 (2001), 011208] is adopted by adding forcing terms into the resulting microdynamic evolution equation. The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form. The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion. The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented. It is found the Knudsen minimum can be captured in all three statistics studied. The results also indicate distinct characteristics of the effects of quantum statistics.

A code that models signals produced by charge-exchange reactions between fast ions and injected neutral beams in tokamak plasmas is described. With the fast-ion distribution function as input, the code predicts the efflux to a neutral particle analyzer (NPA) diagnostic and the photon radiance of Balmer-alpha light to a fast-ion Dα (FIDA) diagnostic. Reactions with both the primary injected neutrals and with the cloud of secondary “halo” neutrals that surround the beam are treated. Accurate calculation of the fraction of neutrals that occupy excited atomic states (the collisional-radiative transition equations) is an important element of the code. Comparison with TRANSP output and other tests verify the solutions. Judicious selection of grid size and other parameters facilitate efficient solutions. The output of the code has been validated by FIDA measurements on DIII-D but further tests are warranted.

A method for gyrokinetic simulation of low frequency (lower than the cyclotron frequency) magnetic compressional modes in general geometry is presented. The gyrokinetic-Maxwell system of equations is expressed fully in terms of the compressional component of the magnetic perturbation, δB∥, with finite Larmor radius effects. This introduces a “gyro-surface” averaging of δB∥ in the gyrocenter equations of motion, and similarly in the perpendicular Ampere’s law, which takes the form of the perpendicular force balance equation. The resulting system can be numerically implemented by representing the gyro-surface averaging by a discrete sum in the configuration space. For the typical wavelength of interest (on the order of the gyroradius), the gyro-surface averaging can be reduced to averaging along an effective gyro-orbit. The phase space integration in the force balance equation can be approximated by summing over carefully chosen samples in the magnetic moment coordinate, allowing for an efficient numerical implementation.

The probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.

In this work we consider a general notion of distributional sensitivity, which measures the variation in solutions of a given physical/mathematical system with respect to the variation of probability distribution of the inputs. This is distinctively different from the classical sensitivity analysis, which studies the changes of solutions with respect to the values of the inputs. The general idea is measurement of sensitivity of outputs with respect to probability distributions, which is a well-studied concept in related disciplines. We adapt these ideas to present a quantitative framework in the context of uncertainty quantification for measuring such a kind of sensitivity and a set of efficient algorithms to approximate the distributional sensitivity numerically. A remarkable feature of the algorithms is that they do not incur additional computational effort in addition to a one-time stochastic solver. Therefore, an accurate stochastic computation with respect to a prior input distribution is needed only once, and the ensuing distributional sensitivity computation for different input distributions is a post-processing step. We prove that an accurate numericalmodel leads to accurate calculations of this sensitivity, which applies not just to slowly-converging Monte-Carlo estimates, but also to exponentially convergent spectral approximations. We provide computational examples to demonstrate the ease of applicability and verify the convergence claims.

The low temperature properties of double exchange model in triangular lattice are investigated via truncated polynomial expansion method (TPEM), which reduces the computational complexity and enables parallel computation. We found that for the half-filling case a stable 120° spin configuration phase occurs owing to the frustration of triangular lattice and is further stabilized by antiferromagnetic (AF) su-perexchange interaction, while a transition between a stable ferromagnetic (FM) phase and a unique flux phase with small finite-size effect is induced by AF superexchange interaction for the quarter-filling case.

In this paper, we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume (SV) context; more specifically the emphasis is on handling equations containing third derivative terms. This formulation is based on the LDG (Local Discontinuous Galerkin) flux discretization method, originally employed for viscous equations containing second derivatives. A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation. The Fourier analysis was utilized for two purposes: firstly to eliminate all the unstable SV partitions, secondly to obtain the optimal SV partition. Numerical experiments are performed to illustrate the capability of this formulation. Since this formulation is extremely local, it can be easily parallelized and a h-p adaptation is relatively straightforward to implement. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.

In this paper we propose a development of the finite difference method, called the tailored finite point method, for solving steady magnetohydrodynamic (MHD) duct flow problems with a high Hartmann number. When the Hartmann number is large, the MHD duct flow is convection-dominated and thus its solution may exhibit localized phenomena such as the boundary layer. Most conventional numerical methods can not efficiently solve the layer problem because they are lacking in either stability or accuracy. However, the proposed tailored finite point method is capable of resolving high gradients near the layer regions without refining the mesh. Firstly, we devise the tailored finite point method for the scalar inhomogeneous convection-diffusion problem, and then extend it to the MHD duct flow which consists of a coupled system of convection-diffusion equations. For each interior grid point of a given rectangular mesh, we construct a finite-point difference operator at that point with some nearby grid points, where the coefficients of the difference operator are tailored to some particular properties of the problem. Numerical examples are provided to show the high performance of the proposed method.

A high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.

The paper is concerned with the numerical solution of Schrödinger equations on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.

We investigate the critical properties of the Ising S = 1/2 and S = 1 model on (3,4,6,4) and (34,6) Archimedean lattices. The system is studied through the extensive Monte Carlo simulations. We calculate the critical temperature as well as the critical point exponents γ/ν, β/ν, and ν basing on finite size scaling analysis. The calculated values of the critical temperature for S = 1 are kBTC/J=1.590(3), and kBTC/J=2.100(4) for (3,4,6,4) and (34,6) Archimedean lattices, respectively. The critical exponents β/ν, γ/ν, and 1/ν, for S=1 are β/ν=0.180(20), γ/ν=1.46(8), and 1/ν=0.83(5), for (3,4,6,4) and 0.103(8), 1.44(8), and 0.94(5), for (34,6) Archimedean lattices. Obtained results differ from the Ising S = 1/2 model on (3,4,6,4), (34,6) and square lattice. The evaluated effective dimensionality of the system for S = 1 are Deff=1.82(4), for (3,4,6,4), and Deff = 1.64(5) for (34,6).