Semyon Tsynkov

Compact 2D and 3D sixth order schemes for the Helmholtz equation with variable wave number

Several studies have presented compact fourth order accurate finite difference approximation for the Helmholtz equation in two or three dimensions. Several of these formulae allow for the wave number k to be variable. Other papers have extended this further to include variable coefficients within the Laplacian which models non-homogeneous materials in electromagnetism. Later papers considered more accurate compact sixth order methods but these were restricted to constant k. In this paper we extend these compact sixth order schemes to variable k in both two and three dimensions. Results on 2D and 3D problems with known analytic solutions verify the sixth order accuracy. We demonstrate that for large wave numbers, the second order scheme cannot produce comparable results with reasonable grid sizes.

Active Shielding and Control of Noise

We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also c...

Artificial boundary conditions for the numerical solution of external viscous flow problems

In this paper we describe an algorithm for the nonlocal artificial boundary conditions setting at the external boundary of a computational domain while numerically solving unbounded viscous compressible flow problems past the finite bodies. Our technique is based on the usage of generalized Calderon projection operators and the application of the difference potentials method. Some computational results are presented.

Inverse source problem and active shielding for composite domains

The problem of active shielding (AS) for a multiply connected domain consists of constructing additional sources of the field (e.g., acoustic) so that all individual subdomains can either communicate freely with one another or otherwise be shielded from their peers. This problem can be interpreted as a special inverse source problem for the differential equation (or system) that governs the field. In the paper, we obtain general solution for a discretized composite AS problem and show that it reduces to solving a collection of auxiliary problems for simply connected domains. c

The Method of Difference Potentials for the Helmholtz Equation Using Compact High Order Schemes

The method of difference potentials was originally proposed by Ryaben’kii and can be interpreted as a generalized discrete version of the method of Calderon’s operators in the theory of partial differential equations. It has a number of important advantages; it easily handles curvilinear boundaries, variable coefficients, and non-standard boundary conditions while keeping the complexity at the level of a finite-difference scheme on a regular structured grid. The method of difference potentials assembles the overall solution of the original boundary value problem by repeatedly solving an auxiliary problem. This auxiliary problem allows a considerable degree of flexibility in its formulation and can be chosen so that it is very efficient to solve.Compact finite difference schemes enable high order accuracy on small stencils at virtually no extra cost. The scheme attains consistency only on the solutions of the differential equation rather than on a wider class of sufficiently smooth functions. Unlike standard high order schemes, compact approximations require no additional boundary conditions beyond those needed for the differential equation itself. However, they exploit two stencils—one applies to the left-hand side of the equation and the other applies to the right-hand side of the equation.We shall show how to properly define and compute the difference potentials and boundary projections for compact schemes. The combination of the method of difference potentials and compact schemes yields an inexpensive numerical procedure that offers high order accuracy for non-conforming smooth curvilinear boundaries on regular grids. We demonstrate the capabilities of the resulting method by solving the inhomogeneous Helmholtz equation with a variable wavenumber with high order (4 and 6) accuracy on Cartesian grids for non-conforming boundaries such as circles and ellipses.

External flow computations using global boundary conditions

We numerically integrate the compressible Navier-Stokes equations by means of a finite volume technique on the domain exterior to an airfoil. The curvilinear grid we use for discretization of the Navier-Stokes equations is obviously finite, it covers only a certain bounded region around the airfoil, consequently, we need to set some artificial boundary conditions at the external boundary of this region. The artificial boundary conditions we use here are non-local in space. They are constructed specifically for the case of steady-state solution. In constructing the artificial boundary conditions, we linearize the Navier-Stokes equations around the far-field solution and apply the difference potentials method. The resulting global conditions are implemented together with a pseudotime multigrid iteration procedure for achieving the steady state. The main goal of this paper is to describe the numerical procedure itself, therefore, we primarily emphasize the computation of artificial boundary conditions and the combined usage of these artificial boundary conditions and the original algorithm for integrating the Navier-Stokes equations. The underlying theory that justifies the proposed numerical techniques will accordingly be addressed more briefly.

A Compact Fourth Order Scheme for the Helmholtz Equation in Polar Coordinates

In many problems, one wishes to solve the Helmholtz equation in cylindrical or spherical coordinates which introduces variable coefficients within the differentiated terms. Fourth order accurate methods are desirable to reduce pollution and dispersion errors and so alleviate the points-per-wavelength constraint. However, the variable coefficients renders existing fourth order finite difference methods inapplicable. We develop a new compact scheme that is provably fourth order accurate even for these problems. The resulting system of finite difference equations is solved by a separation of variables technique based on the FFT. Moreover, in the r direction the unbounded domain is replaced by a finite domain, and an exact artificial boundary condition is specified as a closure. This global boundary condition fits naturally into the inversion of the linear system. We present numerical results that corroborate the fourth order convergence rate for several scattering problems.

Active Control of Sound for Composite Regions

We present a methodology for the active control of time-harmonic wave fields, e.g., acoustic disturbances, in composite regions. This methodology extends our previous approach developed for the case of arcwise connected regions. The overall objective is to eliminate the effect of all outside field sources on a given domain of interest, i.e., to shield this domain. In this context, active shielding means introducing additional field sources, called active controls, that generate the annihilating signal and cancel out the unwanted component of the field. As such, the problem of active shielding can be interpreted as a special inverse source problem for the governing differential equation or system. For a composite domain, not only do the controls prevent interference from all exterior sources, but they can also enforce a predetermined communication pattern between the individual subdomains (as many as desired). In other words, they either allow the subdomains to communicate freely with one another or otherw...

Long-time numerical computation of wave-type solutions driven by moving sources

A bolted steel staircase is erected by placing side stringers spaced apart and attached to the upper and lower floors. A series of stair pans are placed on brackets, the brackets having been welded to the stringers in the fabrication location. The nuts are then hand-tightened on bolts which are welded to the brackets. A set of flat metal wedges, each having an angled slot, is driven between the vertical bolts and lances on the sides of the stair pans in order to align the stair pans and drive them against the stringers. After the wedges are driven, the nuts are fully tightened.

On SAR Imaging through the Earth's Ionosphere

We analyze the effect of dispersion of radio waves in the Earths ionosphere on the performance (image resolution) of spaceborne synthetic aperture radars (SARs). We describe the electromagnetic propagation in the framework of a scalar model for the transverse field subject to weak anomalous dispersion due to the cold plasma. Random contributions to the refraction index are accounted for by the Kolmogorov model of ionospheric turbulence. A key consideration used when analyzing the statistics of waves is normalization of the probability distributions for long propagation distances. The ionospheric phenomena, both deterministic and random, are shown to affect the azimuthal resolution of a SAR sensor stronger than the range resolution; also, the effect of randomness appears weaker than that of the baseline dispersion. Specific quantitative estimates are provided for some typical values of the key parameters. Probing on two carrier frequencies is identified as a possible venue for reducing the ionospheric distortions.