Time-dependent Numerical Method Research Articles

Stability and Accuracy Analysis of $$\theta $$ Scheme for a Convolutional Integro-Differential Equation

Spatially long-range interactions for linearly elastic media resulting in dispersion relations are modelled by an integro-differential equation of convolution type (IDE) that incorporate non-local effects. This type of IDE is nonstandard (hence, it is almost impossible to obtain exact solutions) and plays an important role in modeling various applied science and engineering problems. In this article, such an IDE describing a linear elastic wave phenomenon has been studied. First, a discrete equivalent of the model IDE in space is proposed and then a class of forward backward average one step \(\theta \) scheme for the semi-discrete time dependent numerical method has been developed. Further, stability and accuracy of the developed method has been analyzed rigorously.

Finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains

A finite difference time-dependent numerical method for the wave equation, supported by recently derived novel elliptic grids, is analyzed. The method is successfully applied to single and multiple two-dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries. The new grids have nearly uniform cell area ( J -grids) and nearly uniform grid line spacing ( α γ -grids). Numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state. The restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing. As a consequence, moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost. Also, numerical solutions for wave problems inside annular regions of complex shapes are obtained. The use of the new grids results in late time stability in contrast with other classical finite difference time-dependent methods.

Time-Dependent Numerical Method with Boundary-Conforming Curvilinear Coordinates Applied to Wave Interactions with Prototypical Antennas

A new unstaggered finite-difference time-dependent technique to accurately model scattring from prototypical 2D antenna structures is devised. The unbounded boundary value problems defining these phenomena are redefined over bounded domains using appropriate radiation operators over finite artificial boundaries. Generalized curvilinear coordinates are generated such that physical boundaries correspond to coordinate lines. A numerical procedure to generate almost orthogonal, boundary-conforming, fine grids over these bounded regions is developed. Once the governing equations are written in terms of the new curvilinear coordinates, a time-dependent numerical method is applied to obtain time harmonic steady-state solutions to these problems. The electric field wave amplitude as well as the wave pattern inside the waveguide and the scattered field from the prototypical antennas are obtained. Accuracy and computational cost are compared when almost orthogonal and nonorthogonal grids are in use. An optical theorem for a flanged waveguide antenna with perfect electrical conductor walls is derived. It is verified that the approximate solutions obtained by application of the time-dependent numerical method with boundary-conforming, curvilinear coordinates satisfy the optical theorem.

Nonlinear development of flow patterns in an annulus with decelerating inner cylinder

Decelerating Taylor vortex flow between two concentric cylinders is investigated by using the time-dependent numerical method. The lengths of the rotating inner cylinder and the stationary outer cylinder are finite. We focus on the mode transitions from well-developed flows during the gradual decrease in the angular velocity of the inner cylinder. In the range of the Reynolds number from 50 to 1000 and the range of the aspect ratio Γ from 2.6 to 7.2, the exchanges of flow pattern from normal secondary modes to primary modes are clarified, and the Reynolds numbers at which the flow modes exchange are determined. The reduction of the number of cells in the flow of the normal mode begins with the weakening of a pair of counter-rotating cells with an inward radial flow at their boundary. An anomalous cell has two extra cells at corners of the end wall and the cylinder walls. In the transient process from the flow of the anomalous mode, extra cells grow and merge into a normal cell, and a saddle point appears in the working fluid. As the merged cell enlarges, the saddle point vanishes and the global normal flow mode appears. The result shows good agreement with experimental observations.

Numerical simulation method for elastic-plastic dynamic response for hypervelocity impact phenomena

This study presents a time-dependent numerical method for impact in planar or cylindrical symmetry. We use Eulerian finite-difference scheme, Tilloston's Metallic Equation of state, von Mises Yield criterion, for calculating the large deformation of elastic-plastic high velocity impact. Failure, cavitation and melting of solids are accounted for. The present model treats the formation and evolution of a crater, the deformation of the projectile and the deformation and dynamical response of the target. A two-stage gas gun was employed to experimentally study the phenomena of hypervelocity impact. Good agreement is obtained between the present computational results and craters obtained in experiments of polyethylene/aluminium impacts. The relation of crater shape and penetration depth to dynamic parameters of the projectile and the target is discussed. The Multi Purpose Graphics System (MPGS) is used to describe the calculation results with color graphics.

Acoustic instability in cosmic ray mediated shocks

The acoustic instability was examined in cosmic ray dominated media that can amplify sound waves shorter than the scale height of the cosmic-ray pressure. The effects of the instability on the particle distribution have been studied using a time-dependent numerical method in which the diffusion-advection transport equation for the particle distribution function is solved self-consistently with the hydrodynamic conservation equations

Determination of Internal Gas Flows by a Transient Numerical Technique

A time-dependent numerical method is developed for inviscid mixed subsonic-supersonic flows with shocks. The analysis is based on the Lax-Wendroff procedure but adds special treatments of the boundary conditions, a mapping of the physical contours and a damping term to provide stable solutions for planar and axisymmetric passages of arbitrary shape. Both transient and steady-state flows are computed with special emphasis being placed on the analysis of rapidly accelerating flows in axisymmetric ducts with sharp wall curvature. Results obtained compare favorably with experimental data and with prior analyses in those restricted regions where they have been applicable. In particular, excellent agreement is achieved between the present theory and experiment for the previously unsolved case of mixed flows in nozzles involving rapid subsonic contractions, sharp wall curvature at the throat and shocks in the supersonic flow portion. Here normal comparison parameters such as centerline and wall Mach number distributions, sonic line position and discharge coefficient are matched to a high degree of accuracy.