Linear Finite Difference Research Articles

On the relation between order of accuracy, convergence rate and spectral slope for linear numerical methods applied to multiscale problems

AbstractThe relation between order of accuracy and convergence rate for simple linear finite difference schemes for differentiation and advection is examined theoretically and empirically. For sufficiently smooth functions, i.e. those with sufficiently steep spectral slope, the convergence rate is given by the order of accuracy. For less smooth functions, with shallower spectral slope, differentiation and advection behave slightly differently: the convergence rate of a finite difference derivative is determined entirely by the spectral slope, while the convergence rate of a finite difference advection scheme is determined by an interaction between the spectral slope and the order of accuracy. © Crown copyright 2007. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.

Simulating control rod and fuel assembly motion using moving meshes

A prerequisite for designing a transient simulation experiment which includes the motion of control and fuel assemblies is the careful verification of a steady state model which computes k eff versus assembly insertion distance. Previous studies in nuclear engineering have usually approached the problem of the motion of control rods with the use of nonlinear nodal models. Nodal methods employ special approximations for the leading and trailing cells of the moving assemblies to avoid the rod cusping problem which results from the naive volume weighted cell cross-section approximation. A prototype framework called the MOOSE has been developed for modeling moving components in the presence of diffusion phenomena. A linear finite difference model is constructed, solutions for which are computed by SLEPc, a high performance parallel eigenvalue solver. Design techniques for the implementation of a patched non-conformal mesh which links groups of sub-meshes that can move relative to one another are presented. The generation of matrices which represent moving meshes which conserve neutron current at their boundaries, and the performance of the framework when applied to model reactivity insertion experiments is also discussed.

Numerical methods for nonlinear second-order hyperbolic partial differential equations. I. Time-linearized finite difference methods for 1-D problems

A three-parameter family of finite difference methods for one-dimensional hyperbolic equations with damping and nonlinearities based on the introduction of a new dependent variable and three implicitness parameters for the time discretization and the diffusion and reaction terms, is presented and shown to result in two-time level semi-discrete equations. Second-order accurate spatial discretizations and time linearization of the nonlinear source terms are shown to result in three-time level linear finite difference equations which are second-order accurate in time whenever the reaction and diffusion terms are allocated in the proportion 1:2:1 to the past, current and future time levels. Second-order accurate in time, time-linearized, three-point compact operator methods for the spatial derivatives which only involve three time levels and three grid points and are fourth-order accurate in space, are also presented, and their linear stability analyzed. An exhaustive comparison between analytical and numerical results indicates that the accuracy of time-linearized methods increases as the time step and grid spacing are decreased, and is nearly independent of the order of the spatial discretization when there is a sufficient number of grid points to resolve the wave structure, and the best accuracy is achieved when the three implicitness parameters are equal to 1/2. It is also shown that the accuracy of time-linearized methods is a strong function of the wave speed; as the wave speed is increased and the wave equation tends to a parabolic one, the methods are still three-time level techniques which may not preserve the positivity of the solution, whereas a Crank–Nicolson discretization of the corresponding parabolic equation does preserve this property provided that there are no mathematical incompatibilities between the initial and the boundary conditions. It is also shown that, for one-dimensional wave equations with monotonic kinetics and time-dependent forcing, there are very few differences between second- and first-order time-accurate discretizations of the diffusion terms provided that the time step and grid size are sufficiently small, and the discretization of the reaction terms plays a less important rule than that of the diffusion terms.

Line vortices in a three-dimensional ordered Josephson medium at a small pinning parameter

The structure and energy of a line vortex whose axis is aligned with the symmetry axis of a finite-thickness slab indefinitely long in two directions is calculated by solving a set of linear finite-difference equations. Fluxoid quantization conditions in cells near the center of the vortex serve as boundary conditions. An exact solution is approached by iterations in phase stepwise discontinuities that cannot be considered small. A close similarity between the configuration under study and a periodic sequence (chain) of vortices makes it possible to allow for the effect of the domain boundary on the structure and energy of the vortex. It is shown that, at any width of the slab, one can find a pinning parameter value so small that the vortex cannot be viewed as solitary and contributions from other vortices should be taken into account in calculation. Proceeding in this way, one can find the structure and energy of the vortex however small the pinning parameter is. The total energy of the vortex is its intrinsic energy plus the sum of its energies of interaction with other members of the chain. In turn, the intrinsic energy is the sum of the energies of the small discrete core and quasi-continuous outer shell. It is demonstrated that the energy of the core is a linear function of the pinning parameter and is comparable to the energy of the shell.

A finite difference method for simulating transverse vibrations of an axially moving viscoelastic string

A finite difference method is presented to simulate transverse vibrations of an axially moving string. By discretizing the governing equation and the equation of stress-strain relation at different frictional knots, two linear sparse finite difference equation systems are obtained. The two explicit difference schemes can be calculated alternatively, which make the computation much more efficient. The numerical method makes the nonlinear model easier to deal with and of truncation errors, O(Δt2+Δx2). It also shows quite good stability for small initial values. Numerical examples are presented to demonstrate the efficiency and the stability of the algorithm, and dynamic analysis of a viscoelastic string is given by using the numerical results.

On the Stability of Linear Stochastic Finite-Difference Systems with Almost-Periodic Coefficients

On the Stability of Linear Stochastic Finite-Difference Systems with Almost-Periodic Coefficients

Unconditionally stable nearly PML algorithm for linear dispersive media

Unconditional stable alternating direction implicit (ADI) formulations of the nearly perfectly matched layer (NPML) are presented for truncating linear dispersive finite difference time domain (FDTD) grids. The digital signal processing (DSP) algorithms developed for digital filters are used to implement the frequency dependent property of the media into the ADI-FDTD algorithm. The formulations have the advantage of simplicity of ADI-FDTD implementation. Numerical examples carried out for linear Debye and Lorentz dispersive domains are included to validate the proposed formulations.

A numerical method for simulating transverse vibrations of an axially moving string

A modified finite difference method is presented to simulate transverse vibrations of an axially moving string. By discretizing the governing equation and the stress–strain relation at different frictional knots, two linear sparse finite difference equations are obtained, which can be computed alternatively. The numerical method makes the non-linear model easier to deal with and of small truncation errors. It also shows stable for small initial values, so it can be used in analyzing the non-linear vibration of viscoelastic moving string efficiently. A practical way of testing the precise of the numerical results is given by using a conservative quantity. Numerical examples are presented and dynamical analysis is given by using the numerical results.

Seismic source mechanisms for quarry blasts: modelling observed Rayleigh and Love wave radiation patterns from a Texas quarry

SUMMARY A theoretical understanding of the mechanisms by which quarry blasts excite seismic waves is useful in understanding how quarry blast discriminants may be transported from one region to another. An experiment in Texas with well-placed seismic stations and a cooperative blasting engineer has shed light on some of the physical mechanisms of seismic excitation at short periods (0.1‐3 Hz). Azimuthal radiation patterns of the 0.2‐3 Hz Rayleigh and Love waves are diagnostic of two proposed mechanisms for non-isotropic radiation from quarry blasts. Observations show that the Love and Rayleigh wave radiation patterns depend upon the orientation of the quarry benches. Two possible mechanisms for non-isotropic radiation are (1) the lateral throw of spalled material and (2) the presence of the topographic bench in the quarry. The spall of material can be modelled by vertical and horizontal forces applied to the free surface with time functions proportional to the derivative of the momentum of the spalled material. We use wavenumber integration synthetics to model the explosion plus spall represented by seismic moment tensor sources plus point forces. The resulting synthetics demonstrate that the magnitude of the SH (Love) compared with the SV (fundamental Rayleigh or Rg) in the short period band (0.5‐3 Hz) may be explained by the spall mechanism. Nearly all of the available mass must participate in the spall with an average velocity of 2‐5 m s −1 to provide sufficient impulse to generate the observed Love waves. Love wave radiation patterns from such a mechanism are consistent with the spall mechanism. We modelled the effects of the topographic bench using 3-D linear finite-difference calculations to compute progressive elastic wavefields from explosion sources behind the quarry bench. These 3-D calculations show SH radiation patterns consistent with observations while the SV radiation patterns are not consistent with observations. We find that the radiation patterns from the explosion behind the 3-D bench cannot be modelled by a modified moment tensor. The 3-D effects of the bench are more complicated than the representation by a moment tensor with a single reduced horizontal couple. The 3-D finite-difference synthetics exhibit strong azimuthal asymmetry and polarity reversals in the outgoing P-SV waves (P, S and Rg) radiated behind the bench for V p/V s ratios between 2 and 3. Both mechanisms may contribute to the non-isotropic radiation patterns but the spall mechanism is the simplest physical mechanism that explains the bulk of the observations. Adjustments to the time functions for the horizontal force, the vertical force and the explosion source may further refine the remaining differences between prediction and the observations.

Numerical Consistency of Metric Terms in Terrain-Following Coordinates

In numerically integrating the equations of motion in terrain-following coordinates, care must be taken in treating the metric terms that arise due to the sloping coordinate surfaces. In particular, metric terms that appear in the advection and pressure-gradient operators should be represented in a manner such that they exactly cancel when transformed back to Cartesian coordinates. Noncancellation of these terms can lead to spurious forcing at small scales on the numerical grid. This effect is demonstrated for a mountain wave flow problem through analytic solutions to the linear finite-difference equations. Further confirmation is provided through numerical simulations with a two-dimensional prototype version of the Weather Research and Forecasting (WRF) model, and with the Canadian Mesoscale Compressible Community (MC2) model.

A stable, convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation

We propose a stable, convergent, conservative and linear finite difference scheme to solve numerically the Cahn-Hilliard equation. The proposed scheme realizes both linearity and stability. We show uniqueness, existence and convergence of the solution to the scheme. Numerical examples demonstrate the effectiveness of the proposed scheme.

On Accuracy of Adaptive Grid Methods for Captured Shocks

The accuracy of two grid adaptation strategies, grid redistribution and local grid refinement, is examined by solving the 2-D Euler equations for the supersonic steady flow around a cylinder. Second- and fourth-order linear finite difference shock-capturing schemes, based on the Lax–Friedrichs flux splitting, are used to discretize the governing equations. The grid refinement study shows that for the second-order scheme, neither grid adaptation strategy improves the numerical solution accuracy compared to that calculated on a uniform grid with the same number of grid points. For the fourth-order scheme, the dominant first-order error component is reduced by the grid adaptation, while the design-order error component drastically increases because of the grid nonuniformity. As a result, both grid adaptation techniques improve the numerical solution accuracy only on the coarsest mesh or on very fine grids that are seldom found in practical applications because of the computational cost involved. Similar error behavior has been obtained for the pressure integral across the shock. A simple analysis shows that both grid adaptation strategies are not without penalties in the numerical solution accuracy. Based on these results, a new grid adaptation criterion for captured shocks is proposed.

A new measurement technique for the estimation of core shear strain in closed sandwich structures

Sandwich structures have been widely used for many years in applications such as aircraft panels, marine-craft hulls, racing car bodies and spacecraft solar arrays. Most sandwich panel designs include a lightweight core such as paper honeycomb or closed cell foam encased between two face plates, and in the case of aircraft panels constructed from carbon fibre reinforced plastics, the core is bevelled and edge pan plies are included to totally enclose the core. This type of design restricts access to the core making it almost impossible for the engineer to measure the shear strain developed in the core during in-service static or dynamic loading. This paper introduces a new measurement technique whereby the shear strain in the core can be estimated from face plate measurements using a linear finite difference approximation. The estimation method is presented and supported by calculations on a statically loaded sandwich beam. Static and dynamic experiments were conducted in order to validate the technique using a honeycomb sandwich beam instrumented with strain gauges on the core and face plates. The results showed excellent agreement between measured and estimated core shear strain for sandwich configurations with thin face plates, such as those encountered in aircraft and marine-craft constructions.

Synthesis of complete rational orthonormal bases with prescribed asymptotic order

In this paper, a method to construct complete rational orthonormal model sets with prescribed asymptotic order is presented for continuous-time systems. Two special cases that illustrate the application of the method are studied. In the first case, this method is shown to produce the generalized Laguerre functions which form a complete model set for H 2 and have asymptotic order two. In both cases, this method is equivalent to calculating Toeplitz determinants generated by a rational function. The latter problem is reduced to finding solution of a linear finite-difference equation.

The configurational dependence of binding free energies: a Poisson-Boltzmann study of Neuraminidase inhibitors.

The linear finite difference Poisson-Boltzmann (FDPB) equation is applied to the calculation of the electrostatic binding free energies of a group of inhibitors to the Neuraminidase enzyme. An ensemble of enzyme-inhibitor complex conformations was generated using Monte Carlo simulations and the electrostatic binding free energies of subtly different configurations of the enzyme-inhibitor complexes were calculated. It was seen that the binding free energies calculated using FDPB depend strongly on the configuration of the complex taken from the ensemble. This configurational dependence was investigated in detail in the electrostatic hydration free energies of the inhibitors. Differences in hydration energies of up to 7 kcal mol(-1) were obtained for root mean square (RMS) structural deviations of only 0.5 A. To verify the result, the grid size and parameter dependence of the calculated hydration free energies were systematically investigated. This showed that the absolute hydration free energies calculated using the FDPB equation were very sensitive to the values of key parameters, but that the configurational dependence of the free energies was independent of the parameters chosen. Thus just as molecular mechanics energies are very sensitive to configuration, and single-structure values are not typically used to score binding free energies, single FDPB energies should be treated with the same caution.

Tunnelling in soils – ground movements, and damage to buildings in Workington, UK

Tunnelling through soils results in ground loss, causing surface settlements and transverse movements. Where the tunnel drive passes below an existing structure, it is important to estimate the effects upon the structure. However, the free ground deformations should not simply be imposed upon a structure, because the structure contributes to stiffening of the ground. A computational soil-structure interaction analysis is required, to otain detailed stress–deformation response. First, linear finite element and Lagrangian finite difference methods are used to estimate ground movements due to a tunnel in free ground, and the results are compared with values based on empirical equations. The two linear methods and an additional hybrid FE method are then used to assess with soil-structure interaction; two cases of a typical short wall and a long wall lying across the route of tunnels of different depths. The results support the validity of the hybrid method which is used to estimate interactive ground settlements for comparison with a reported case of tunnelling below a building in central London. A more detailed case study is then undertaken to assess building damage caused by gross settlements during tunnelling in mixed soils, at a site in Workington, west Cumbria. A survey allowed estimation of free ground movements. Analysis by the hybrid method on the soils plus uncracked structures indicated intolerable horizontal stresses. Re-analysis with major cracks introduced into the structures resulted in close agreement between measured and computed settlements.

The solution of finite-difference equations

In every textbook which deals with second order linear finite-difference equations, methods will be found for solving such equations when the coefficients are constants. In the author’s experience it is unknown for textbooks to treat second order linear difference equations with variable coefficients. Yet such equations certainly occur in practical applications. It is the present intention to remedy this defect in the literature. It will be seen that, by the methods discussed here, we can always reduce the solution of a linear second order difference equation to that of a first order equation.

An iterative finite difference beam propagation method for modeling second-order nonlinear effects in optical waveguides

An iterative finite difference beam propagation method based on the Crank-Nicholson scheme is presented to simulate continuous wave (CW) second-order nonlinear effects in optical waveguides with the depletion of the pump wave taken into account. This method is an extension of the linear finite difference beam propagation method and preserves the same order of accuracy. Comparisons with the previously published explicit finite difference beam propagation method and the rectangular approximation method are presented. Quasi-phase matched difference frequency generation in AlGaAs and quasi-phase-matched second harmonic generation in LiNbO/sub 3/ are considered in the evaluation, showing that one iteration for the IFD-BPM is sufficient for the simulation with good accuracy and without increasing much computation time.

A linear scheme for simulating conduction heat transfer problems with nonlinear boundary conditions

A scheme for constructing the solution of conduction heat transfer problems with nonlinear boundary conditions is presented through the solution of problems with linear boundary conditions. The proposed procedure provides an efficient and easy way for carrying out numerical simulations of nonlinear heat transfer problems. In order to illustrate the procedure a particular problem is simulated by means of a (linear) finite difference scheme.

Parallel beam propagation method for the analysis of second harmonic generation

We extend the linear parallel explicit finite difference beam propagation method to model the nonlinear second harmonic generation process in three-dimensional waveguides. It has been concluded from testing this method that it is accurate as well as very efficient.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>