Discrete Dispersion Relation Research Articles

Accuracy assessment of the super compact and combined compact schemes for spatial differencing of a two-layer oceanic model: Presentation of linear inertia-gravity and Rossby waves

This paper is devoted to accuracy assessment of the sixth-order super compact finite difference method (SCFDM) and the sixth-order combined compact finite difference method (CCFDM) for spatial differencing of the linearized two-layer shallow water equations on the Arakawa’s A- E and Randall’s Z numerical grids. Two general discrete dispersion relations on different numerical grids, for inertia-gravity and Rossby waves, are derived. These general relations can be used for evaluation of the performance of any desired numerical scheme. For both inertia-gravity and Rossby waves, minimum error generally occurs on Z grid using either the sixth-order SCFDM or CCFDM methods. For the Randall’s Z grid, the sixth-order CCFDM exhibits a substantial improvement, for the frequency of the barotropic and baroclinic modes of the linear inertia-gravity waves of the two layer shallow water model, over the sixth-order SCFDM. For the Rossby waves, the sixth-order SCFDM shows improvement, for the barotropic and baroclinic modes, over the sixth-order CCFDM method except on Arakawa’s C grid.

A fast convolution approach to the transformation of surface gravity waves: Linear waves in 1DH

The transformation of irrotational surface gravity waves in an inviscid fluid can be studied by time stepping the kinematic and dynamic surface boundary conditions. This requires a closure providing the normal surface particle velocity in terms of the surface velocity potential or its tangential derivative. A convolution integral giving this closure as an explicit expression is derived for linear 1D waves over a mildly sloping bottom. The model has exact linear dispersion and shoaling properties. A discrete numerical model is developed for a spatially staggered uniform grid. The model involves a spatial derivative which is discretized by an arbitrary-order finite-difference scheme. Error control is attained by solving the discrete dispersion relation a priori and model results make a perfect match to this prediction. A procedure is developed by which the computational effort is minimized for a specific physical problem while adapting the numerical parameters under the constraint of a predefined tolerance of damping and dispersion error. Two computational examples show that accurate irregular-wave transformation on the kilometre scale can be computed in seconds. Thus, the method makes up a highly efficient basis for a forthcoming extension that includes nonlinearity at arbitrary order. The relation to Boussinesq equations, mild-slope wave equations, boundary integral equations and spectral methods is briefly discussed.

Analysis of a consistency recovery method for the 1D convection–diffusion equation using linear finite elements

AbstractFor residual‐based stabilization methods such as streamline‐upwind Petrov–Galerkin (SUPG) and finite calculus (FIC), the higher‐order derivatives of the residual that appear in the stabilization term vanish when simplicial elements are used. The sub‐grid scale method using orthogonal sub‐scales (OSS) attempts to recover the lost consistency by using a fine‐scale projected residual in the stabilization term. The FIC method may also be cast into an OSS form with very little manipulation using an auxiliary convective projection equation. This paper discusses the gain/loss by recovering the consistency of the discrete residual in the stabilization terms via the form that includes the convective projection (as in the OSS method). We present the von Neumann analysis of the FIC method with recovered consistency (FIC_RC) for the 1D convection–diffusion problem and we compare it with the standard Bubnov–Galerkin linear finite element method and FIC/SUPG methods. The transient analysis is done by examining the discrete dispersion relation of the stabilization methods. The spectral results for the semi‐discrete and fully discrete problem are presented with time integration done by the trapezoidal and second‐order backward differencing formula schemes. The effect of lumping the effective mass matrix T is considered relative to using a consistent form. The effect of refinement in space and time is also discussed. Finally, an optimal expression for the stabilization parameter for the FIC_RC method on a uniform grid and for the steady state is given and its performance in the transient mode is discussed. Copyright © 2008 John Wiley & Sons, Ltd.

Dispersion Analysis of Discontinuous Galerkin Schemes Applied to Poincaré, Kelvin and Rossby Waves

A technique for analyzing dispersion properties of numerical schemes is proposed. The method is able to deal with both non dispersive or dispersive waves, i.e. waves for which the phase speed varies with wavenumber. It can be applied to unstructured grids and to finite domains with or without periodic boundary conditions. We consider the discrete version L of a linear differential operator ?. An eigenvalue analysis of L gives eigenfunctions and eigenvalues (l i ,? i ). The spatially resolved modes are found out using a standard a posteriori error estimation procedure applied to eigenmodes. Resolved eigenfunctions l i 's are used to determine numerical wavenumbers k i 's. Eigenvalues' imaginary parts are the wave frequencies ? i and a discrete dispersion relation ? i =f(k i ) is constructed and compared with the exact dispersion relation of the continuous operator ?. Real parts of eigenvalues ? i 's allow to compute dissipation errors of the scheme for each given class of wave. The method is applied to the discontinuous Galerkin discretization of shallow water equations in a rotating framework with a variable Coriolis force. Such a model exhibits three families of dispersive waves, including the slow Rossby waves that are usually difficult to analyze. In this paper, we present dissipation and dispersion errors for Rossby, Poincare and Kelvin waves. We exhibit the strong superconvergence of numerical wave numbers issued of discontinuous Galerkin discretizations for all families of waves. In particular, the theoretical superconvergent rates, demonstrated for a one dimensional linear transport equation, for dissipation and dispersion errors are obtained in this two dimensional model with a variable Coriolis parameter for the Kelvin and Poincare waves.

Discrete dispersion relations and the breaking of Lorentz invariance

We present a spacetime discretization of the Klein–Gordon equation, whose quadratic corrections to the continuum dispersion relation can be interpreted as Lorentz-invariance violating (LIV) effects. This discretization is based on a time-smearing procedure which, while violating micro-causality, does not preclude the emergence of macroscopically causal signals. The propagation of these signals is governed by a modified Klein–Gordon equation resumming non-local effects at all orders in the ratio of the wavelength to the LIV length scale.

Dispersive properties of high-order Nédélec/edge element approximation of the time-harmonic Maxwell equations.

The dispersive behaviour of high-order Nédélec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency omega on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where omega h is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1<< omega h, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly.

Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number

The dispersive properties of high order finite element schemes are analyzed in the setting of the Helmholtz equation, and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order. It is shown that the numerical dispersion displays three different types of behavior, depending on the size of the order of the method relative to the mesh-size and the wave number. Quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes. All estimates are fully explicit and are shown to be sharp. Limits are obtained, where transitions between the different regimes occur, and used to provide guidelines for the selection of the mesh-size and the polynomial order in terms of the wave number so that the dispersion error is controlled.

On the use of boundary-integral equation methods for unsteady free-surface flows

The Mixed Eulerian-Lagrangian Methods (MEL) forfree-surface potential flows solved by boundary-integral equations (BIEs) is considered, and the diffusion and dispersion errors are studied in the discrete linearized problem. The diffusion error is the base for the stability analysis of the scheme; both the errors give indications on the accuracy of the numerical solution. The study is divided into two steps: comparison of the discrete dispersion relation with the analytical solution and coupling with different time-integration schemes. In particular, a stability analysis of the Runge-Kutta and Taylor-expansion schemes, previously not given in the literature, is addressed. It is shown that MEL methods based on first- and second-order explicit Runge-Kutta and Taylor-expansion schemes are unstable, regardless of the technique adopted to discretize the BIEs. Higher-order Runge-Kutta and Taylor-expansion schemes lead to conditionally stable methods. Known results for explicit, implicit and explicit-implicit Euler schemes are recovered by the present analysis. The theoretical predictions of the errors are confirmed for two different boundary-element techniques: a high-order panel method based on B-Splines to solve for the velocity potential and a spectrally-accurate method based on the Euler-McLaurin summation formula to solve directly for the velocity field.

Electrostatic degrees of freedom in non-Maxwellian plasma

Detailed measurements of the ion velocity distribution function are used to test representations of the electrostatic degrees of freedom of slightly non-Maxwellian plasmas. It is found that fluid theory does not describe the data very well because there exist multiple closely spaced kinetic electrostatic modes. New wave branches appear that theoretically should persist as weakly damped modes even with Te∼Ti. Both a sum over discrete dispersion relations and the Case–Van Kampen spectral representation can be used to provide working descriptions of the data, but the latter has certain advantages.

Phase-Accuracy Comparisons and Improved Far-Field Estimates for 3-D Edge Elements on Tetrahedral Meshes

Edge-element methods have proved very effective for 3-D electromagnetic computations and are widely used on unstructured meshes. However, the accuracy of standard edge elements can be criticised because of their low order. This paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3-D edge-element approximations in comparison to other methods. Scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements, and a new formulation of the far-field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations.

Discrete Dispersion Relations and Taylor-Galerkin FEM for Transport of Dispersed Fronts

A time and space accurate method for computing the movement of an adsorbing contaminant in a carrier fluid is presented. The method uses a finite element analog to a previously described finite difference approach which can accurately describe movement of highly convected, but dispersing fronts. An operator splitting scheme between convection and dispersion is employed which provides fourth order time accuracy on the convection step. The higher accuracy for the convection step is also accompanied with belter mass conservation properties than is common for most algorithms which model purely convective problems. The method is demonstrated on problems varying in one dimension by applications to pure convection-dispersion, the nonlinear Burger's equation, convection-reaction equations, and convection-dispersion through an adsorbing medium. The model predictions are found to provide accurate and stable results.

Discrete perimeter plasmon excitations of electron gases in a quasi-one-dimensional wire: ring geometry

A self-consistent-field theory is applied to electron gases confined to a quasi-one-dimensional wire with a ring geometry. An analytical expression of the dynamic dielectric function is obtained. Collective excitations of the electron gases are studied. Different from the usual two-dimensional or quasi-one-dimensional plasmon modes, novel perimeter plasmon modes of the discrete dispersion relation due to angular subband transitions are found.

On steady and unsteady ship wave patterns

The properties of steady and unsteady ship waves propagating on a free surface discretized with panels are studied. The wave propagation is characterized by an explicit discrete dispersion relation which allows the systematic analysis of the distortion of the wave pattern due to discretization and the derivation of a stability criterion to be met by the numerical algorithm. The conclusions of the study are applied to a panel method used for the computation of steady and time-harmonic free-surface flows past elementary singularities and a ship hull.

Numerical noise in ocean and estuarine models

Approximate methods for solving the shallow water equations may lead to solutions exhibiting large fictitious, numerically-induced oscillations. The analysis of the discrete dispersion relation and modal solutions of small wavelengths provides a powerful technique for assessing the sensitivity of alternative numerical schemes to irregular data which may lead to such oscillatory numerical noise. For those schemes where phase speed vanishes at a finite wavenumber or there are multiple roots for wavenumber, oscillation modes can exist which are uncoupled from the dynamics of the problem. The discrete modal analysis approach is used here to identify two classes of spurious oscillation modes associated respectively with the two different asymptotic limits corresponding to estuarine and large scale ocean models. The analysis provides further insight into recent numerical results for models which include large spatial scales and Coriolis acceleration.