Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming

We study the preconditioning of symmetric indefinite linear systems of equations that arise in interior point solution of linear optimization problems. The preconditioning method that we study exploits the block structure of the augmented matrix to design a similar block structure preconditioner to improve the spectral properties of the resulting preconditioned matrix so as to improve the convergence rate of the iterative solution of the system. We also propose a two-phase algorithm that takes advantage of the spectral properties of the transformed matrix to solve for the Newton directions in the interior-point method. Numerical experiments have been performed on some LP test problems in the NETLIB suite to demonstrate the potential of the preconditioning method discussed.

Temporary and Permanent Buyout Prices in Online Auctions

Increasingly used in online auctions, buyout prices allow bidders to instantly purchase the item listed. We distinguish two types: a temporary buyout option disappears if a bid above the reserve price is made; a permanent one remains throughout the auction or until it is exercised. In a model featuring time-sensitive bidders with uniform valuations and Poisson arrivals but endogenous bidding times, we focus on finding temporary and permanent buyout prices maximizing the seller's discounted revenue, and examine the relative benefit of using each type of option in various environments. We characterize equilibrium bidder strategies in both cases and then solve the problem of maximizing seller's utility by simulation. Our numerical experiments suggest that buyout options may significantly increase a seller’s revenue. Additionally, while a temporary buyout option promotes early bidding, a permanent option gives an incentive to the bidders to bid late, thus leading to concentrated bids near the end of the auction.

A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

Benchmarking 2D hydraulic models for urban flooding

This paper describes benchmark testing of six two-dimensional (2D) hydraulic models (DIVAST, DIVASTTVD, TUFLOW, JFLOW, TRENT and LISFLOOD-FP) in terms of their ability to simulate surface flows in a densely urbanised area. The models are applied to a 1·0 km × 0·4 km urban catchment within the city of Glasgow, Scotland, UK, and are used to simulate a flood event that occurred at this site on 30 July 2002. An identical numerical grid describing the underlying topography is constructed for each model, using a combination of airborne laser altimetry (LiDAR) fused with digital map data, and used to run a benchmark simulation. Two numerical experiments were then conducted to test the response of each model to topographic error and uncertainty over friction parameterisation. While all the models tested produce plausible results, subtle differences between particular groups of codes give considerable insight into both the practice and science of urban hydraulic modelling. In particular, the results show that the terrain data available from modern LiDAR systems are sufficiently accurate and resolved for simulating urban flows, but such data need to be fused with digital map data of building topology and land use to gain maximum benefit from the information contained therein. When such terrain data are available, uncertainty in friction parameters becomes a more dominant factor than topographic error for typical problems. The simulations also show that flows in urban environments are characterised by numerous transitions to supercritical flow and numerical shocks. However, the effects of these are localised and they do not appear to affect overall wave propagation. In contrast, inertia terms are shown to be important in this particular case, but the specific characteristics of the test site may mean that this does not hold more generally.

Can the dynamical impact of the stratosphere on the troposphere be described by large-scale adjustment to the stratospheric PV distribution?

Recent numerical experiments have demonstrated that the state of the stratosphere has a dynamical impact on the state of the troposphere. To account for such an effect, a number of mechanisms have been proposed in the literature, all of which amount to a large-scale adjustment of the troposphere to potential vorticity (PV) anomalies in the stratosphere. This paper analyses whether a simple PV adjustment suffices to explain the actual dynamical response of the troposphere to the state of the stratosphere, the actual response being determined by ensembles of numerical experiments run with an atmospheric general-circulation model. For this purpose, a new PV inverter is developed. It is shown that the simple PV adjustment hypothesis is inadequate. PV anomalies in the stratosphere induce, by inversion, flow anomalies in the troposphere that do not coincide spatially with the tropospheric changes determined by the numerical experiments. Moreover, the tropospheric anomalies induced by PV inversion are on a larger scale than the changes found in the numerical experiments, which are linked to the Atlantic and Pacific storm-tracks. These findings imply that the impact of the stratospheric state on the troposphere is manifested through the impact on individual synoptic-scale systems and their self-organization in the storm-tracks. Changes in these weather systems in the troposphere are not merely synoptic-scale noise on a larger scale tropospheric response, but an integral part of the mechanism by which the state of the stratosphere impacts that of the troposphere.

The role of nonhydrostatic dynamics in controlling development of a surface ocean front

Numerical studies of surface ocean fronts forced by inhomogeneous buoyancy loss show nonhydrostatic convective plumes coexisting with baroclinic eddies. The character of the vertical overturning depends sensitively on the treatment of the vertical momentum equation in the model. It is less well known how the frontal evolution over scales of O(10 km) is affected by these dynamics. Here, we compare highly resolved numerical experiments using nonhydrostatic and hydrostatic models and the convective-adjustment parametrization. The impact of nonhydrostatic processes on average cross-frontal transfer is weak compared to the effect of the O(1 km) scale baroclinic motions. For water-mass distribution and formation rate nonhydrostatic dynamics have similar influence to the baroclinic eddies although adequate resolution of the gradients in forcing fluxes is more important. The overall implication is that including nonhydrostatic surface frontal dynamics in ocean general circulation models will have only a minor effect on scales of O(1 km) and greater.

The effect of mechanical stirring on horizontal convection

An important experimental result, as yet poorly understood, is that mechanical stirring can significantly enhance the strength of horizontal convection. A contentious issue is whether this necessarily implies that the mechanical stirring replaces the buoyancy forcing as the main source of energy driving the observed overturning circulation, as has been suggested for the Atlantic meridional overturning circulation (AMOC). In this paper, rigorous energetics considerations and idealized numerical experiments reveal that the rate at which the surface buoyancy forcing supplies energy to the fluid, as measured by the production rate of available potential energy G(APE), does not solely depend upon the buoyancy forcing, as is often implicitly assumed, but also upon the vertical stratification, such that the deeper the thermocline depth, the larger G(APE). This suggests that mechanical stirring enhances horizontal convection because it causes more energy to be extracted from the buoyancy forcing. It does so by enhancing turbulent mixing, which allows surface heating to reach greater depths, which increases the thermocline depth and hence G(APE). This paper therefore proposes a new hypothesis, namely that mechanically stirred horizontal convection and the AMOC are best described as mechanically controlled heat engines.

Why the western Pacific subtropical high has extended westward since the late 1970s

The western Pacific subtropical high (WPSH) is closely related to Asian climate. Previous examination of changes in the WPSH found a westward extension since the late 1970s, which has contributed to the inter-decadal transition of East Asian climate. The reason for the westward extension is unknown, however. The present study suggests that this significant change of WPSH is partly due to the atmosphere's response to the observed Indian Ocean-western Pacific (IWP) warming. Coordinated by a European Union's Sixth Framework Programme, Understanding the Dynamics of the Coupled Climate System (DYNAMITE), five AGCMs were forced by identical idealized sea surface temperature patterns representative of the IWP warming and cooling. The results of these numerical experiments suggest that the negative heating in the central and eastern tropical Pacific and increased convective heating in the equatorial Indian Ocean/ Maritime Continent associated with IWP warming are in favor of the westward extension of WPSH. The SST changes in IWP influences the Walker circulation, with a subsequent reduction of convections in the tropical central and eastern Pacific, which then forces an ENSO/Gill-type response that modulates the WPSH. The monsoon diabatic heating mechanism proposed by Rodwell and Hoskins plays a secondary reinforcing role in the westward extension of WPSH. The low-level equatorial flank of WPSH is interpreted as a Kelvin response to monsoon condensational heating, while the intensified poleward flow along the western flank of WPSH is in accord with Sverdrup vorticity balance. The IWP warming has led to an expansion of the South Asian high in the upper troposphere, as seen in the reanalysis.

Approximate factorization constraint preconditioners for saddle-point matrices

We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ?.( b(u)? 2u), where generically b(u) := |u|? for any given ? ? (0,?). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ? 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.

Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems

We consider conjugate-gradient like methods for solving block symmetric indefinite linear systems that arise from saddle-point problems or, in particular, regularizations thereof. Such methods require preconditioners that preserve certain sub-blocks from the original systems but allow considerable flexibility for the remaining blocks. We construct a number of families of implicit factorizations that are capable of reproducing the required sub-blocks and (some) of the remainder. These generalize known implicit factorizations for the unregularized case. Improved eigenvalue clustering is possible if additionally some of the noncrucial blocks are reproduced. Numerical experiments confirm that these implicit-factorization preconditioners can be very effective in practice.

Effects of fluctuating daily surface fluxes on the time-mean oceanic circulation

The effect of fluctuating daily surface fluxes on the time-mean oceanic circulation is studied using an empirical flux model. The model produces fluctuating fluxes resulting from atmospheric variability and includes oceanic feedbacks on the fluxes. Numerical experiments were carried out by driving an ocean general circulation model with three different versions of the empirical model. It is found that fluctuating daily fluxes lead to an increase in the meridional overturning circulation (MOC) of the Atlantic of about 1 Sv and a decrease in the Antarctic circumpolar current (ACC) of about 32 Sv. The changes are approximately 7% of the MOC and 16% of the ACC obtained without fluctuating daily fluxes. The fluctuating fluxes change the intensity and the depth of vertical mixing. This, in turn, changes the density field and thus the circulation. Fluctuating buoyancy fluxes change the vertical mixing in a non-linear way: they tend to increase the convective mixing in mostly stable regions and to decrease the convective mixing in mostly unstable regions. The ACC changes are related to the enhanced mixing in the subtropical and the mid-latitude Southern Ocean and reduced mixing in the high-latitude Southern Ocean. The enhanced mixing is related to an increase in the frequency and the depth of convective events. As these events bring more dense water downward, the mixing changes lead to a reduction in meridional gradient of the depth-integrated density in the Southern Ocean and hence the strength of the ACC. The MOC changes are related to more subtle density changes. It is found that the vertical mixing in a latitudinal strip in the northern North Atlantic is more strongly enhanced due to fluctuating fluxes than the mixing in a latitudinal strip in the South Atlantic. This leads to an increase in the density difference between the two strips, which can be responsible for the increase in the Atlantic MOC.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.