Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the $p$-version

Plane wave discontinuous Galerkin (PWDG) methods are a class of Trefftz-type methods for the spatial discretization of boundary value problems for the Helmholtz operator $-\Delta-\omega^2$, $\omega>0$. They include the so-called ultra weak variational formulation from [O. Cessenat and B. Després, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. This paper is concerned with the a priori convergence analysis of PWDG in the case of $p$-refinement, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased. For convex domains in two space dimensions, we derive convergence rates, employing mesh skeleton-based norms, duality techniques from [P. Monk and D. Wang, Comput. Methods Appl. Mech. Engrg., 175 (1999), pp. 121–136], and plane wave approximation theory.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Trefftz discontinuous Galerkin methods for acoustic scattering on locally refined meshes

We extend the a priori error analysis of Trefftz-discontinuous Galerkin methods for time-harmonic wave propagation problems developed in previous papers to acoustic scattering problems and locally refined meshes. To this aim, we prove refined regularity and stability results with explicit dependence of the stability constant on the wave number for non convex domains with non connected boundaries. Moreover, we devise a new choice of numerical flux parameters for which we can prove L2-error estimates in the case of locally refined meshes near the scatterer. This is the setting needed to develop a complete hp-convergence analysis.

The application of Regional Climate Model output for the simulation of high-mountain permafrost scenarios

This study investigates the possibilities and limitations of using Regional Climate Model (RCM) output for the simulation of alpine permafrost scenarios. It focuses on the general problem of scale mismatch between RCMs and impact models and, in particular, the special challenges that arise when driving an impact model in topographically complex high-mountain environments with the output of an RCM. Two approaches are introduced that take into account the special difficulties in such areas, and thus enable the use of RCM for alpine permafrost scenario modelling. Intended as an initial example, they are applied at the area of Corvatsch (Upper Engadine, Switzerland) in order to demonstrate and discuss the application of the two approaches, rather than to provide an assessment of future changes in permafrost occurrence. There are still many uncertainties and inaccuracies inherent in climate and impact models, which increase when driving one model with the output of the other. Nevertheless, our study shows that the use of RCMs offers new and promising perspectives for the simulation of high-mountain permafrost scenarios

Validation of Canopy Height Profile methodology for small-footprint full-waveform airborne LiDAR data in a discontinuous canopy environment

A Canopy Height Profile (CHP) procedure presented in Harding et al. (2001) for large footprint LiDAR data was tested in a closed canopy environment as a way of extracting vertical foliage profiles from LiDAR raw-waveform. In this study, an adaptation of this method to small-footprint data has been shown, tested and validated in an Australian sparse canopy forest at plot- and site-level. Further, the methodology itself has been enhanced by implementing a dataset-adjusted reflectance ratio calculation according to Armston et al. (2013) in the processing chain, and tested against a fixed ratio of 0.5 estimated for the laser wavelength of 1550nm. As a by-product of the methodology, effective leaf area index (LAIe) estimates were derived and compared to hemispherical photography-derived values. To assess the influence of LiDAR aggregation area size on the estimates in a sparse canopy environment, LiDAR CHPs and LAIes were generated by aggregating waveforms to plot- and site-level footprints (plot/site-aggregated) as well as in 5m grids (grid-processed). LiDAR profiles were then compared to leaf biomass field profiles generated based on field tree measurements. The correlation between field and LiDAR profiles was very high, with a mean R2 of 0.75 at plot-level and 0.86 at site-level for 55 plots and the corresponding 11 sites. Gridding had almost no impact on the correlation between LiDAR and field profiles (only marginally improvement), nor did the dataset-adjusted reflectance ratio. However, gridding and the dataset-adjusted reflectance ratio were found to improve the correlation between raw-waveform LiDAR and hemispherical photography LAIe estimates, yielding the highest correlations of 0.61 at plot-level and of 0.83 at site-level. This proved the validity of the approach and superiority of dataset-adjusted reflectance ratio of Armston et al. (2013) over a fixed ratio of 0.5 for LAIe estimation, as well as showed the adequacy of small-footprint LiDAR data for LAIe estimation in discontinuous canopy forests.

Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

Energy consistent discontinuous Galerkin methods for the Navier–Stokes–Korteweg system

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods are consistent with the energy dissipation of the continuous PDE systems. - See more at: http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02792-0/home.html#sthash.rwTIhNWi.dpuf

A Posteriori analysis of discontinuous Galerkin schemes for systems of hyperbolic conservation laws

In this work we construct reliable a posteriori estimates for some semi- (spatially) discrete discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

Energy consistent discontinuous Galerkin methods for a quasi-incompressible diffuse two phase flow model

We design consistent discontinuous Galerkin finite element schemes for the approximation of a quasi-incompressible two phase flow model of Allen–Cahn/Cahn–Hilliard/Navier–Stokes–Korteweg type which allows for phase transitions. We show that the scheme is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to discrete equivalents of those effects already causing dissipation on the continuous level, that is, there is no artificial numerical dissipation added into the scheme. In this sense the methods are consistent with the energy dissipation of the continuous PDE system.

Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective

We study discontinuous Galerkin approximations of the p-biharmonic equation for p∈(1,∞) from a variational perspective. We propose a discrete variational formulation of the problem based on an appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a semicontinuity argument. We also present numerical experiments aimed at testing the robustness of the method.

Change escalation processes and complex adaptive systems: from incremental reconfigurations to discontinuous restructuring

This study examines when “incremental” change is likely to trigger “discontinuous” change, using the lens of complex adaptive systems theory. Going beyond the simulations and case studies through which complex adaptive systems have been approached so far, we study the relationship between incremental organizational reconfigurations and discontinuous organizational restructurings using a large-scale database of U.S. Fortune 50 industrial corporations. We develop two types of escalation process in organizations: accumulation and perturbation. Under ordinary conditions, it is perturbation rather than the accumulation that is more likely to trigger subsequent discontinuous change. Consistent with complex adaptive systems theory, organizations are more sensitive to both accumulation and perturbation in conditions of heightened disequilibrium. Contrary to expectations, highly interconnected organizations are not more liable to discontinuous change. We conclude with implications for further research, especially the need to attend to the potential role of managerial design and coping when transferring complex adaptive systems theory from natural systems to organizational systems.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Perturbation growth at the convective scale for CSIP IOP18

The Met Office Unified Model is run for a case observed during Intensive Observation Period 18 (IOP18) of the Convective Storms Initiation Project (CSIP). The aims are to identify the physical processes that lead to perturbation growth at the convective scale in response to model-state perturbations and to determine their sensitivity to the character of the perturbations. The case is strongly upper-level forced but with detailed mesoscale/convective-scale evolution that is dependent on smaller-scale processes. Potential temperature is perturbed within the boundary layer. The effects on perturbation growth of both the amplitude and typical scalelength of the perturbations are investigated and perturbations are applied either sequentially (every 30 min throughout the simulation) or at specific times. The direct effects (within one timestep) of the perturbations are to generate propagating Lamb and acoustic waves and produce generally small changes in cloud parameters and convective instability. In exceptional cases a perturbation at a specific gridpoint leads to switching of the diagnosed boundary-layer type or discontinuous changes in convective instability, through the generation or removal of a lid. The indirect effects (during the entire simulation) are changes in the intensity and location of precipitation and in the cloud size distribution. Qualitatively different behaviour is found for strong (1K amplitude) and weak (0.01K amplitude) perturbations, with faster growth after sunrise found only for the weaker perturbations. However, the overall perturbation growth (as measured by the root-mean-square error of accumulated precipitation) reaches similar values at saturation, regardless of the perturbation characterisation.

Pb-210-Ra-226 and Ra-228-Th-232 systematics in young arc lavas: implications for magma degassing and ascent rates

New data show that island arc rocks have (Pb-210/Ra-226)(o) ratios which range from as low as 0.24 up to 2.88. In contrast, (Ra-22S/Th-232) appears always within error of I suggesting that the large Ra-226-excesses observed in arc rocks were generated more than 30 years ago. This places a maximum estimate on melt ascent velocities of around 4000 m/year and provides further confidence that the Ra-226 excesses reflect deep (source) processes rather than shallow level alteration or seawater contamination. Conversely, partial melting must have occurred more than 30 years prior to eruption. The Pb-210 deficits are most readily explained by protracted magma degassing. Using published numerical models, the data suggest that degassing occurred continuously for periods up to several decades just prior to eruption but no link with eruption periodicity was found. Longer periods are required if degassing is discontinuous, less than 100% efficient or if magma is recharged or stored after degassing. The long durations suggest much of this degassing occurs at depth with implications for the formation of hydrothermal and copper-porphyry systems. A suite of lavas erupted in 1985-1986 from Sangeang Api volcano in the Sunda arc are characterised by deficits of Pb-210 relative to Ra-226 from which 6-8 years of continuous Rn-222 degassing would be inferred from recent numerical models. These data also form a linear (Pb-210)/Pb-(Ra-226)/Pb array which might be interpreted as a 71-year isochron. However, the array passes through the origin suggesting displacement downwards from the equiline in response to degassing and so the slope of the array is inferred not to have any age significance. Simple modelling shows that the range of (Ra-226)/Pb ratios requires thousands of years to develop consistent with differentiation occurring in response to cooling at the base of the crust. Thus, degassing post-dated, and was not responsible for magma differentiation. The formation, migration and extraction of gas bubbles must be extremely efficient in mafic magma whereas the higher viscosity of more siliceous magmas retards the process and can lead to Pb-210 excesses. A possible negative correlation between (Pb-210/Ra-226)(o) and SO2 emission rate requires further testing but may have implications for future eruptions. (C) 2004 Elsevier B.V. All rights reserved.

Floodplain environmental change during the Younger Dryas and Holocene in Northwest Europe: Insights from the lower Kennet Valley, south central England

Many lowland rivers across northwest Europe exhibit broadly similar behavioural responses to glacial-interglacial transitions and landscape development. Difficulties exist in assessing these, largely because the evidence from many rivers remains limited and fragmentary. Here we address this issue in the context of the river Kennet, a tributary of the Thames, since c. 13,000 cal BP. Some similarities with other rivers are present, suggesting that regional climatic shifts are important controls. The Kennet differs from the regional pattern in a number of ways. The rate of response to sudden climatic change, particularly at the start of the Holocene and also mid-Holocene forest clearance, appears very high. This may reflect abrupt shifts between two catchment scale hydrological states arising from contemporary climates, land use change and geology. Stadial hydrology is dominated by nival regimes, with limited winter infiltration and high spring and summer runoff. Under an interglacial climate, infiltration is more significant. The probable absence of permafrost in the catchment means that a lag between the two states due to its gradual decay is unlikely. Palaeoecology, supported by radiocarbon dates, suggests that, at the very start of the Holocene, a dramatic episode of fine sediment deposition across most of the valley floor occurred, lasting 500-1000 years. A phase of peat accumulation followed as mineral sediment supply declined. A further shift led to tufa deposition, initially in small pools, then across the whole floodplain area, with the river flowing through channels cut in tufa and experiencing repeated avulsion. Major floods, leaving large gravel bars that still form positive relief features on the floodplain, followed mid-Holocene floodplain stability. Prehistoric deforestation is likely to be the cause of this flooding, inducing a major environmental shift with significantly increased surface runoff. Since the Bronze Age, predominantly fine sediments were deposited along the valley with apparently stable channels and vertical floodplain accretion associated with soil erosion and less catastrophic flooding. The Kennet demonstrates that, while a general pattern of river behaviour over time, within a region, may be identifiable, individual rivers are likely to diverge from this. Consequently, it is essential to understand catchment controls, particularly the relative significance of surface and subsurface hydrology. (c) 2005 Elsevier B.V. All rights reserved.