A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: Model development and analysis

International audienceA high‐order finite volume scheme is developed to numerically integrate a fully nonlinear and weakly dispersive set of Boussinesq‐type equations (the so‐called Serre equations) (J. Fluid Mech. 1987; 176:117–134; Surveys Geophys. 2004; 25(3–4):315–337). The choice of this discretization strategy is motivated by the fact that this particular set of equations is recasted in a convenient quasi‐conservative form. Cell face values are reconstructed using implicit compact schemes (J. Comput. Phys. 1999; 156:137–180; J. Comput. Phys. 2004; 198:535–566) and time integration is performed with the help of a four‐stage Runge–Kutta method. Numerical properties of the proposed scheme are investigated both, analytically using linear spectral analysis, and numerically for highly nonlinear cases. The numerical analysis indicates that the newly developed scheme has wider stability regions and better spectral resolution than most of the previously published numerical methods used to handle equivalent set of equations. Moreover, it was also noticed that the use of mixed‐order strategies to discretize convective and dispersive terms may result in an important overall reduction of the spectral resolution of the scheme. Additionally, there is some numerical evidence, which seems to indicate that the incorporation of a high‐order dispersion correction term as given by Madsen et al. (Coastal Eng. 1991; 15:371–388) may introduce instability in the syste

Target Element Sizes For Finite Element Tidal Models From A Domain-wide, Localized Truncation Error Analysis Incorporating Botto

A new methodology for the determination of target element sizes for the construction of finite element meshes applicable to the simulation of tidal flow in coastal and oceanic domains is developed and tested. The methodology is consistent with the discrete physics of tidal flow, and includes the effects of bottom stress. The method enables the estimation of the localized truncation error of the nonconservative momentum equations throughout a triangulated data set of water surface elevation and flow velocity. The method\u27s domain-wide applicability is due in part to the formulation of a new localized truncation error estimator in terms of complex derivatives. More conventional criteria that are often used to determine target element sizes are limited to certain bathymetric conditions. The methodology developed herein is applicable over a broad range of bathymetric conditions, and can be implemented efficiently. Since the methodology permits the determination of target element size at points up to and including the coastal boundary, it is amenable to coastal domain applications including estuaries, embayments, and riverine systems. These applications require consideration of spatially varying bottom stress and advective terms, addressed herein. The new method, called LTEA-CD (localized truncation error analysis with complex derivatives), is applied to model solutions over the Western North Atlantic Tidal model domain (the bodies of water lying west of the 60° W meridian). The convergence properties of LTEACD are also analyzed. It is found that LTEA-CD may be used to build a series of meshes that produce converging solutions of the shallow water equations. An enhanced version of the new methodology, LTEA+CD (which accounts for locally variable bottom stress and Coriolis terms) is used to generate a mesh of the WNAT model domain having 25% fewer nodes and elements than an existing mesh upon which it is based; performance of the two meshes, in an average sense, is indistinguishable when considering elevation tidal signals. Finally, LTEA+CD is applied to the development of a mesh for the Loxahatchee River estuary; it is found that application of LTEA+CD provides a target element size distribution that, when implemented, outperforms a high-resolution semi-uniform mesh as well as a manually constructed, existing, documented mesh

Numerical Dispersion in Non-Hydrostatic Modeling of Long-Wave Propagation.

Ph.D. Thesis. University of Hawaiʻi at Mānoa 2018

Balance model for equatorial long waves

Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field without some kind of diagnostic balance relation. As a result, estimates of Kelvin wave amplitudes can be poor. Our goal is to find a balance model that includes Kelvin waves for planetary-scale motions. Using asymptotic methods, we derive a balance model for the weakly nonlinear equatorial shallow-water equations. Specifically we adopt the ‘slaving’ method proposed by Warn et al. (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 723–739), which avoids secular terms in the expansion and thus can in principle be carried out to any order. Different from previous approaches, our expansion is based on a long-wave scaling and the slow dynamics is described using the height field instead of potential vorticity. The leading-order model is equivalent to the truncated long-wave model considered previously (e.g. Heckley & Gill, Q. J. R. Meteorol. Soc., vol. 110, 1984, pp. 203–217), which retains Kelvin waves in addition to equatorial Rossby waves. Our method allows for the derivation of higher-order models which significantly improve the representation of Rossby waves in the isotropic limit. In addition, the ‘slaving’ method is applicable even when the weakly nonlinear assumption is relaxed, and the resulting nonlinear model encompasses the weakly nonlinear model. We also demonstrate that the method can be applied to more realistic stratified models, such as the Boussinesq model

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

An isogeometric finite element formulation for phase transitions on deforming surfaces

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn-Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation. Both PDEs can be efficiently discretized using $C^1$-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise $C^1$-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-$\alpha$ scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton-Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to the text, added supplementary movie file

Analysis, Modeling, And Simulation Of The Tides In The Loxahatchee River Estuary (Southeastern Florida).

Recent cooperative efforts between the University of Central Florida, the Florida Department of Environmental Protection, and the South Florida Water Management District explore the development of a two-dimensional, depth-integrated tidal model for the Loxahatchee River estuary (Southeastern Florida). Employing a large-domain approach (i.e., the Western North Atlantic Tidal model domain), two-dimensional tidal flows within the Loxahatchee River estuary are reproduced to provide: 1) recommendations for the domain extent of an integrated, surface/groundwater, three-dimensional model; 2) nearshore, harmonically decomposed, tidal elevation boundary conditions. Tidal simulations are performed using a two-dimensional, depth-integrated, finite element-based code for coastal and ocean circulation, ADCIRC-2DDI. Multiple variations of an unstructured, finite element mesh are applied to encompass the Loxahatchee River estuary and different spatial extents of the Atlantic Intracoastal Waterway (AIW). Phase and amplitude errors between model output and historical data are quantified at five locations within the Loxahatchee River estuary to emphasize the importance of including the AIW in the computational domain. In addition, velocity residuals are computed globally to reveal significantly different net circulation patterns within the Loxahatchee River estuary, as depending on the spatial coverage of the AIW