A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin (RKDG2) methods solving the non-linear shallow water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a non-destructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well-balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme -- even on unstructured grids -- and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, super-linear convergence, mass-conservation, well-balancedness, and stability are verified

MATHEMATICAL CURIOSITIES ABOUT DIVISION OF INTEGERS

As mathematics educators, our focus of attention is mainly placed on the learning and teaching of mathematics. But, as we study phenomena of mathematical learning and teaching, we often come across intriguing mathematical phenomena that capture our interest. We find ourselves often bouncing mathematical ideas back and forth, not just looking for (new/better) ways of teaching or presenting a mathematical concept, but also of uncovering and discovering potential understandings of the concept. These mathematical issues we encounter represent for us a significant aspect of our work, and are also very stimulating. One of these issues arose for us as we were tackling issues of division of numbers and of conventions relating to the remainder; issues that are, mathematically speaking, as we hope to communicate, very interesting and thought provoking. Thus, we explore four different avenues/curiosities about division, where operations with positive and negative numbers are considered, as well as the meaning one can draw out of these operations

Triangular grids

A novel wetting and drying treatment for second‐order Runge‐Kutta discontinuous Galerkin methods solving the nonlinear shallow‐water equations is proposed. It is developed for general conforming two‐dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a nondestructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well balancing with an innovative velocity‐based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme, even on unstructured grids, and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near‐realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, superlinear convergence, mass conservation, well balancedness, and stability are verified

Revisión histórica y actualización del inventario de humedales salinos de Monegros Sur. Base para una propuesta RAMSAR

12 Pags.- 6 Figs. Esta publicación de la ULPGC está basada en una contribución de las autoras con el mismo título al XIII Congreso Nacional de Tecnologías de la Información Geográfica celebrado ese mismo año 2008 en Las Palmas de Gran Canaria.[ES] En el marco nacional e internacional, el inventario de humedales es un objetivo de los planes estratégicos de conserva-ción. En este contexto se ha reconocido la importancia de disponer de inventarios amplios, de designar sitios RAMSAR, y documentar e identificar los humedales perdidos o los susceptibles de restauración. Este trabajo presenta un inventario actualizado de los humedales salinos de Monegros Sur y describe su evolución desde 1927. El desarrollo de los sistemas de información geográfica (SIG) y la teledetección han permitido una mejor y mayor utilización de los datos espaciales. Para el inventario se ha empleado el análisis cartográfico basado en un SIG utilizando imágenes de satélite, fotografías aéreas, mapas, documentos inéditos, y estudios sobre el terreno. La vegetación, el régimen hídrico y la geomorfología han sido los elementos fundamentales empleados para reconocer estos humedales. La presión humana ha favorecido su modifi-cación en número, tamaño y forma, y es el principal factor de control del estado de conservación de los elementos paisajís-ticos que los caracterizan. La información obtenida contribuye al conocimiento y conservación de nuestro patrimonio natural y ayudará a establecer las bases para proponer su inclusión en la lista RAMSAR.[EN] Wetlands inventory is a key objective in conservation policies in national and international frameworks. In this context the importance of getting comprehensive inventories and designating RAMSAR sites has been recognized. These strategies require identifying the wetland loss, and their degradation and restoration possibilities. In this work we present the up-dated inventory of Monegros saline wetlands and its historical evolution from 1927 to nowadays. The development of geo-graphic information systems (GIS) and remote sensing techniques have allowed a better and major exploitation of the spa-tial information. Our wetlands inventory was accomplished by means of a cartographic analysis based on a GIS and incor-porating satellite images, aerial photographs, maps, unpublished documents, and ground data. Vegetation, water regime, and geomorphology have been the three main features used for wetlands recognition. Human pressure yielded the degra-dation of wetlands in number, size, and shape, and is the main controlling factor of the landscape’s conservation status. The obtained information allows to establish the base for proposing them as RAMSAR site, contributing to the knowledge and conservation of our natural heritage.Este trabajo ha sido financiado por el MICINN a través del proyecto AGL2006-01283. M. Domínguez dis-fruta de una beca predoctoral INIA (BOE de 02/12/2005).Peer reviewe

Curricular Change in Institutional Context: A Profile of the SUMMIT-P Institutions

There is a national call to improve the mathematics curricula in the first two undergraduate years to improve student success and engagement. But curricular change happens in an institutional context: Who are the students, and what do they need to succeed? What is the climate for change? Does the department regularly revise its courses and curriculum? Is it common for different departments to collaborate on curricular change? What supports or obstacles does the department, college, or university have for changing the curriculum? Who are the institutional stakeholders, and what practices build their buy-in? In the SUMMIT-P project, nine different institutions ranging from small private colleges to mid-sized state universities to large public universities and a community college worked on changing the undergraduate mathematics curricula in the first two years. This paper examines the context at each institution in the project. We hope that other institutions looking to follow in our collaboration with the partner disciplines on revising the introductory mathematics curriculum at their institution will find a familiar context in one (or more) of these institutions. We include a list of questions that programs can use to examine their own institutional context

Metrics for Performance Quantification of Adaptive Mesh Refinement

Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities for geophysical simulations that exhibit strongly localised features such as is the case for tsunami, hurricane or typhoon prediction. Using the example of a shallow water solver, this study explores a set of metrics as a tool to distinguish the performance of numerical methods using adaptively refined versus uniform meshes independent of computational architecture or implementation. These metrics allow us to quantify how a numerical simulation benefits from the use of adaptive mesh refinement. The type of meshes we are focusing on are adaptive triangular meshes that are non-uniform and structured. Refinement is controlled by physics-based indicators that capture relevant physical processes and determine the areas of mesh refinement and coarsening. The proposed performance metrics take into account a number of characteristics of numerical simulations such as numerical errors, spatial resolution, as well as computing time. Using a number of test cases we demonstrate that correlating different quantities offers insight into computational overhead, the distribution of numerical error across various mesh resolutions as well as the evolution of numerical error and run-time per degree of freedom

An adaptive discontinuous Galerkin method for the simulation of hurricane storm surge

Numerical simulations based on solving the 2D shallow water equations using a discontinuous Galerkin (DG) discretisation have evolved to be a viable tool for many geophysical applications. In the context of flood modelling, however, they have not yet been methodologically studied to a large extent. Systematic model testing is non-trivial as no comprehensive collection of numerical test cases exists to ensure the correctness of the implementation. Hence, the first part of this manuscript aims at collecting test cases from the literature that are generally useful for storm surge modellers and can be used to benchmark codes. On geographic scale, hurricane storm surge can be interpreted as a localised phenomenon making it ideally suited for adaptive mesh refinement (AMR). Past studies employing dynamic AMR have exclusively focused on nested meshes. For that reason, we have developed a DG storm surge model on a triangular and dynamically adaptive mesh. In order to increase computational efficiency, the refinement is driven by physics-based refinement indicators capturing major model sensitivities. Using idealised numerical test cases, we demonstrate the model’s ability to correctly represent all source terms and reproduce known variability of coastal flooding with respect to hurricane characteristics such as size and approach speed. Finally, the adaptive mesh significantly reduces computing time with no effect on storm waves measured at discrete wave gauges just off the coast which shows the model’s potential for use as a robust simulation tool for real-time predictions