Dynamic p-enrichment schemes for multicomponent reactive flows

We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called \emph{fixed tolerance schemes}, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called \emph{dioristic schemes}, rely on time-evolving bounds on the local variation in the solution. Each class of $p$-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in "areas of highest interest." The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested over a pair of model problems arising in coastal hydrology. The first is a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. The second is a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.Comment: 29 pages, 7 figures, 3 table

A staggered semi-implicit hybrid finite volume / finite element scheme for the shallow water equations at all Froude numbers

We present a novel staggered semi-implicit hybrid FV/FE method for the numerical solution of the shallow water equations at all Froude numbers on unstructured meshes. A semi-discretization in time of the conservative Saint-Venant equations with bottom friction terms leads to its decomposition into a first order hyperbolic subsystem containing the nonlinear convective term and a second order wave equation for the pressure. For the spatial discretization of the free surface elevation an unstructured mesh of triangular simplex elements is considered, whereas a dual grid of the edge-type is employed for the computation of the depth-averaged momentum vector. The first stage of the proposed algorithm consists in the solution of the nonlinear convective subsystem using an explicit Godunov-type FV method on the staggered grid. Next, a classical continuous FE scheme provides the free surface elevation at the vertex of the primal mesh. The semi-implicit strategy followed circumvents the contribution of the surface wave celerity to the CFL-type time step restriction making the proposed algorithm well-suited for low Froude number flows. The conservative formulation of the governing equations also allows the discretization of high Froude number flows with shock waves. As such, the new hybrid FV/FE scheme is able to deal simultaneously with both, subcritical as well as supercritical flows. Besides, the algorithm is well balanced by construction. The accuracy of the overall methodology is studied numerically and the C-property is proven theoretically and validated via numerical experiments. The solution of several Riemann problems attests the robustness of the new method to deal also with flows containing bores and discontinuities. Finally, a 3D dam break problem over a dry bottom is studied and our numerical results are successfully compared with numerical reference solutions and experimental data

Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography

A well-designed numerical method for the shallow water equations (SWE) should ensure well-balancedness, nonnegativity of water heights, and entropy stability. For a continuous finite element discretization of a nonlinear hyperbolic system without source terms, positivity preservation and entropy stability can be enforced using the framework of algebraic flux correction (AFC). In this work, we develop a well-balanced AFC scheme for the SWE system including a topography source term. Our method preserves the lake at rest equilibrium up to machine precision. The low-order version represents a generalization of existing finite volume approaches to the finite element setting. The high-order extension is equipped with a property-preserving flux limiter. Nonnegativity of water heights is guaranteed under a standard CFL condition. Moreover, the flux-corrected space discretization satisfies a semi-discrete entropy inequality. New algorithms are proposed for realistic simulation of wetting and drying processes. Numerical examples for well-known benchmarks are presented to evaluate the performance of the scheme

Mathematical and numerical modelling of dispersive water waves

Fecha de lectura de Tesis: 4 diciembre 2018.En esta tesis doctoral se expone en primer lugar una visión general del modelado de ondas dispersivas para la simulación de procesos tsunami-génicos. Se deduce un nuevo sistema bicapa con propiedades de dispersión mejoradas y un nuevo sistema hiperbólico. Además se estudian sus respectivas propiedades dispersivas, estructura espectral y ciertas soluciones analíticas. Así mismo, se ha diseñado un nuevo modelo de viscosidad sencillo para la simulación de los fenómenos físicos relacionados con la ruptura de olas en costa. Se establecen los resultados teóricos requeridos para el diseño de esquemas numéricos de tipo volúmenes finitos y Galerkin discontinuo de alto orden bien equilibrados para sistemas hiperbólicos no conservativos en una y dos dimensiones. Más adelante, los esquemas numéricos propuestos para los sistemas de presión no hidrostática introducidos se describen. Se pueden destacar diferentes enfoques y estrategias. Por un lado, se diseñan esquemas de volúmenes finitos implícitos de tipo proyección-corrección en mallas decaladas y no decaladas. Por otro lado, se propone un esquema numérico de tipo Galerkin discontinuo explícito para el nuevo sistema de EDPs hiperbólico propuesto. Para permitir simulaciones en tiempo real, una implementación eficiente en GPU de los métodos es llevado a cabo y algunas directrices sobre su implementación son dados. Los esquemas numéricos antes mencionados se han aplicado a test de referencia académicos y a situaciones físicas más desafiantes como la simulación de tsunamis reales, y la comparación con datos de campo. Finalmente, un último capítulo es dedicado a medir la influencia al considerar efectos dispersivos en la simulación de transporte y arrastre de sedimentos. Para ello, se deduce un nuevo sistema de dos capas de aguas someras, se diseña un esquema numérico y se muestran algunos test académicos y de validación, que ofrecen resultados prometedores

Multilevel and Local Timestepping Discontinuous Galerkin Methods for Magma Dynamics

Discontinuous Galerkin (DG) method is presented for numerical modeling of melt migration in a chemically reactive and viscously deforming upwelling mantle column. DG methods for both advection and elliptic equations provide a robust and efficient solution to the problems of melt migration in the asthenospheric upper mantle. Assembling and solving the elliptic equation is the major bottleneck in these computations. To address this issue, adaptive mesh refinement and local timestepping methods have been proposed to significantly improve the computational wall time. The robustness of DG methods is demonstrated through two benchmark problems by modeling detailed structure of high-porosity dissolution channels and compaction-dissolution waves

Numerical simulation of flooding from multiple sources using adaptive anisotropic unstructured meshes and machine learning methods

Over the past few decades, urban floods have been gaining more attention due to their increase in frequency. To provide reliable flooding predictions in urban areas, various numerical models have been developed to perform high-resolution flood simulations. However, the use of high-resolution meshes across the whole computational domain causes a high computational burden. In this thesis, a 2D control-volume and finite-element (DCV-FEM) flood model using adaptive unstructured mesh technology has been developed. This adaptive unstructured mesh technique enables meshes to be adapted optimally in time and space in response to the evolving flow features, thus providing sufficient mesh resolution where and when it is required. It has the advantage of capturing the details of local flows and wetting and drying front while reducing the computational cost. Complex topographic features are represented accurately during the flooding process. This adaptive unstructured mesh technique can dynamically modify (both, coarsening and refining the mesh) and adapt the mesh to achieve a desired precision, thus better capturing transient and complex flow dynamics as the flow evolves. A flooding event that happened in 2002 in Glasgow, Scotland, United Kingdom has been simulated to demonstrate the capability of the adaptive unstructured mesh flooding model. The simulations have been performed using both fixed and adaptive unstructured meshes, and then results have been compared with those published 2D and 3D results. The presented method shows that the 2D adaptive mesh model provides accurate results while having a low computational cost. The above adaptive mesh flooding model (named as Floodity) has been further developed by introducing (1) an anisotropic dynamic mesh optimization technique (anisotropic-DMO); (2) multiple flooding sources (extreme rainfall and sea-level events); and (3) a unique combination of anisotropic-DMO and high-resolution Digital Terrain Model (DTM) data. It has been applied to a densely urbanized area within Greve, Denmark. Results from MIKE 21 FM are utilized to validate our model. To assess uncertainties in model predictions, sensitivity of flooding results to extreme sea levels, rainfall and mesh resolution has been undertaken. The use of anisotropic-DMO enables us to capture high resolution topographic features (buildings, rivers and streets) only where and when is needed, thus providing improved accurate flooding prediction while reducing the computational cost. It also allows us to better capture the evolving flow features (wetting-drying fronts). To provide real-time spatio-temporal flood predictions, an integrated long short-term memory (LSTM) and reduced order model (ROM) framework has been developed. This integrated LSTM-ROM has the capability of representing the spatio-temporal distribution of floods since it takes advantage of both ROM and LSTM. To reduce the dimensional size of large spatial datasets in LSTM, the proper orthogonal decomposition (POD) and singular value decomposition (SVD) approaches are introduced. The performance of the LSTM-ROM developed here has been evaluated using Okushiri tsunami as test cases. The results obtained from the LSTM-ROM have been compared with those from the full model (Fluidity). Promising results indicate that the use of LSTM-ROM can provide the flood prediction in seconds, enabling us to provide real-time flood prediction and inform the public in a timely manner, reducing injuries and fatalities. Additionally, data-driven optimal sensing for reconstruction (DOSR) and data assimilation (DA) have been further introduced to LSTM-ROM. This linkage between modelling and experimental data/observations allows us to minimize model errors and determine uncertainties, thus improving the accuracy of modelling. It should be noting that after we introduced the DA approach, the prediction errors are significantly reduced at time levels when an assimilation procedure is conducted, which illustrates the ability of DOSR-LSTM-DA to significantly improve the model performance. By using DOSR-LSTM-DA, the predictive horizon can be extended by 3 times of the initial horizon. More importantly, the online CPU cost of using DOSR-LSTM-DA is only 1/3 of the cost required by running the full model.Open Acces

Non-intrusive reduced order models and their applications

Reduced order models (ROMs) have become prevalent in many fields of physics as they offer the potential to simulate dynamical systems with substantially increased computation efficiency in comparison to standard techniques. Among the model reduction techniques, the proper orthogonal decomposition (POD) method has proven to be an efficient means of deriving a reduced basis for high-dimensional flow systems. The intrusive ROM (IROM) is normally derived by the POD and Galerkin projection methods. The IROM is appealing for non-linear and linear model reductions and has been successfully applied to numerous research fields. However, IROMs suffer from instability and non-linearity efficiency issues. In addition, they can be complex to code because they are intrusive. In most cases the source code describing the physical system has to be modified in order to generate the reduced order model. These modifications can be complex, especially in legacy codes, or may not be possible if the source code is not available (e.g. in some commercial software). To circumvent these shortcomings, non-intrusive approaches have been introduced into ROMs. The Non-Intrusive ROM (NIROM) is independent of the original physical system. The key contribution of this thesis are: Firstly, three novel NIROMs have been presented in this thesis: POD/Taylor series, POD-Smolyak and POD-RBF (radial basis function). Secondly, two NIROMs with varying material properties have been presented. Thirdly, these newly developed NIROMs were implemented and tested under the framework of an unstructured mesh finite element model (FLUIDITY) and a combined finite-discrete element method based solid model (Y2D). Fourthly, these NIROMs have been used to construct ROMs for multi-scale 3-D free surface flows, multi-phase porous media flows, fluid-structure interaction and blasting problems.Open Acces