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Towards whole program generation of quadrature-free discontinuous Galerkin methods for the shallow water equations [article]

Sara Faghih-Naini, Sebastian Kuckuk, Vadym Aizinger, Daniel Zint, Roberto Grosso, Harald Köstler
2019 arXiv   pre-print
In this paper, we show how code for an advanced quadrature-free discontinuous Galerkin (DG) discretized shallow water equation solver can be generated.  ...  Our contributions include: A new quadrature-free discontinuous Galerkin formulation, an extension of the class of supported computational grids, and an extension of our toolchain allowing to evaluate discrete  ...  Acknowledgements The authors acknowledge financial support by the German Research Foundation (DFG) through grants AI 117/6-1, KO 4641/1-1, and GR 1107/3-1.  ... 
arXiv:1904.08684v1 fatcat:yags2rcsubcudp5vhvthoswyby

Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction

Boris Bonev, Jan S. Hesthaven, Francis X. Giraldo, Michal A. Kopera
2018 Journal of Computational Physics  
We present a novel high-order discontinuous Galerkin discretization for the spherical shallow water equations, able to handle wetting/drying and non-conforming, curved meshes in a well-balanced manner.  ...  Using the strong form of the discontinuous Galerkin discretization, we achieve a splitting of the well-balanced condition into individual problems for the flux and volume terms, which has significant advantages  ...  In this work, we present a novel method for the simulation of the shallow water equations on the surface of a rotating sphere using the discontinuous Galerkin method.  ... 
doi:10.1016/j.jcp.2018.02.008 fatcat:bgsxwp444vcp3ndtytjtqzfegq

Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms

A. Duran, F. Marche
2014 Computers & Fluids  
In the recent years, several improvements have been made in the quality of the discontinuous Galerkin approximations for the Shallow Water equations.  ...  We consider in this work the discontinuous Galerkin discretization of the nonlinear Shallow Water equations on unstructured triangulations.  ...  Scientifique) program LEFE-MANU (Méthodes Mathématiques et Numériques), project SOLi.  ... 
doi:10.1016/j.compfluid.2014.05.031 fatcat:mtx2tavulzelvcksvplfvsqgxe

An unstructured grid morphodynamic model with a discontinuous Galerkin method for bed evolution

Ethan J. Kubatko, Joannes J. Westerink, Clint Dawson
2006 Ocean Modelling  
model that uses a discontinuous Galerkin (DG) method for the solution of the sediment continuity equation.  ...  A new unstructured grid two-dimensional, depth-integrated (2DDI), morphodynamic model is presented for the prediction of morphological evolutions in shallow water.  ...  Acknowledgment This work was supported by the US Army Engineer Research and Development Center under contracts DACW 42-00-C-0006 and W912HZ-05-0022 under the Coastal Inlets Research Program with Dr.  ... 
doi:10.1016/j.ocemod.2005.05.005 fatcat:mow5waaedzfyjm3npboocd4gbe

ADER-DG with a-posteriori finite-volume limiting to simulate tsunamis in a parallel adaptive mesh refinement framework

Leonhard Rannabauer, Michael Dumbser, Michael Bader
2018 Computers & Fluids  
In this paper we present the application of the ADER Discontinuous Galerkin method with an a-posteriori Finite-Volume limiter on the simulation of tsunamis.  ...  Goal is to obtain a method of high-order convergence in deep water areas while being able to handle wetting & drying at coast lines.  ...  the program new Conceptual Approaches to Modeling and Simulation of Complex Systems under grant agreement No. 88479 .  ... 
doi:10.1016/j.compfluid.2018.01.031 fatcat:zxinyqojlrczzhx62jewewr6py

Meshless techniques for anisotropic diffusion

Annamaria Mazzia, Giorgio Pini, Flavio Sartoretto
2014 Applied Mathematics and Computation  
Transport processes are common in geoscience applications, and find their way into models of, e.g., the atmosphere, oceans, shallow water, subsurface, seismic inversion, and deep earth.  ...  The latter would allow better use of parallel computers, since time-stepping is essentially a serial process. Moreover, it would be good for the methods to be of high order accuracy.  ...  We present complex scenarios of multicomponent reactive transport in 2-D and in 3-D. These scenarios are further developments of the setting given by the MoMaS benchmark.  ... 
doi:10.1016/j.amc.2014.03.032 fatcat:c527226gyfgbffnq4p67qxd7wi

A Review of Element-Based Galerkin Methods for Numerical Weather Prediction: Finite Elements, Spectral Elements, and Discontinuous Galerkin

Simone Marras, James F. Kelly, Margarida Moragues, Andreas Müller, Michal A. Kopera, Mariano Vázquez, Francis X. Giraldo, Guillaume Houzeaux, Oriol Jorba
2015 Archives of Computational Methods in Engineering  
Phys. 82, 2003], this review discusses EBG methods as a viable numerical approach for the next-generation NWP models.  ...  As resolutions increase, global models are moving towards fully nonhydrostatic dynamical cores, with the local and global models using the same governing equations; therefore we have reached a point where  ...  Finite elements and Galerkin methods in general (finite volumes included) are free of all the drawbacks of methods that are not flexible with regard to the grid.  ... 
doi:10.1007/s11831-015-9152-1 fatcat:tmjzwcxorfgt5fdcrlf6kb24ny

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

Arnaud Duran, Fabien Marche
2015 Communications in Computational Physics  
AbstractWe describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models.  ...  Using the free surface instead of the water height as a conservative variable, the models are recasted under apre-balancedformulation and discretized using a nodal expansion basis.  ...  program LEFE-MANU (Méthodes Mathématiques et Numériques), project SOLi.  ... 
doi:10.4208/cicp.150414.101014a fatcat:lfnw3v5luzgrbkouj2nvgpstme

A well-balanced spectral volume scheme with the wetting–drying property for the shallow-water equations

Luca Cozzolino, Renata Della Morte, Giuseppe Del Giudice, Anna Palumbo, Domenico Pianese
2012 Journal of Hydroinformatics  
In this paper, we present a spectral-volume method for the approximate solution of the one-dimensional shallow-water equations, which is third-order accurate in wet regions, far from discontinuities, and  ...  The shallow-water equations are widely used to model surface water bodies, such as lakes, rivers and the swash zone in coastal flows.  ...  THE 1D SPECTRAL VOLUME METHOD FOR THE SHALLOW-WATER EQUATIONS The SV method is a high-order numerical method, recently developed for the solution of conservation laws on unstruc- tured grids (Wang  ... 
doi:10.2166/hydro.2012.035 fatcat:fesvfyemobemznx24ybxy4zcwy

A Compact High Order Space-Time Method for Conservation Laws

Shuangzhang Tu, Gordon W. Skelton, Qing Pang
2011 Communications in Computational Physics  
Galerkin (DG) finite element method, are the two key ingredients of the new scheme.  ...  For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS).  ...  The authors are grateful to the School of Engineering and the Department of Computer Engineering at Jackson State University for their support.  ... 
doi:10.4208/cicp.050309.110510a fatcat:pa746rep2jbtdpli3eg3pcq23q

Nektar++: Enhancing the capability and application of high-fidelity spectral/hp element methods

David Moxey, Chris D. Cantwell, Yan Bao, Andrea Cassinelli, Giacomo Castiglioni, Sehun Chun, Emilia Juda, Ehsan Kazemi, Kilian Lackhove, Julian Marcon, Gianmarco Mengaldo, Douglas Serson (+5 others)
2019 Computer Physics Communications  
This release also incorporates a number of numerical method developments -in particular: the method of moving frames (MMF), which provides an additional approach for the simulation of equations on embedded  ...  solvers for partial differential equations using the high-order spectral/hp element method.  ...  Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper  ... 
doi:10.1016/j.cpc.2019.107110 fatcat:cfkq2f7z3jbtzayhsnn5uwcswe

A hybrid spectral/DG method for solving the phase-averaged ocean wave equation: Algorithm and validation

B. Yildirim, George Em Karniadakis
2012 Journal of Computational Physics  
Application of high-order methods to phase-resolving methods (fully nonlinear water waves, Boussinesq, shallow water equations, etc.) have recently received attention with DG schemes proposed and implemented  ...  ; mild-slope equation models, Boussinesq models, and shallow water equations.  ...  This equation is the ancestor of phase-averaged models, which were later modified to take into account shoaling and refraction in shallow water and physical sources (wind generation) and sinks (white-capping  ... 
doi:10.1016/j.jcp.2012.04.013 fatcat:3svleoikyfeznfsqrvbdjj3cky

The mathematics of finite elements and applications

1989 International Journal of Engineering Science  
of upwinding for the constitutive equations.  ...  That theory has been shown to produce good -"--" results for general elliptic systems and general hp-finite element methods.  ...  The model is based on shallow water theory.  ... 
doi:10.1016/0020-7225(89)90012-8 fatcat:z6hknn325re4tbssbqqzmcketa

Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods [article]

David Moxey, Chris D. Cantwell, Yan Bao, Andrea Cassinelli, Giacomo Castiglioni, Sehun Chun, Emilia Juda, Ehsan Kazemi, Kilian Lackhove, Julian Marcon, Gianmarco Mengaldo, Douglas Serson (+5 others)
2019 arXiv   pre-print
This release also incorporates a number of numerical method developments - in particular: the method of moving frames, which provides an additional approach for the simulation of equations on embedded  ...  Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp  ...  Some examples of simulations that can be achieved under this approach include conservational laws [63] , the diffusion equation [64] , shallow water equations (SWEs) [65] , and Maxwell's equations  ... 
arXiv:1906.03489v2 fatcat:nlz3erpuzjenvbjetfbdv4nxc4

A novel staggered semi-implicit space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations [article]

Francesco Lohengrin Romeo, Michael Dumbser, Maurizio Tavelli
2020 arXiv   pre-print
A new high order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible flows on unstructured triangular grids in two space  ...  This choice allows the construction of an efficient, quadrature-free and memory saving algorithm.  ...  For their computation, we have used a quadrature-free approach in a program employing symbolic computation.  ... 
arXiv:2003.06891v1 fatcat:i7jjjd7kevaubce7wv2xukbjdm
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