Discontinuous galerkin methods for spectral wavecirculation. We first develop, implement, verify and validate a dg spectral wave model, which allows for the implementation of unstructured meshes in geographic space and the utility of adaptive, higherorder. May 19, 2015 operational and research nwp codes currently use a wide range of numerical methods. Multilevel preconditioning of discontinuousgalerkin spectral. This program has been developed for teaching purposes only. This method employs the level set procedure to describe the timeevolving interface. Staggeredgrid finite difference method, spectral element method, interiorpenalty discontinuous galerkin method, and isogeometric method. Polynomial degrees for mate 1,2 and 3 can be set independently one from the other for discontinuous galerkin spectral element approxiamtion. The spectral element method uses a tensor product space spanned by nodal basis functions associated with gausslobatto points. Pdf a discontinuous galerkin spectral element method for.
The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. A novel spacetime discontinuous galerkin dg spectral element method is presented to solve the one dimensional stefan problem in an eulerian coordinate system. In order to run the simulation download main notebook files elasticwave. Discontinuousgalerkin spectralelement approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. The following example is an extended version of the one found in kopriva 3. Discontinuous galerkin spectral element approximations for cfd d. Spectral element method because of this, u can be approximately.
Skewsymmetric, entropy conservative discontinuous galerkin spectral element method. Spectral integral method and spectral element method domain. The traditional dgsem belongs to the collocationtype nodal discontinuous galerkin method which is computationally efficient on onedimensional and tensorproduct grids. This code, written by vinh phu nguyen, implements one and two dimensional element free galerkin efg method which is one of the most common meshfree methods for elasticity. Multilevel preconditioning of discontinuousgalerkin spectral element methods part i. Application of discontinuous galerkin spectral method on. Discontinuous galerkin methods on shaperegular and anisotropic meshes emmanuil h. Contribute to ashimrijalspectralelementdiscontinuousgalerkin development by creating an account on github. Discontinuous galerkin methods on shaperegular and. Polynomial approximation degrees for mate 1,2 and 3 must be the same if you are using spectral element approximation. We present a spectralelement discontinuous galerkin lattice boltzmann method for solving singlephase incompressible. Explicit highorder discontinuous galerkin spectral element. Spacetime discontinuous galerkin finite element methods. Texsor3dg an explicit 3d cartesian discontinous galerkin.
The lobattogalerkin method described by young is identical to sem, while the hcgm is equivalent to these methods. The main contributions of this work are listed in the following. Yousu hussaini department of mathematics, florida state university, tallahassee, fl, 32306, usa. The derivation of hybridizable discontinuous galerkin methods. A petrovgalerkin spectral element method for fractional. It also implements partition of unity based enrichment for weak and strong discontinuities. A type of continuousgalerkin finite element method with explicit local and global conservation, and a diagonal mass matrix a typical continuousgalerkin formulation of a problem starts by replacing the fields u with a piecewise polynomial function.
Ge 263 computational geophysics the spectral element method jeanpaul ampuero. Abstract the spectral element method sem is a high order numerical method for solving partial di. The discontinuous galerkin scheme differs from the classical finite element method with respect to the mass matrix, which is defined as being local to the generating element. First, there is the hybridcollocationgalerkin method hcgm, which applies collocation at the interior lobatto points and uses a galerkinlike integral procedure at element interfaces. The discontinuous galerkin method is well suited for equations, for which the functional spaces, where the solution lies, are dif cult to discretize. Apply the basic ideas underlying discontinuous galerkin methods. To overcome this fault, the discontinuous spectral element method dsem was developed. This makes it possible for the discontinuous galerkin scheme to temporally evolve the unknown solution coefficients by using simple explicit methods. We present spectralhp discontinuous galerkin methods for modelling weakly nonlinear and dispersive water waves, described by a set of depthintegrated boussinesq equations, on. The spacetime method is combined with two di erent approaches for treating problems with discontinuous solutions. A hybridizable discontinuous galerkin method for steadystate. The discretization uses a skewsymmetric formulation of the one dimensional shallow water equations and was shown to exactly preserve the. It provides a practical framework for the development of highorder accurate methods using unstructured grids. Apply how the dgfem methods are used as building blocks in the simulation of phenomena descibed by partial differential equations.
A spacetime discontinuous galerkin spectral element method for nonlinear hyperbolic problems chaoxu pei, mark sussman, m. Hesthaven is a professor of applied mathematics at brown university. Seismic wave propagation in 2d acoustic or elastic media using the following methods. Comparison between the 4 different models and methods presented in the planve wave tutorial. Murmany nasa ames research center, mo ett field, ca, usa direct numerical simulation dns of turbulent compressible ows is performed using a higherorder spacetime discontinuousgalerkin niteelement method. Identify and exploit the properties and structutre of the underlying problem. Spectral integral method and spectral element method. Spectralhp discontinuous galerkin methods for modelling.
Spectral integral method and spectral element method domain decomposition method for electromagnetic field analysis by yun lin department of electrical and computer engineering duke university date. Jacobs san diego state university san diego, ca 92182 september 30, 2011 1 code description 1. Only recently various finite volume 8, 21, 17, finite element femcontinuous galerkin 15, 35, pseudo spectral 17, discontinuous galerkin 18 and residual distribution 40 schemes have. Spectral elements in elastodynamics with discontinuous. Multiply the governing equations by a square integrable test function. Georgoulis christ church doctor of philosophy hilary term 2003 we consider the hpversion interior penalty discontinuous galerkin. The discontinuous galerkin spectral element methods for. A spacetime discontinuous galerkin spectral element method for the stefan problem chaoxu pei department of mathematics florida state university tallahassee, fl, 32306, usa mark sussman and m. Dns of flows over periodic hills using a discontinuous. Dns of flows over periodic hills using a discontinuous galerkin spectral element method laslo t. Computer methods in applied mechanics and engineering 349, 149166. A validation study using this method to perform dns of flow in. Discontinuous galerkin an overview sciencedirect topics. Matlab database partial differential equations finite element method discontinuous galerkin method.
Thus the method combines advantages of the spectral methods and the fem, i. File containing physical coordinates of monitored point. Dns of flows over periodic hills using a discontinuousgalerkin spectralelement method laslo t. Discontinuous galerkin dg methods have a long history in computational physics and engineering to. Much like the continuous galerkin cg method, the discontinuous galerkin dg method is a finite element method formulated relative to a weak formulation of a particular model system. In this work we apply the highorder discontinuous galerkin spectral element method dgsem with explicit rungekutta time integration to a classical square duct channel flow problem, which is a widely used benchmark case for turbulent flows. Finite element implementations software oriented c. Discontinuous galerkin spectral element approach that extends to arbitrarily high orders of spatial and temporal accuracy and is implemented in a computationally efficient manner on a modern high performance computer architecture. Course on an introduction to discontinuous galerkin. Maxfem maxfem is an open software package for electromagnetic simulation by using finite element methods.
Speed spectral elements in elastodynamics with discontinuous galerkin is an opensource code for the simulation of seismic wave propagation in threedimensional complex media. In the streaming step, we employ highorder spectralelement discretizations using a tensor prod. In an effort to classify existing dg methods for elliptic problems, arnold et al. Murmany nasa ames research center, mo ett field, ca, usa direct numerical simulation dns of turbulent compressible ows is performed using a higherorder spacetime discontinuous galerkin nite element method. They also developed a highly accurate discontinuous sem for time and space fractional advection equation in 38. A spacetime discontinuous galerkin dg spectral element method is presented for the solution to nonlinear hyperbolic problems. Minev department of mechanical engineering, eindhoven university of technology, po box 5, nl5600 mb eindhoven, the netherlands summary. Department of aerospace engineering, delft univeristy of technology, kluyverweg 2, 2629 hs delft, the netherlands. Computational infrastructure for geodynamics software. Even though windwaves and circulation are modeled separately, it is important to account for their interactions by coupling their respective models. The discontinuous galerkin spectral element method dgsem is highly attractive for both dns and les of turbulent flows due to its low dispersion and dissipation errors, but also because of its. Introduction discontinuous galerkin methods have been extensively studied for tetrahedral meshes e. The authors in design an arbitrarily highorder accurate nodal discontinuous galerkin spectral element type method. High performance computational platform in python for the spectral galerkin method spectraldnsshenfun.
However, it is imperative that one would need more number of elements if the flow gradients are made stronger as in hypersonic cases. A spectral element discontinuous galerkin method for the. In this dissertation we use discontinuousgalerkin dg methods to couple spectral wave and circulation models to. Construction of modern robust nodal discontinuous galerkin. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. Introduction to discontinuous galerkin spectral element.
A comparison of two entropy stable discontinuous galerkin. This paper presents a practical algorithm for constructing high order discontinuous galerkin spectral element methods dgsem on mixed triangular and quadrilateral grids. The derivation of hybridizable discontinuous galerkin. For discretisation in time, constant and linear finite elements are provided. Discontinuous galerkin spectral element approximations for cfd. Speed is jointly developed by mox the laboratory for modeling and scientific computing, department of mathematics and dica department of civil and environmental. In 41, zayernouri and karniadakis developed a spectral and spectral element method for fodes with an exponential accuracy. Explicit highorder discontinuous galerkin spectral. Concurrently, other discontinuous galerkin formulations for parabolic and elliptic problems were proposed 27. Yousuff hussaini department of mathematics florida state university tallahassee, fl, 32306, usa communicated by jie shen abstract.
To deal with the prior unknown interface, a backward transformation and a forward. Operational and research nwp codes currently use a wide range of numerical methods. Aug 12, 2014 an explicit 3d cartesian discontinous galerkin spectral element compressible navierstokes solver andrew c. Multilevel preconditioning of discontinuousgalerkin. Unlike traditional cg methods that are conforming, the dg method works over a trial space of functions that are only piecewise continuous, and thus often comprise more inclusive. A hybridizable discontinuous galerkin method for steady. This is precisely why the high order accurate methods such as discontinuous galerkin and fast spectral are useful. The discontinuous galerkin method is somewhere between a finite element and a finite volume method and has many good features of both. Divide the domain into k nonoverlapping grid cells. Introduction to discontinuous galerkin spectral element methods. Spectralhp discontinuous galerkin methods for modelling 2d. Geometrically conforming meshes kolja brix, martin campos pintox, claudio canutojj and wolfgang dahmen bericht nr.
Spacetime discontinuous galerkin finite element methods j. To construct a nodal discontinuous galerkin spectral element method for the skewsymmetric form of the shallow water equations on each element we do the following. Discontinuous spectral element method for radiative heat. In contrast, the pversion finite element method spans a space of high order polynomials by nodeless basis functions, chosen approximately orthogonal for numerical stability. Construction of modern robust nodal discontinuous galerkin spectral element methods for the compressible navierstokes equations. In this paper, we introduce a new class of discontinuous galerkin methods for the stokes equations. The full text of this article hosted at is unavailable due to technical difficulties. Decoupling the collision step from the streaming step o. In applied mathematics, discontinuous galerkin methods dg methods form a class of numerical methods for solving differential equations.
Discontinuous galerkin discretization for elliptic problems, interior penalty method, locally re ned meshes. Finite difference and discontinuous galerkin finite element. A discontinuous galerkin fast spectral method for the full. Development of a new fast and accurate c0continuous petrov galerkin spectral element method, employing local basistest functions, where the. Pozrikidis, introduction to finite and spectral element methods using matlab, 2nd edition, chapman and hallcrc, 2014.
Analysis of a discontinuous least squares spectral element method. Discontinuous galerkin spectral element approximations for. However, similar to the fem, the numerical instability still exists in the sem. Analysis of a discontinuous least squares spectral element. A spacetime discontinuous galerkin spectral element method. Speed is jointly developed by mox the laboratory for modeling and scientific computing, department of mathematics and dica department of civil and environmental engineering at politecnico. A spacetime discontinuous galerkin spectral element. This results in less computational cost, while we show that the accuracy is still preserved. A type of continuous galerkin finite element method with explicit local and global conservation, and a diagonal mass matrix a typical continuous galerkin formulation of a problem starts by replacing the fields u with a piecewise polynomial function. Higherorder discontinuous galerkin method for pyramidal.
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