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Dispersive behaviour of high order finite element schemes for the one-way wave equation

Mark Ainsworth
2014 Journal of Computational Physics  
We study the ability of high order numerical methods to propagate discrete waves at the same speed as the physical waves in the case of the one-way wave equation.  ...  A detailed analysis of the finite element method is presented including an explicit form for the discrete dispersion relation and a complete characterisation of the numerical Bloch waves admitted by the  ...  [2] that, in the case of the second order wave equation, the spectral element method propagates waves with an accuracy superior to that of the finite element scheme.  ... 
doi:10.1016/j.jcp.2013.11.003 fatcat:3avjiwunvjf4lefjbgrtyz252a

A finite element method for the one-dimensional extended Boussinesq equations

M. Walkley, M. Berzins
1999 International Journal for Numerical Methods in Fluids  
The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured  ...  The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J.  ...  ACKNOWLEDGMENTS The authors would like to thank the EPSRC and HR Wallingford for funding in the form of a CASE studentship for MW. The authors also thank A.J. Cooper, J.V. Smallman, J. Lawson and N.  ... 
doi:10.1002/(sici)1097-0363(19990130)29:2<143::aid-fld779>3.0.co;2-5 fatcat:j5wjnzhk4jgxbdhh5asusw5tz4

Two-dimensional dispersion analyses of finite element approximations to the shallow water equations

J. H. Atkinson, J. J. Westerink, R. A. Luettich
2004 International Journal for Numerical Methods in Fluids  
wave continuity equation (GWCE) reformulation of the harmonic shallow water equations on a number of grid conÿgurations.  ...  In this paper, we explore the application of two-dimensional dispersion analysis to cluster based and Galerkin ÿnite element-based discretizations of the primitive shallow water equations and the generalized  ...  The undesirable, high wave number response has a wavelength on the order of twice the grid spacing and is therefore referred to as a 2 x wave.  ... 
doi:10.1002/fld.701 fatcat:yabuvofrnvgthiooxmpum2b3xm

Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations

D. Sármány, M. A. Botchev, J. J. W. van der Vegt
2007 Journal of Scientific Computing  
Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed.  ...  It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.  ...  Acknowledgements This research was supported by the Dutch government through the national program BSIK: knowledge and research capacity, in the ICT project BRICKS (http://www.bsik-bricks.nl), theme MSV1  ... 
doi:10.1007/s10915-007-9143-y fatcat:4mvwrhr3xzfvtjn5lk6yenfoce

Reprint of: Physics-Based Computational Methods for Aero-Acoustics

G. Gabard, R.J. Astley, P. Gamallo, G. Kennedy
2010 Procedia IUTAM  
The methods described in this paper include the Green's function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method.  ...  For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green's functions or plane waves  ...  Acknowledgements: Some of the work presented in this paper was supported in part by EPSRC research grants (GR/R90758/01, CNA/07/65) and by Rolls-Royce plc.  ... 
doi:10.1016/j.piutam.2010.10.020 fatcat:6lqlhm2tb5cg3j7kdev6r7bpwm

Physics-based computational methods for aero-acoustics

G. Gabard, R.J. Astley, P. Gamallo, G. Kennedy
2010 Procedia Engineering  
The methods described in this paper include the Green's function discretisation, the partition of unity finite element method and the wave-based discontinuous Galerkin method.  ...  For instance, instead of using standard polynomials or Chebyshev polynomials to interpolate the solution, physics-based methods generally use canonical solutions such as Green's functions or plane waves  ...  Acknowledgements: Some of the work presented in this paper was supported in part by EPSRC research grants (GR/R90758/01, CNA/07/65) and by Rolls-Royce plc.  ... 
doi:10.1016/j.proeng.2010.09.020 fatcat:gruyo5r2frhcbai4icioyum2m4

Dispersion Analysis of Discontinuous Galerkin Schemes Applied to Poincaré, Kelvin and Rossby Waves

P.-E. Bernard, E. Deleersnijder, V. Legat, J.-F. Remacle
2007 Journal of Scientific Computing  
Real parts of eigenvalues λ i 's allow to compute dissipation errors of the scheme for each given class of wave.  ...  A technique for analyzing dispersion properties of numerical schemes is proposed.  ...  The present study was carried out within the scope of the project "A second-generation model of the ocean system" funded by the Communauté Française de Belgique, as Actions de Recherche Concertées, under  ... 
doi:10.1007/s10915-007-9156-6 fatcat:zo7lbo52ffet3kgrveppofqg64

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

M. Ainsworth, P. Monk, W. Muniz
2006 Journal of Scientific Computing  
We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second order wave equation and a more general family of schemes  ...  We derive the dispersion error for various schemes and conjecture on the generalization to higher order approximation in space.  ...  The research of PM was supported by a grant from AFOSR and by an EPSRC visitor grant to the University of Reading, UK.  ... 
doi:10.1007/s10915-005-9044-x fatcat:ndsdr6ybfzek7g5uxst25f6bh4

Dispersion of elastic waves in the contact–impact problem of a long cylinder

D. Gabriel, J. Plešek, R. Kolman, F. Valeš
2010 Journal of Computational and Applied Mathematics  
Numerical dispersion of two-dimensional finite elements was studied.  ...  In accordance with the results of the dispersion analysis it was demonstrated that the quadratic elements showed better accuracy than the linear ones.  ...  Acknowledgements This work was supported by the Grant Agency of the Czech Republic under grant numbers GA101/07/1471, GA101/09/1630 and GA101/06/0213 in the framework of AV0Z 20760514.  ... 
doi:10.1016/j.cam.2009.08.043 fatcat:6wqehqo64zbzbnznh2zygylqoy

Evaluation of the Spectral Finite Element Method with the Theory of Phononic Crystals

Nicolás Guarín-Zapata, Juan Gomez
2015 Journal of Computational Acoustics  
The effects in the dispersion properties of the lumping process for the mass matrices resulting from the classical finite element method are also investigated, since that is a common practice when the  ...  We evaluated the performance of the classical and spectral finite element method in the simulation of elastodynamic problems.  ...  The Spectral Element Method The Spectral Finite Element Method (SFEM) is a finite element technique based on high order Lagrange interpolation but with a non-equidistant nodal distribution.  ... 
doi:10.1142/s0218396x15500046 fatcat:esnva4626jduzcjgmjdke6s6em

Numerical noise suppression for wave propagation with finite elements in first-order form by an extended source term

R Shamasundar, W A Mulder
2018 Geophysical Journal International  
If, instead of the usual second-order form, the first-order form of the wave equation is considered, this is no longer true.  ...  S U M M A R Y Finite elements can, in some cases, outperform finite-difference methods for modelling wave propagation in complex geological models with topography.  ...  Dispersive behaviour of high order finite element schemes for the one-way wave equation, J. Comput. Phys., 259, 1-10. Anderson, J.E., Brytik, V. & Ayeni, G., 2015.  ... 
doi:10.1093/gji/ggy337 fatcat:2sxw2efuzjgjbg6meemak3plqi

A high-order Petrov-Galerkin finite element method for the classical Boussinesq wave model

Paulo Avilez-Valente, Fernando J. Seabra-Santos
2009 International Journal for Numerical Methods in Fluids  
A high-order Petrov-Galerkin finite element scheme is presented to solve the one-dimensional depthintegrated classical Boussinesq equations for weakly non-linear and weakly dispersive waves.  ...  Based on the order of the analytical terms of the Boussinesq model and on the order of the numerical discretization, it is concluded that the scheme is fourth-order accurate in terms of phase velocity.  ...  ACKNOWLEDGEMENTS The authors would like to thank the two anonymous reviewers for their helpful comments.  ... 
doi:10.1002/fld.1846 fatcat:ofawdcar7jemzfs52osf3p24im

Computational modelling of severe gradients with mass operator constructions

C.A.J. Fletcher
1988 Computers and Mathematics with Applications  
A generalized mass operator construction is developed from the Gederkin finite element method and used to compute problems with severe interned gradients as simple models for the frontal process.  ...  When combined with Crank-Nicolson time differencing both schemes provide optimal choices for generating accurate solutions with severe gradients, primarily by controlling dispersion errors.  ...  The extension of the Galerkin finite element method to generalize the mass operators provide a mechanism for controlling dispersion errors.  ... 
doi:10.1016/0898-1221(88)90022-3 fatcat:zhqhdrc4pnbqjp6m7ezytofflu

Choice of function spaces for thermodynamic variables in mixed finite-element methods

Thomas Melvin, Tommaso Benacchio, John Thuburn, Colin Cotter
2018 Quarterly Journal of the Royal Meteorological Society  
We study the dispersion properties of three choices for the buoyancy space in a mixed finite-element discretization of geophysical fluid flow equations.  ...  Discrete dispersion relations of the two-dimensional linear gravity wave equations are computed.  ...  ACKNOWLEDGEMENTS The authors would like to thank Nigel Wood for comments on an early draft of the paper.  ... 
doi:10.1002/qj.3268 fatcat:uh6wd535lvdold37eodaaoc64y

Revisiting the spectral analysis for high-order spectral discontinuous methods

Julien Vanharen, Guillaume Puigt, Xavier Vasseur, Jean-François Boussuge, Pierre Sagaut
2017 Journal of Computational Physics  
Christophe Bogey for the fruitful discussions on the spectral behaviour of his numerical methods. His feedback on computational cost using his own (optimised) CFD code is highly appreciated.  ...  Acknowledgements We would like to thank the anonymous referees for their careful reading and helpful remarks. We gratefully acknowledge Dr.  ...  Depending on the polynomial degree p, p + 1 waves are solutions of the linear set of equations.  ... 
doi:10.1016/j.jcp.2017.02.043 fatcat:7fbxqfcfvvgyvellzlj2ulazhi
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