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A space-time discontinuous Galerkin method for linear hyperbolic PDE's with high frequencies

Yıl 2020, Cilt: 69 Sayı: 1, 213 - 231, 30.06.2020

Şuayip Toprakseven

https://doi.org/10.31801/cfsuasmas.544522

Öz

The main purpose of this paper is to describe a space-time discontinuous Galerin (DG) method based on an extended space-time approximation
space for the linear first order hyperbolic equation that contains a high frequency component. We extend the space-time DG spaces of tensor-product of
polynomials by adding trigonometric functions in space and time that capture
the oscillatory behavior of the solution. We construct the method by combining the basic framework of the space-time DG method with the extended finite
element method. The basic principle of the method is integrating the features
of the partial differential equation with the standard space-time spaces in the
approximation. We present error analysis of the space-time DG method for
the linear first order hyperbolic problems. We show that the new space-time
DG approximation has an improvement in the convergence compared to the
space-time DG schemes with tensor-product polynomials. Numerical examples verify the theoretical findings and demonstrate the effects of the proposed
method.

Anahtar Kelimeler

Discontinuous Galerkin finite element methods space-time discontinuous Galerkin methods hyperbolic problems high frequency solutions

Kaynakça

  • Huttunen, J.M.J., Huttunen, T., Malinen, M. and Kaipio, J.P., Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping, Phys. Med. Biol., 51 (2006), 1011-1032.
  • Chirputkar, S.U. and Qian, D., Coupled Atomistic/Continuum Simulation based on an Extended Space-Time Finite Element Method, CMES, 850 (2008), 1-18.
  • Garrido, P.L., Goldstein, S., Lukkarinen, J. and Tumulka, R., Paradoxial reflection in quantum mechanics (Preprint, 2008).
  • Hughes, TJR., Franca, LP. and Mallet, M., A new finite-element formulation for Computational Fluid-Dynamics 6. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering, 63, (1987),97-112.
  • Hulbert, GM. and Hughes, TJR., Space-time finite-element methods for 2nd-order hyperbolic-equations, Computer Methods In Applied Mechanics And Engineering, 84, (1990), 327-348.
  • French, DA., A space-time finite element method for the wave equation, Computer Methods in Applied Mechanics and Engineering, 107, (1993),145-157.
  • Johnson, C., Discontinuous Galerkin finite element methods for second order hyperbolic problems, Computer Methods in Applied Mechanics and Engineering, 107, (1993), 117-129.
  • Grote, M. J., Schneebeli, A. and Schotzau D., Discontinuous Galerkin finite element method for the wave equation. SIAM Journal on Numerical Analysis, 44, (2006), 2408-2431.
  • Johnson, C., Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.
  • Falk, RS. and Richter, GR., Explicit finite element methods for symmetric hyperbolic equations, SIAM Journal on Numerical Analysis, 36, (1999), 935-952
  • Monk, P. and Richter, GR., A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, Journal of Scientific Computing, 22-23, (2005), 443-477
  • Demkowicz, L.F. and Gopalakrishnan, J., An overview of the discontinuous Petrovâ€"Galerkin method. In Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, Springer, (2014), 149-180.
  • Van der Vegt, J.J.W. and Van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow, part I. General formulation, J. Comput. Phys. 182, (2002), 546-585
  • Van der Vegt, J. J. W., Space-time discontinuous Galerkin finite element methods, VKI Lecture Series. Von Karman Institute for Fluid Dynamics, (2006), 1-37.
  • Feistaur, M., Hajek, J. and Vadlenka, K.S., Space-time discontinuous Galerkin method for solving nonstationary linear convection-diffusion-reaction problems, Appl. Math., 52, (2007), 197-234.
  • Hulbert, GM. and Hughes, TJR., Space-time finite element methods for second-order hyperbolic equations. Computer Methods in Applied Mechanics and Engineering, 84, (1990), 327-348.
  • Klaij, C., Van der Vegt, J.J.W. and Van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput Phys, 217, (2006), 589-611.
  • Sudirham, J.J., Van der Vegt, J.J.W. and Van Damme, R.M.J., Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math. 56, (2006), 1491-1518.
  • Feistaur, M., Kucera, V., Najzar, K. and Prokopova, J., Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math., 117, (2011), 251-288.
  • Rhebergen, S., Bokhove, O. and Van der Vegt, J.J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 227, (2008), 1887-1922.
  • Ambati, V.R. and Bokhove, O., Space-time discontinuous Galerkin discretization of rotating shallow water equations, J. Comput. Phys., 225, (2007), 1233-1261.
  • Cessenat, O. and Despres, B., Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional Helmholtz problem, SIAM Journal on Numerical Analysis, 35, (1998), 255-299.
  • Banjai, L., Georgoulis, E.H. and Lijoka, O., A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation, SIAM J. Numer. Anal., 55(1), (2017), 63-86
  • Farhat, C., Harari, I. and Franca, LP., The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering, 190, (2001), 6455-6479.
  • Chessa, J. and Belytschko, T., Arbitrary discontinuities in space-time finite elements by level sets and X-FEM, International Journal for Numerical Methods in Engineering, 61,(2014), 2595-2614.
  • Moes, N., Dolbow, J. and Belytschko, T., A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46, (1999), 131-150.
  • Melenk, JM. and Babuska, I., The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 139, (1996), 289-314
  • Toprakseven, S., Error Analysis of Extended Discontinuous Galerkin (XDG) Method. 2004, (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/
  • Johnson, C. and Pitkâranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46, (1986), 1-26.
  • Petersen, S., Farhat, C. and Tezaur, R., A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain, Int. J. Numer. Meth. Eng., 78, (2009), 275-295.
  • Brenner, S.C. and Scott, L.R., The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York, 2008.
  • Cockburn, B., Dong, B. and Guzman, J., Optimal convergence of the original DG method for the transport-reaction equation on special meshes, Institute for Mathematics and its Applications, University of Minnesota, IMA Preprint Series, 2147 (2006).
  • Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math.Comp., 50 (1988), 75-88.

Yıl 2020, Cilt: 69 Sayı: 1, 213 - 231, 30.06.2020

Şuayip Toprakseven

https://doi.org/10.31801/cfsuasmas.544522

Öz

Kaynakça

  • Huttunen, J.M.J., Huttunen, T., Malinen, M. and Kaipio, J.P., Determination of heterogeneous thermal parameters using ultrasound induced heating and MR thermal mapping, Phys. Med. Biol., 51 (2006), 1011-1032.
  • Chirputkar, S.U. and Qian, D., Coupled Atomistic/Continuum Simulation based on an Extended Space-Time Finite Element Method, CMES, 850 (2008), 1-18.
  • Garrido, P.L., Goldstein, S., Lukkarinen, J. and Tumulka, R., Paradoxial reflection in quantum mechanics (Preprint, 2008).
  • Hughes, TJR., Franca, LP. and Mallet, M., A new finite-element formulation for Computational Fluid-Dynamics 6. Convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems. Computer Methods in Applied Mechanics and Engineering, 63, (1987),97-112.
  • Hulbert, GM. and Hughes, TJR., Space-time finite-element methods for 2nd-order hyperbolic-equations, Computer Methods In Applied Mechanics And Engineering, 84, (1990), 327-348.
  • French, DA., A space-time finite element method for the wave equation, Computer Methods in Applied Mechanics and Engineering, 107, (1993),145-157.
  • Johnson, C., Discontinuous Galerkin finite element methods for second order hyperbolic problems, Computer Methods in Applied Mechanics and Engineering, 107, (1993), 117-129.
  • Grote, M. J., Schneebeli, A. and Schotzau D., Discontinuous Galerkin finite element method for the wave equation. SIAM Journal on Numerical Analysis, 44, (2006), 2408-2431.
  • Johnson, C., Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.
  • Falk, RS. and Richter, GR., Explicit finite element methods for symmetric hyperbolic equations, SIAM Journal on Numerical Analysis, 36, (1999), 935-952
  • Monk, P. and Richter, GR., A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, Journal of Scientific Computing, 22-23, (2005), 443-477
  • Demkowicz, L.F. and Gopalakrishnan, J., An overview of the discontinuous Petrovâ€"Galerkin method. In Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, Springer, (2014), 149-180.
  • Van der Vegt, J.J.W. and Van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flow, part I. General formulation, J. Comput. Phys. 182, (2002), 546-585
  • Van der Vegt, J. J. W., Space-time discontinuous Galerkin finite element methods, VKI Lecture Series. Von Karman Institute for Fluid Dynamics, (2006), 1-37.
  • Feistaur, M., Hajek, J. and Vadlenka, K.S., Space-time discontinuous Galerkin method for solving nonstationary linear convection-diffusion-reaction problems, Appl. Math., 52, (2007), 197-234.
  • Hulbert, GM. and Hughes, TJR., Space-time finite element methods for second-order hyperbolic equations. Computer Methods in Applied Mechanics and Engineering, 84, (1990), 327-348.
  • Klaij, C., Van der Vegt, J.J.W. and Van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput Phys, 217, (2006), 589-611.
  • Sudirham, J.J., Van der Vegt, J.J.W. and Van Damme, R.M.J., Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math. 56, (2006), 1491-1518.
  • Feistaur, M., Kucera, V., Najzar, K. and Prokopova, J., Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math., 117, (2011), 251-288.
  • Rhebergen, S., Bokhove, O. and Van der Vegt, J.J.W., Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 227, (2008), 1887-1922.
  • Ambati, V.R. and Bokhove, O., Space-time discontinuous Galerkin discretization of rotating shallow water equations, J. Comput. Phys., 225, (2007), 1233-1261.
  • Cessenat, O. and Despres, B., Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional Helmholtz problem, SIAM Journal on Numerical Analysis, 35, (1998), 255-299.
  • Banjai, L., Georgoulis, E.H. and Lijoka, O., A Trefftz polynomial space-time discontinuous Galerkin method for the second order wave equation, SIAM J. Numer. Anal., 55(1), (2017), 63-86
  • Farhat, C., Harari, I. and Franca, LP., The discontinuous enrichment method, Computer Methods in Applied Mechanics and Engineering, 190, (2001), 6455-6479.
  • Chessa, J. and Belytschko, T., Arbitrary discontinuities in space-time finite elements by level sets and X-FEM, International Journal for Numerical Methods in Engineering, 61,(2014), 2595-2614.
  • Moes, N., Dolbow, J. and Belytschko, T., A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46, (1999), 131-150.
  • Melenk, JM. and Babuska, I., The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering, 139, (1996), 289-314
  • Toprakseven, S., Error Analysis of Extended Discontinuous Galerkin (XDG) Method. 2004, (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/
  • Johnson, C. and Pitkâranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., 46, (1986), 1-26.
  • Petersen, S., Farhat, C. and Tezaur, R., A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain, Int. J. Numer. Meth. Eng., 78, (2009), 275-295.
  • Brenner, S.C. and Scott, L.R., The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York, 2008.
  • Cockburn, B., Dong, B. and Guzman, J., Optimal convergence of the original DG method for the transport-reaction equation on special meshes, Institute for Mathematics and its Applications, University of Minnesota, IMA Preprint Series, 2147 (2006).
  • Richter, G. R., An optimal-order error estimate for the discontinuous Galerkin method, Math.Comp., 50 (1988), 75-88.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Research Article
Yazarlar

Şuayip Toprakseven 0000-0003-3901-9641

Yayımlanma Tarihi 30 Haziran 2020
Gönderilme Tarihi 25 Mart 2019
Kabul Tarihi 8 Ekim 2019
Yayımlandığı Sayı Yıl 2020 Cilt: 69 Sayı: 1

Kaynak Göster

APA Toprakseven, Ş. (2020). A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(1), 213-231. https://doi.org/10.31801/cfsuasmas.544522
AMA Toprakseven Ş. A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Haziran 2020;69(1):213-231. doi:10.31801/cfsuasmas.544522
Chicago Toprakseven, Şuayip. “A Space-Time Discontinuous Galerkin Method for Linear Hyperbolic PDE’s With High Frequencies”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, sy. 1 (Haziran 2020): 213-31. https://doi.org/10.31801/cfsuasmas.544522.
EndNote Toprakseven Ş (01 Haziran 2020) A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 1 213–231.
IEEE Ş. Toprakseven, “A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 69, sy. 1, ss. 213–231, 2020, doi: 10.31801/cfsuasmas.544522.
ISNAD Toprakseven, Şuayip. “A Space-Time Discontinuous Galerkin Method for Linear Hyperbolic PDE’s With High Frequencies”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/1 (Haziran 2020), 213-231. https://doi.org/10.31801/cfsuasmas.544522.
JAMA Toprakseven Ş. A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:213–231.
MLA Toprakseven, Şuayip. “A Space-Time Discontinuous Galerkin Method for Linear Hyperbolic PDE’s With High Frequencies”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 69, sy. 1, 2020, ss. 213-31, doi:10.31801/cfsuasmas.544522.
Vancouver Toprakseven Ş. A space-time discontinuous Galerkin method for linear hyperbolic PDE’s with high frequencies. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(1):213-31.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

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