Computing and Visualization in Science

, Volume 15, Issue 2, pp 61–73 | Cite as

A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations

Article

Abstract

In this work we consider Runge–Kutta discontinuous Galerkin methods for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.

List of symbols

\(\alpha \)

Angle of attack

\(D_{r,s,t}\)

Differentiation matrices

\(\mathcal {E}\)

Young’s modulus

\(E\)

Total energy

\(\epsilon \)

Strain tensor

\(\eta \)

Artificial viscosity coefficient

\(F\)

Flux tensor

\(F^*\)

Numerical flux function

\(g\)

Deformation on curved boundary

\(\varvec{\hat{f}}\)

Modal expansion coefficients

\(\varvec{f}\)

Nodal values

\(h\)

Linear elasticity solution

\(\mathcal {I}_{cub}\)

Cubature interpolation matrix

\(\mathcal {I}_{g}\)

Gauss point interpolation matrix

\(J_i^{cub}\)

Jacobian at \(i\)th cubature node

\(J_i^{g}\)

Jacobian at \(i\)th Gauss node

\(\kappa \)

Coefficient for smoothness indicator

\(l_j\)

Lagrange polynomials

\(\lambda , \mu \)

Lamé parameter

\(M\)

Mass matrix

\(M_{\infty }\)

Free stream Mach number

\(M_{\partial \Omega }\)

Face mass matrix

\(N_g\)

Number of Gauss nodes

\(N_{cub}\)

Number of cubature nodes

\(N_{p}\)

Number of collocation nodes

\(n\)

Number of components of \(U\)

\(\nu \)

Poisson’s ratio

\(\Omega \)

Domain of interest

\(\Omega _k\)

\(k\)th element

\(\Omega _c\)

Region of curved elements

\(p\)

Degree of basis polynomials

\(\mathfrak {p}\)

Pressure

\(\psi _i\)

Modal basis functions

\(\Psi _k\)

Transformation functions

\(q_h\)

Vector of auxiliary variables

\(\rho \)

Density

\(\varvec{r}\)

Coordinate in reference element

\(\varvec{r}_i\)

Collocation points

\(\varvec{r}_i^{{cub}}\)

Cubature nodes in reference element

\(\varvec{r}_i^{{g}}\)

Gauss nodes in reference element

\(S_{x,y,z}\)

Stiffness matrices

\(\sigma \)

Stress tensor

\(S_k\)

Smoothness indicator in element k

\(\mathcal {S}\)

NURB surface of the geometry

\(\mathcal {T}\)

Reference element

\(u\)

Velocity in x direction

\(U\)

Unknown function

\(U_h\)

Vector of unknown values

\(U^{{cub}}_h\)

Unknown values at cubature nodes

\(U^{{g}}_h\)

Unknown values at Gauss nodes

\(\mathcal {V}_h\)

Global ansatz space

\(\mathcal {V}_h^k\)

Local ansatz space

\(v\)

Velocity in y direction

\(V\)

Vandermonde matrix

\(w\)

Velocity in z direction

\(W_i^{{cub}}\)

Cubature weights

\(W_i^g\)

Gauss quadrature

\(\varvec{x}\)

Coordinate in physical mesh

References

  1. 1.
    Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos, Report LA-UR-73-479 (1973)Google Scholar
  2. 2.
    Cockburn, B., Shu, C.W.: The Runge–Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. RAIRO, Modélisation Mathématique et Analyse Numérique 25, 337–361 (1991)MathSciNetMATHGoogle Scholar
  3. 3.
    Cockburn, B., Shu, C.W.: for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)MathSciNetMATHGoogle Scholar
  4. 4.
    Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)MathSciNetMATHGoogle Scholar
  6. 6.
    Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Biswas, R., Devine, K.D., Flaherty, J.E.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–283 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bassi, F., Rebay, S.: A high-order discontinuous Galerkin finite element method solution of the 2d Euler equations. J. Comput. Phys. 138, 251–285 (1997)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dwight, R.P.: Robust mesh deformation using the linear elasticity equations. In :Computational Fluid Dynamics 2006, pp. 401–406. Springer, Berlin (2009)Google Scholar
  10. 10.
    Persson, P.O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, 112 (2006)Google Scholar
  11. 11.
    Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181, 186–221 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Klockner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228, 7863–7882 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2008) Google Scholar
  14. 14.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2009)CrossRefGoogle Scholar
  15. 15.
    Warburton, T.: An explicit construction of interpolation nodes on the simplex. J. Eng. Math. 56, 247–262 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Grundmann, A., Möller, H.M.: Invariant integration formulas for the N-simplex by combinatorial methods. SIAM J. Numer. Anal. 15, 282–290 (1978)Google Scholar
  17. 17.
    Cools, R.: Monomial cubature rules since “Stroud”: a compilation-part 2. J. Comput. Appl. Math. 112, 21–27 (1999)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Carpenter, M.H., Kennedy, C.A.: Fourth-Order 2n-Storage Runge–Kutta Schemes. Nasa Report TM, 109112 (1994)Google Scholar
  19. 19.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. Lect. Notes Comput. Sci. Eng. 11, 89–102 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    NVIDIA Corporation: C Programming Guide v4.0. Nvidia Corp (2011)Google Scholar
  21. 21.
    Liebmann, M., Douglas, C., Haase, G., Horvath, Z.: Large scale simulations of the Euler equations on GPU clusters. In: Guo, Q., Guo Y., (eds.) Proceedings of DCABES 2010, Hongkong, pp. 50–54. IEEE Computer Society (2010)Google Scholar
  22. 22.
    Warburton, T.: MIni DG Code. http://www.caam.rice.edu/timwar/RMMC/MIDG.html. (2008). (Online). Accessed 15 Nov 2010
  23. 23.
    Blazek, J.: Computational Fluid Dynamics: Principles and Applications. Elsevier, Amsterdam (2001)Google Scholar
  24. 24.
    Schmitt, V., Charpin, F.: Pressure Distributions on the ONERA-M6-wing at Transonic Mach Numbers. Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138 (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Universität TrierTrierGermany
  2. 2.Department of AeronauticsImperial College LondonLondonUK

Personalised recommendations