Computing and Visualization in Science

, Volume 15, Issue 2, pp 61–73

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

Article

DOI: 10.1007/s00791-013-0197-0

Cite this article as:
Siebenborn, M., Schulz, V. & Schmidt, S. Comput. Visual Sci. (2012) 15: 61. doi:10.1007/s00791-013-0197-0

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

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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