A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements
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The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stable for all orders of accuracy. The new schemes can easily be extended to quadrilateral elements via the construction of tensor product bases. However, for triangular elements, such a construction is not possible. Since numerical simulations over complicated geometries often require the computational domain to be tessellated with simplex elements, the development of stable FR schemes on simplex elements is highly desirable. In this article, a new class of energy stable FR schemes for triangular elements is developed. The schemes are parameterized by a single scalar quantity, which can be adjusted to provide an infinite range of linearly stable high-order methods on triangular elements. Von Neumann stability analysis is conducted on the new class of schemes, which allows identification of schemes with increased explicit time-step limits compared to the collocation based nodal DG method. Numerical experiments are performed to confirm that the new schemes yield the optimal order of accuracy for linear advection on triangular grids.
KeywordsHigh-order methods Flux reconstruction Nodal discontinuous Galerkin method Triangular elements Stability
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- 1.Carpenter, M.H., Kennedy, C.: Fourth-order 2n-storage Runge-Kutta schemes. Technical Report TM 109112, NASA, NASA Langley Research Center (1994) Google Scholar
- 2.Castonguay, P., Liang, C., Jameson, A.: Simulation of transitional flow over airfoils using the spectral difference method. In: 40th AIAA Fluid Dynamics Conference, Chicago, IL, June 28–July 1 (2010). AIAA Paper, 2010-4626 Google Scholar
- 9.Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007) Google Scholar
- 10.Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. In: 18th AIAA Computational Fluid Dynamics Conference, Miami, FL, Jun 25–28 (2007). AIAA Paper, 4079 Google Scholar
- 11.Huynh, H.T.: A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion. In: 47th AIAA Aerospace Sciences Meeting, Orlando, FL, Jan 5–8 (2009). AIAA Paper, 403 Google Scholar
- 18.Raviart, P.A., Thomas, J.M.: A mixed hybrid finite element method for the second order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lectures Notes in Mathematics. Springer, Berlin (1977) Google Scholar
- 19.Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479 (1973) Google Scholar