×

A new class of high-order energy stable flux reconstruction schemes for triangular elements. (English) Zbl 1457.65101

Summary: 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.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs

Software:

LDG2
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carpenter, M.H., Kennedy, C.: Fourth-order 2n-storage Runge-Kutta schemes. Technical Report TM 109112, NASA, NASA Langley Research Center (1994)
[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
[3] Cockburn, B., Shu, C.W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52(186), 411–435 (1989) · Zbl 0662.65083
[4] Cockburn, B., Shu, C.W.: The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér. 25(3), 337–361 (1991) · Zbl 0732.65094
[5] Cockburn, B., Shu, C.W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001) · Zbl 1065.76135 · doi:10.1023/A:1012873910884
[6] 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(1), 90–113 (1989) · Zbl 0677.65093 · doi:10.1016/0021-9991(89)90183-6
[7] 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(190), 545–581 (1990) · Zbl 0695.65066
[8] Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comput. 6(4), 345–390 (1991) · Zbl 0742.76059 · doi:10.1007/BF01060030
[9] Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, Berlin (2007) · Zbl 1134.65068
[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
[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
[12] Jameson, A.: A proof the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45(1–3), 348–358 (2010) · Zbl 1203.65198 · doi:10.1007/s10915-009-9339-4
[13] Kannan, R., Wang, Z.J.: A study of viscous flux formulations for a p-multigrid spectral volume Navier Stokes solver. J. Sci. Comput. 41(2), 165–199 (2009) · Zbl 1203.65160 · doi:10.1007/s10915-009-9269-1
[14] Kannan, R., Wang, Z.J.: LDG2: A variant of the LDG flux formulation for the spectral volume method. J. Sci. Comput. 46(2), 314–328 (2011) · Zbl 1259.76037 · doi:10.1007/s10915-010-9391-0
[15] Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244–261 (1996) · Zbl 0847.76069 · doi:10.1006/jcph.1996.0091
[16] Liang, C., Kannan, R., Wang, Z.J.: A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids. Comput. Fluids 38, 254–265 (2009) · Zbl 1237.76113 · doi:10.1016/j.compfluid.2008.02.004
[17] Liu, Y., Vinokur, M., Wang, Z.J.: Spectral difference method for unstructured grids i: Basic formulation. J. Comput. Phys. 216, 780–801 (2006) · Zbl 1097.65089 · doi:10.1016/j.jcp.2006.01.024
[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) · Zbl 0362.65089
[19] Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479 (1973)
[20] Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981) · Zbl 0474.65066 · doi:10.1016/0021-9991(81)90128-5
[21] Van den Abeele, K., Lacor, C., Wang, Z.J.: On the stability and accuracy of the spectral difference method. J. Sci. Comput. 37(2), 162–188 (2008) · Zbl 1203.65132 · doi:10.1007/s10915-008-9201-0
[22] Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47(1), 50–72 (2011) · Zbl 1433.76094 · doi:10.1007/s10915-010-9420-z
[23] Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228(21), 8161–8186 (2009) · Zbl 1173.65343 · doi:10.1016/j.jcp.2009.07.036
[24] Wang, Z.J., Liu, Y., May, G., Jameson, A.: Spectral difference method for unstructured grids II: Extension to the Euler equations. J. Sci. Comput. 32, 45–71 (2007) · Zbl 1151.76543 · doi:10.1007/s10915-006-9113-9
[25] Zhang, M., Shu, C.W.: An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math. Models Methods Appl. Sci. 13(3), 395–414 (2003) · Zbl 1050.65094 · doi:10.1142/S0218202503002568
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.