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A conservative spectral element method for the approximation of compressible fluid flow. (English) Zbl 1274.76271
Summary: A method to approximate the Euler equations is presented. The method is a multi-domain approximation, and a variational form of the Euler equations is found by making use of the divergence theorem. The method is similar to that of the discontinuous Galerkin method of B. Cockburn and C. W. Shu, but the implementation is constructed through a spectral, multi-domain approach. The method is introduced and is shown to be a conservative scheme. A numerical example is given for the expanding flow around a point source as a comparison with the method proposed by D. A. Kopriva.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Software:
HLLE
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References:
[1] Bassi F., Rebay S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131 (1997), 267-279, 1997 · Zbl 0871.76040
[2] Cockburn B., Shu C. W.: TVB Runga-Kutta local projection discontinuous Galerkin finite-element method for conservation laws II: General framework. Math. Comp. 52 (1989) · Zbl 0662.65083
[3] Cockburn B., Shu C. W.: TVB Runga-Kutta local projection discontinuous Galerkin finite-element method for conservation laws III: One dimensional systems. J. Comput. Phys. 84 (1989), 90 · Zbl 0677.65093
[4] Cockburn B., Shu C. W.: TVB Runga-Kutta local projection discontinuous Galerkin finite-element method for conservation laws IV: The multidimensional case. Math. Comp. 54 (1990) · Zbl 0695.65066
[5] Courant R., Friedrichs K. O.: Supersonic Flow and Shock Waves. Applied Mathematical Sciences. Springer-Verlag, New York 1948 · Zbl 0365.76001
[6] Gordon W. J., Hall C. A.: Transfinite element methods: Blending-function interpolation over arbitrary curved element domains. Numer. Math. 21 (1973), 109-129 · Zbl 0254.65072
[7] Harten A., Lax P. D., Leer B. Van: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review 25 (1983), 1, 35-61 · Zbl 0565.65051
[8] Hesthaven J. S.: A stable penalty method for the compressible Navier-Stokes equations II: One dimensional domain decomposition schemes, to appea. · Zbl 0882.76061
[9] Hesthaven J. S.: A stable penalty method for the compressible Navier-Stokes equations III: Multi dimensional domain decomposition schemes, to appea. · Zbl 0957.76059
[10] Hesthaven J. S., Gottlieb D.: A stable penalty method for the compressible Navier-Stokes equations. I. Open boundary conditions. SIAM J. Sci. Statist. Comput 17 (1996), 3, 579-612 · Zbl 0853.76061
[11] Kopriva D. A.: A Conservative Staggered Grid Chebychev Multi-Domain Method for Compressible Flows. II: A Semi-Structured Method. NASA Contractor Report ICASE Report No. 96-15, ICASE, NASA Langley Research Center, 1996 · Zbl 0866.76064
[12] Kopriva D. A., Kolias J. H.: A conservative staggered grid Chebychev multi-domain method for compressible flows. J. Comput. Phys. 125 (1996), 1, 244-261 · Zbl 0847.76069
[13] Rumsey C., Leer B. van, Roe P. L.: A multidimensional flux function with applications to the Euler and Navier-Stokes equations. J. Comput. Phys. 105 (1993), 306-323 · Zbl 0767.76039
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