On the boundary treatment in spectral methods for hyperbolic systems. (English) Zbl 0626.65123
Motivated by gas pipeline problems (under slowly varying conditions, without shocks), the authors consider the correct approximation of boundary conditions in spectral methods with Chebyshev collocation and implicit time stepping. Converting a special linear hyperbolic system to characteristic variables, discretizing and taking into account there the outgoing resp. incoming characteristics and converting back to physical variables, they obtain a stable discretization.
It turns out that the correct conditions contain the prescribed conditions in linear combination with the collocated differential equations. This approach is used in connection with the schemes of R. M. Beam and R. F. Warming [ibid. 22, 87-110 (1976; Zbl 0336.76021)] and of A. Lerat [C. R. Acad. Sci., Paris, Sér. A 288, 1033-1036 (1979; Zbl 0439.65075)] for nonlinear equations. Computational results are presented for a gas flow problem of industrial relevance. Here the implicit method admits great time steps and gives satisfactory accuracy in less CPU time than an explicit method.
It turns out that the correct conditions contain the prescribed conditions in linear combination with the collocated differential equations. This approach is used in connection with the schemes of R. M. Beam and R. F. Warming [ibid. 22, 87-110 (1976; Zbl 0336.76021)] and of A. Lerat [C. R. Acad. Sci., Paris, Sér. A 288, 1033-1036 (1979; Zbl 0439.65075)] for nonlinear equations. Computational results are presented for a gas flow problem of industrial relevance. Here the implicit method admits great time steps and gives satisfactory accuracy in less CPU time than an explicit method.
Reviewer: G.Stoyan
MSC:
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 76N15 | Gas dynamics (general theory) |
Keywords:
simulation of slow transients in gas transportation networks; Chebyshev pseudospectral method; collocation method; characteristic variables; gas pipeline problems; without shocks; spectral methods; Chebyshev collocation; implicit time steppingReferences:
| [1] | Battarra, V.; Canuto, C.; Quarteroni, A., Comput. Methods Appl. Mech. Eng., 48, 329 (1985) · Zbl 0542.76092 |
| [2] | Beam, R. M.; Warming, R. F., J. Comput. Phys., 22, 87 (1986) |
| [3] | Bullister, E. T.; Karniadakis, G. E.; Rønquist, E. M.; Patera, A. T., Solution of the Unsteady Navier-Stokes Equations by Spectral Element Methods, (6th International Symposium on Finite Element Methods in Flow Problems. 6th International Symposium on Finite Element Methods in Flow Problems, Antibes, France (16 June 1986)) |
| [4] | Daru, V.; Lerat, A., Analysis of an Implicit Euler Solver, (Angrand, F.; etal., Numerical Methods for the Euler Equations of Fluid Dynamics (1985), SIAM: SIAM Philadelphia), 246 · Zbl 0604.65072 |
| [5] | Friedrichs, K. O., Commun. Pure Appl. Math., 2, 333 (1958) |
| [6] | Funaro, D., I.A.N.-C.N.R. Technical Report No. 452 (1985), Pavia |
| [7] | Funaro, D., I.A.N.-C.N.R. Technical Report No. 509 (1986), Pavia |
| [8] | Gottlieb, D.; Gunzburger, M.; Turkel, E., SIAM J. Numer. Anal., 19, 671 (1982) · Zbl 0481.65072 |
| [9] | Gottlieb, D.; Hussaini, M. Y.; Orszag, S. A., (Voigt, R. G.; etal., Spectral Methods for Partial Differential Equations (1984), SIAM: SIAM Philadelphia), 1 |
| [10] | Gottlieb, D.; Turkel, E., (Brezzi, F., Numerical Methods in Fluid Dynamics. Numerical Methods in Fluid Dynamics, Lecture Notes in Mathematics, Vol. 1127 (1985), Springer: Springer New York), 115 · Zbl 0549.00026 |
| [11] | Lerat, A., C. R. Acad. Sci. Paris Sér. A, 288, 1033 (1979) · Zbl 0439.65075 |
| [12] | Luskin, M., Math. Comput., 35, 1093 (1980) · Zbl 0447.65064 |
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