A spectral collocation method to solve Helmholtz problems with boundary conditions involving mixed tangential and normal derivatives. (English) Zbl 1055.65134

Summary: The performing spectral method, developed by P. Haldenwang, G. Labrasse, S. Abboudi, and M. Deville [J. Comput. Phys. 55, 115–128 (1984; Zbl 0544.65071)], to solve multi-dimensional Helmholtz equations, associated to mixed boundary conditions with constant coefficients, is extended to boundary conditions mixing a first order normal derivative with a second order tangential derivative. The accuracy of the proposed algorithm is evaluated on two test cases for which analytical solutions exist: an academic problem and a physical configuration including an interface with shear viscosity. The procedure is also applied to the research of the Rayleigh–Bénard instability thresholds in closed cavities with thin diffusive walls.


65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
76M22 Spectral methods applied to problems in fluid mechanics
76S05 Flows in porous media; filtration; seepage


Zbl 0544.65071
Full Text: DOI


[1] Davis, S.H., Convection in a box, J. fluid mech, 30, 465-478, (1967) · Zbl 0153.29702
[2] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T., Spectral methods in fluids dynamics, (1977), Springer Berlin
[3] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, (1977), SIAM Philadelphia · Zbl 0412.65058
[4] Haidvogel, D.B.; Zang, T., The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials, J. comput. phys, 30, 167-180, (1979) · Zbl 0397.65077
[5] Buell, J.C.; Catton, I., The effect of wall conduction on the stability of a fluid in a right circular cylinder heated from below, J. heat transfer, 105, 255-260, (1983)
[6] Haldenwang, P.; Labrosse, G.; Abboudi, S.; Deville, M., Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation, J. comput. phys, 55, 115-128, (1984) · Zbl 0544.65071
[7] Platten, J.K.; Legros, J.C., Convection in liquids, (1984), Springer Berlin · Zbl 0545.76048
[8] Dang-Vu, H.; Delcarte, C., An accurate solution of the Poisson equation by the Chebyshev collocation method, J. comput. phys, 104, 211-220, (1993) · Zbl 0765.65107
[9] Batoul, A.; Khallouf, H.; Labrosse, G., Une méthode de résolution directe (pseudospectrale) du problème de Stokes 2D/3D instationnaire. application à la cavité entrainée carrée, C. R. acad. sci. ser. II, 319, 1455, (1994) · Zbl 0816.76062
[10] Regnier, V.C.; Parmentier, P.M.; Lebon, G.; Platten, J.K., Numerical simulations of interface viscosity effects on thermoconvective motion in two-dimensional rectangular boxes, Int. J. heat mass transfer, 14, 2539-2548, (1995) · Zbl 0925.76918
[11] C. Delcarte, G. Labrosse, S. Nguyen, Non-homogeneous boundary conditions in collocation spectral method, in: Proceedings of the ECCOMAS CFD Cnference, Swansea, UK, September 4-7, 2001
[12] Leriche, E.; Labrosse, G., High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties, SIAM J. sci. comput, 22, N4, 1386-1410, (2001) · Zbl 0972.35087
[13] Gatignol, R.; Prud’homme, R., Mechanical and thermomecanical modelling of fluid interfaces, Series on advances in mathematics for applied sciences, vol. 58, (2001), World Scientific Singapore
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.