Abstract
Following ideas of Abgrall, four different implementations of a third-order ENO scheme on general triangulations are described and examined. Two implementations utilize implicit time stepping where the resulting linear systems are solved by means of a preconditioned GMRES method. Two other schemes are constructed using an explicit Adams method in time. Quadratic polynomial recovery is used to result in a formally third-order accurate space discretisation. While one class of implementations makes use of cell averages defined on boxes and thus is close in spirit to the finite volume idea, the second class of methods considered is completely node-based. In this second case the interpretation as a true finite volume recovery is completely lost but the recovery process is much simpler and cheaper than the original one. Although one would expect a consistency error in the finite difference type implementations no such problem ever occurred in the numerical experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Abgrall, Design of an essentially non-oscillatory reconstruction procedure on finite-element-type meshes, Preprint submitted to Math. Comp. (1994).
R. Abgrall, An essentially non-oscillatory reconstruction procedure on finite-element type meshes: Application to compressible flows, Comput. Methods Appl Mech. Engrg. 116, (1994) 95–101.
R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: Analysis and implementation. J. Comput. Phys. 114 (1994) 45–58.
R. Abgrall and F. C. Lafon, ENO schemes on unstructured meshes, in:Lecture Series 1993–04, Computational Fluid Dynamics (von Karman Institute for Fluid Dynamics, 1993).
R. Abgrall and Th. Sonar, On the use of Mühlbach expansions in the recovery step of ENO methods, Numer Math. (1996), to appear.
G. Bruhn, Erhaltungssätze und schwache Lösungen in der Gasdynamik, Math. Methods Appl Sci. 7 (1985) 470–479.
P. G. Ciarlet,The Finite Element Method for Elliptic Problems (North-Holland, 2nd ed., 1987).
O. Friedrich, A new method for generating inner points of triangulations in two dimensions, Comput. Methods Appl. Mech. Engrg 194 (1993) 77–86.
M. Geiben, D. Kröner and M. Rokyta, A Lax-Wendroff type theorem for cell-centered, finite volume schemes in 2-D, Sonderforschungsbereich 256, Rheinische Friedrichs-Wilhelms-Universität Bonn, Preprint No. 278 (April, 1993).
E. Hairer, S. P. Nørsett and G. Wanner,Solving Ordinary Differential Equations I, Springer Series in Computational Mathematics 8 (Springer New York, 1987).
A. Harten, On high-order accurate interpolation for non-oscillatory shock capturing schemes, in:Oscillation Theory, Computation and Methods of Compensated Compactness, IMA Vol. 2 (Springer, Berlin, 1987).
A. Harten and S. R. Chakravarthy, Multi-dimensional ENO schemes for general geometries, ICASE Report No. 91-76 (1991).
A. Harten, S. Osher, B. Engquist and S. R. Chakravarthy, Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl. Numer. Math. 2 (1986) 347–377.
A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231–303.
A. Harten and S. Osher, Uniformly high order accurate nonoscillatory schemes I, SIAM J. Numer. Anal. 24 (1987) 279–309.
B. Heinrich,Finite Difference Methods on Irregular Networks, ISNM Vol. 82 (Birkhäuser, Basel, 1987).
C. Hirsch,Numerical Computation of Internal and External Flows Vol. 2:Computational Methods for Inviscid and Viscous Flows (Wiley, New York, 1990).
P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, in:Regional Conference Series in Applied Mathematics 11 (SIAM, Philadelphia, PA, 1973).
A. Majda,Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences 53 (Springer, New York, 1984).
A. Meister, Ein Beitrag zum DLR-τ-Code: Ein explizites und implizites Finite-Volumen-Verfahren zur Berechnung instationärer Strömungen auf unstrukturierten Gittern, DLR Interner Bericht IB 223-94 A 36, Göttingen (1994).
C. Morrey Jr.,Multiple Integral Problems in the Calculus of Variations and Related Topics, Publications in Mathematics (University of California Press, Berkeley and Los Angeles, 1954).
G. Mühlbach, The general Neville-Aitken algorithm and some applications, Numer. Math. 31 (1978) 97–110.
G. Mühlbach, The general recurrence relation for divided differences and the general Newton interpolation algorithm with applications to trigonometric interpolation, Numer. Math. 32 (1979) 393–408.
Y. Saad and M. H. Schulz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.
Th. Sonar, Multivariate Rekonstruktionsverfahren zur numerischen Berechnung hyperbolischer Erhaltungsgleichungen, DLR Forschungsbericht 95-02 (1995).
P. Vankeirsbilck, Algorithmic developments for the solution of hyperbolic conservation laws on adaptive unstructured grids, Ph.D. thesis, Katholieke Universiteit Leuven, Faculteit Toegepaste Wetenschappen, Afdeling Numerieke Analyse en Toegepaste Wiskunde, Celestijnenlaan 200A, 3001 Leuven (Heverlee), Belgium (1993).
Author information
Authors and Affiliations
Additional information
Communicated by G. Mühlbach
Dedicated to Willi Törnig on the occasion of his 65th birthday
Rights and permissions
About this article
Cite this article
Hietel, D., Meister, A. & Sonar, T. On the comparison of four different implementations of a third-order ENO scheme of box type for the computation of compressible flow. Numer Algor 13, 77–105 (1996). https://doi.org/10.1007/BF02143128
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02143128