On the comparison of four different implementations of a third-order ENO scheme of box type for the computation of compressible flow

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.

[1]  Ami Harten,et al.  On High-Order Accurate Interpolation for Non-Oscillatory Shock Capturing Schemes , 1986 .

[2]  Rémi Abgrall,et al.  Eno schemes on unstructured meshes , 1993 .

[3]  G. Mühlbach.  The general Neville-Aitken-algorithm and some applications , 1978 .

[4]  G. Mühlbach.  The general recurrence relation for divided differences and the general Newton-interpolation-algorithm with applications to trigonometric interpolation , 1979 .

[5]  W. Wendland,et al.  Erhaltungssätze und schwache lösungen in der gasdynamik , 1985 .

[6]  Rémi Abgrall,et al.  An essentially non-oscillatory reconstruction procedure on finite-element type meshes: Application to compressible flows , 1994 .

[7]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[8]  A. Harten,et al.  Multi-Dimensional ENO Schemes for General Geometries , 1991 .

[9]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[10]  C. B. Morrey,et al.  Multiple integral problems in the calculus of variations and related topics , 1960 .

[11]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[12]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[13]  Rémi Abgrall,et al.  On the use of Mühlbach expansions in the recovery step of ENO methods , 1997 .

[14]  B. Heinrich.  Finite Difference Methods on Irregular Networks , 1987 .

[15]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[16]  Oliver Friedrich.  A new method for generating inner points of triangulations in two dimensions , 1993 .

[17]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[18]  A. Majda.  Compressible fluid flow and systems of conservation laws in several space variables , 1984 .

[19]  Rémi Abgrall,et al.  Design of an Essentially Nonoscillatory Reconstruction Procedure on Finite-Element-Type Meshes , 1991 .

[20]  T. Sonar.  Multivariate Rekonstruktionsverfahren zur numerischen Berechnung hyperbolischer Erhaltungsgleichungen , 1995 .

[21]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[22]  S. Osher,et al.  Regular ArticleUniformly High Order Accurate Essentially Non-oscillatory Schemes, III , 1997 .

[23]  C. Hirsch.  Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows , 1990 .

[24]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[25]  S. Osher,et al.  Some results on uniformly high-order accurate essentially nonoscillatory schemes , 1986 .