Abstract
We discuss the design features and mathematical background of an explicit upwind finite-volume method to simulate non-stationary flow of a compressible, inviscid fluid. One of the design goals was the rigorous mathematical justification of each ingredient of the method. The method itself contains elements from finite-difference methods as well as finite-element methods and is formulated in a finite volume framework. The use of well-known algorithmic ingredients in a new framework results in a robust time-accurate scheme. To be able to easily handle complex geometries as well as adaption algorithms a tringale-based formulation was chosen. Numerical tests for two-dimensional flow are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comp. Phys. 43 (1981) 357–372.
S. Osher and F. Solomon, Upwind difference schemes for hyperbolic conservation laws, Math. Comp. 38 (1982) 3390–374.
S.P. Spekreijse, Multigrid solution of the steady Euler equations, Centrum voor Wiskunde en Informatica, Amsterdam (1987).
C-W Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comp. Phys. 77 (1988) 439–471.
S. Osher, Convergence of generalized MUSCL schemes, SIAM J. Num Anal. 22 (1985) 947–961.
L. Fezoui and B. Stoufflet, A. class of implicit upwind schemes for Euler simulations with unstructured meshes, J. Comp. Phys. 84 (1989) 174–206.
M. Křižek and P. Neitaanmäki, Superconvergence phenomenon in the finite element method arising from averaging gradients, Numer. Math. 45 (1984) 105–116
T.J. Barth and D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366 (1989), unpublished.
P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comp. Phys. 54 (1984) 115–173.
B. van Leer, Towards the ultimate conservative difference scheme. V.A. second order sequel to Godunov's method, J. Comp. Phys. 32 (1979) 101–136.
A. Jameson, Computational transonics, Comm. Pure Appl. Math. 61 (1988) 507–549.
P. Alexandroff and H. Hopf,Topologie I (Springer, Berlin-Heidelberg-New York, 1974).
L.J. Durlofsky, B. Engquist and S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comp. Phys. 98 (1992) 64–73.
G. Bruhn, Erhaltungssätze und schwache Lösungen in der Gasdynamik, Math. Meth. Appl. Sci. 7 (1985).
Author information
Authors and Affiliations
Additional information
Communicated by G. Mühlbach
Rights and permissions
About this article
Cite this article
Sonar, T. On the design of an upwind scheme for compressible flow on general triangulations. Numer Algor 4, 135–149 (1993). https://doi.org/10.1007/BF02142743
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02142743
Keywords
- Finite volume methods
- hyperbolic conservation laws
- compressible fluid flow
- method of lines
- triangulations