Abstract
In this paper, the third in a series, the Spectral Volume (SV) method is extended to one-dimensional systems—the quasi-1D Euler equations. In addition, several new partitions are identified which optimize a certain form of the Lebesgue constant, and the performance of these partitions is assessed with the linear wave equation. A major focus of this paper is to verify that the SV method is capable of achieving high-order accuracy for hyperbolic systems of conservation laws. Both steady state and time accurate problems are used to demonstrate the overall capability of the SV method.
REFERENCES
Abgrall, R. (1994). On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. Comp. Phys. 114, 45–58.
Barth, T. J., and Jespersen, D. C. The design and application of upwind schemes on unstructured meshes, AIAA, Paper No. 89-0366.
Barth, T. J., and Frederickson, P. O. (1990). High-order solution of the Euler equations on unstructured grids using quadratic reconstruction, AIAA, Paper No. 90-0013.
Chen, Q., and Babuska, I. (1995). Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comp. Meth. Appl. Mech. Engrg. 128, 405–417.
Cockburn, B., and Shu, C. W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comp. 52, 411–435.
Cockburn, B., Lin, S.-Y., and Shu, C. W. (1989). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems. J. Comp. Phys. 84, 90–113.
Cockburn, B., Hou, S., and Shu, C. W. (1990). TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case. Math. Comp. 54, 545–581.
Colella, P., and Woodward, P. (1984). The piecewise parabolic method for gas-dynamical simulations. J. Comp. Phys. 54.
Friedrich, O. (1998). Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J. Comp. Phys. 144, 194–212.
Godunov, S. K. (1959). A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47, 271.
Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. J. Comp. Phys. 49, 357–393.
Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. (1987). Uniformly high order essentially non-oscillatory schemes III. J. Comp. Phys. 71, 231.
Hu, C., and Shu, C. W. (1999). Weighted essentially non-oscillatory schemes on triangular meshes. J. Comp. Phys. 150, 97–127.
Lax, P. D. (1954). Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math. 7, 159–193.
Roe, P. L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comp. Phys. 43, 357–372.
Shu, C. W. (1988). Total-Variation-Diminishing time discretizations. SIAM J. Sci. Statist. Comput. 9, 1073–1084.
Shu, C. W. (1987). TVB uniformly high-order schemes for conservation laws. Math. Comp. 49, 105–121.
Shu, C. W., and Osher, S. (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J. Comp. Phys. 83, 32.
Wang, Z. J. (2002). Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comp. Phys. 178, 210.
Wang, Z. J., and Liu, Y. (2002). Spectral (finite) volume method for conservation laws on unstructured grids II: Extension to two-dimensional scalar equation. J. Comp. Phys. 179, 665–697.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wang, Z.J., Liu, Y. Spectral (Finite) Volume Method for Conservation Laws on Unstructured Grids III: One Dimensional Systems and Partition Optimization. Journal of Scientific Computing 20, 137–157 (2004). https://doi.org/10.1023/A:1025896119548
Issue Date:
DOI: https://doi.org/10.1023/A:1025896119548