×

zbMATH — the first resource for mathematics

Spectral (finite) volume method for conservation laws on unstructured grids. II: Extension to two-dimensional scalar equation. (English) Zbl 1006.65113
Summary: The framework for constructing a high-order, conservative spectral (finite) volume (SV) method is presented for two-dimensional scalar hyperbolic conservation laws on unstructured triangular grids. Each triangular grid cell forms a spectral volume (SV), and the SV is further subdivided into polygonal control volumes (CVs) to supported high-order data reconstructions. Cell-averaged solutions from these CVs are used to reconstruct a high-order polynomial approximation in the SV. Each CV is then updated independently with a Godunov-type finite volume method and a high-order Runge-Kutta time integration scheme.
A universal reconstruction is obtained by partitioning all SVs in a geometrically similar manner. The convergence of the SV method is shown to depend on how an SV is partitioned. A criterion based on the Lebesgue constant has been developed and used successfully to determine the quality of various partitions. Symmetric, stable, and convergent linear, quadratic, and cubic SVs have been obtained, and many different types of partitions have been evaluated. The SV method is tested for both linear and nonlinear model problems with and without discontinuities.
For part I see Z. J. Wang [ibid. 178, No. 1, 210–251 (2002; Zbl 0997.65115)].

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
Mathematica
PDF BibTeX Cite
Full Text: DOI
References:
[1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. comput. phys., 114, 45, (1994) · Zbl 0822.65062
[2] Atkin, H.L.; Shu, C.-W., Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, Aiaa j., 36, 775, (1998)
[3] Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high-order accuracy, J. comput. phys., 160, 405, (2000) · Zbl 0961.65078
[4] T. J. Barth, and, D. C. Jespersen, The Design and Application of Upwind Schemes on Unstructured Meshes, AIAA Paper 89-0366, 1989.
[5] T. J. Barth, and, P. O. Frederickson, High-Order Solution of the Euler Equations on Unstructured Grids Using Quadratic Reconstruction, AIAA Paper 90-0013, 1990.
[6] Bassi, F.; Rebay, S., High-order accurate discontinuous finite element solution of the 2D Euler equations, J. comput. phys., 138, 251, (1997) · Zbl 0902.76056
[7] Batten, P.; Lambert, C.; Causon, D.M., Positively conservative high-resolution convection schemes for unstructured elements, Int. J. numer. methods eng., 39, 1821, (1996) · Zbl 0884.76048
[8] Bos, L.P., Bounding the Lebesgue functions for Lagrange interpolation in a simplex, J. approx. theory, 39, 43, (1983) · Zbl 0546.41003
[9] Cai, W.; Gottlieb, D.; Shu, C.-W., Essentially nonoscillatory spectral Fourier methods for shock wave calculations, Math. comput., 52, 389, (1989) · Zbl 0666.65067
[10] Casper, J.; Atkins, H.L., A finite volume high-order ENO scheme for two-dimensional hyperbolic systems, J. comput. phys., 106, 62, (1993) · Zbl 0774.65066
[11] Chen, Q.; Babuska, I., Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. methods appl. mech. eng, 128, 405, (1995) · Zbl 0862.65006
[12] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: general framework, Math. comput., 52, 411, (1989) · Zbl 0662.65083
[13] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: one-dimensional systems, J. comput. phys., 84, 90, (1989) · Zbl 0677.65093
[14] Cockburn, B.; Hou, S.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: the multidimensional case, Math. comput., 54, 545, (1990) · Zbl 0695.65066
[15] Colella, P.; Woodward, P., The piecewise parabolic method for gas-dynamical simulations, J. comput. phys., 54, 174, (1984) · Zbl 0531.76082
[16] M. Delanaye, and, Y. Liu, Quadratic Reconstruction Finite Volume Schemes on 3D Arbitrary Unstructured Polyhedral Grids, AIAA Paper 99-3259-CP, 1999.
[17] Friedrich, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, J. comput. phys., 144, 194, (1998) · Zbl 1392.76048
[18] Godunov, S.K., A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mater. sb., 47, 271, (1959) · Zbl 0171.46204
[19] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially nonoscillatory schemes, III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[20] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97, (1999) · Zbl 0926.65090
[21] Jameson, A., Analysis and design of numerical schemes for gas dynamics. 2: artificial diffusion and discrete shock structure, Int. J. comput. fluid dyn., 5, 1, (1995)
[22] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[23] Kopriva, D.A., Multidomain spectral solutions of the Euler gas-dynamics equations, J. comput. phys., 96, 428, (1991) · Zbl 0726.76077
[24] Kopriva, D.A.; Kolias, J.H., A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. comput. phys., 125, 244, (1996) · Zbl 0847.76069
[25] Liou, M.-S., Mass flux schemes and connection to shock instability, J. comput. phys., 160, 623, (2000) · Zbl 0967.76062
[26] Liu, X.D., A maximum principle satisfying modification of triangle based adaptive stencil for the solution of scalar hyperbolic conservation laws, SIAM J. numer. anal., 30, 701, (1993) · Zbl 0791.65068
[27] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200, (1994) · Zbl 0811.65076
[28] Liu, Y.; Vinokur, M., Exact integration of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids, J. comput. phys., 140, 122, (1998) · Zbl 0899.65008
[29] Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. numer. anal., 21, 217, (1984) · Zbl 0592.65069
[30] Patera, A.T., A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. comput. phys., 54, 468, (1984) · Zbl 0535.76035
[31] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[32] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. stat. comput., 9, 1073, (1988) · Zbl 0662.65081
[33] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. comput., 49, 105, (1987) · Zbl 0628.65075
[34] Sidilkover, D.; Karniadakis, G.E., Non-oscillatory spectral element Chebyshev method for shock wave calculations, J. comput. phys., 107, 10, (1993) · Zbl 0781.65083
[35] Steger, J.L.; Warming, R.F., Flux vector splitting of the inviscid gasdynamics equations with application to finite difference methods, J. comput. phys., 40, 263, (1981) · Zbl 0468.76066
[36] van Leer, B., Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, J. comput. phys., 32, 101, (1979) · Zbl 1364.65223
[37] van Leer, B., Flux-vector splitting for the Euler equations, Lecture notes phys., 170, 507, (1982)
[38] Wang, Z.J., Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation, J. comput. phys., 178, 210, (2002) · Zbl 0997.65115
[39] Wolfram, S., Mathematica book, (1999), Wolfram Media and Cambridge Univ. Press New York
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.