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Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation. (English) Zbl 0997.65115
Summary: A high-order, conservative, yet efficient method named the spectral volume (SV) method is developed for conservation laws on unstructured grids. The concept of a “spectral volume” is introduced to achieve high-order accuracy in an efficient manner similar to spectral element and multidomain spectral methods. Each spectral volume is further subdivided into control volumes, and cell-averaged data from these control volumes are used to reconstruct a high-order approximation in the spectral volume. Then Riemann solvers are used to compute the fluxes at spectral volume boundaries. Cell-averaged state variables in the control volumes are updated independently.
Furthermore, total variation diminishing and total variation bounded limiters are introduced in the SV method to remove/reduce spurious oscillations near discontinuities. Unlike spectral element and multidomain spectral methods, the SV method can be applied to fully unstructured grids. A very desirable feature of the SV method is that the reconstruction is carried out analytically, and the reconstruction stencil is always nonsingular, in contrast to the memory and CPU-intensive reconstruction in a high-order \(k\)-exact finite volume method. Fundamental properties of the SV method are studied and high-order accuracy is demonstrated for several model problems with and without discontinuities.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
SHASTA
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[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, P. O. Frederickson, High-Order Solution of the Euler Equations on Unstructured Grids using Quadratic Reconstruction, AIAA Paper No. 90-0013, 1990.
[5] 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
[6] Boris, J.P.; Book, D.L., Flux-corrected transport I: SHASTA, a fluid transport algorithm that works, J. comput. phys., 11, 38, (1973) · Zbl 0251.76004
[7] Book, D.L.; Boris, J.P.; Hain, K., Flux-corrected transport II: generalization of the method, J. comput. phys., 18, 248, (1975) · Zbl 0306.76004
[8] Burbeau, A.; Sagaut, P.; Bruneau, Ch.-H., A problem independent limiter for high order runge – kutta discontinuous Galerkin methods, J. comput. phys., 169, 111, (2001) · Zbl 0979.65081
[9] 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
[10] S. R. Chakravarthy, and, S. Osher, A New Class of High Accuracy TVD Schemes for Hyperbolic Conservation Laws, AIAA Paper No. 85-0363, 1985.
[11] 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
[12] 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
[13] 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
[14] Colella, P.; Woodward, P., The piecewise parabolic method for gas-dynamical simulations, J. comput. phys., 54, 174, (1984) · Zbl 0531.76082
[15] M. Delanaye, and, Y. Liu, Quadratic Reconstruction Finite Volume Schemes on 3D Arbitrary Unstructured Polyhedral Grids, AIAA Paper No. 99-3259-CP, 1999.
[16] 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
[17] Godunov, S.K., A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. sb., 47, 271, (1959) · Zbl 0171.46204
[18] Goodman, J.B.; LeVeque, R.J., On the accuracy of stable schemes for 2D scalar conservation laws, Math. comput., 45, 15, (1985) · Zbl 0592.65058
[19] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357, (1983) · Zbl 0565.65050
[20] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes III, J. comput. phys., 71, 231, (1987) · Zbl 0652.65067
[21] Harten, A., ENO schemes with subcell resolution, J. comput. phys., 83, 148, (1989) · Zbl 0696.65078
[22] O. Hassan, K. Morgan, and, J. Peraire, An Implicit Finite Element Method for High Speed Flows, AIAA Paper No. 90-0402, January 1990.
[23] Isaacson, E.; Keller, H.B., analysis of numerical methods, (1993), Dover New York · Zbl 0168.13101
[24] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97, (1999) · Zbl 0926.65090
[25] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202, (1996) · Zbl 0877.65065
[26] Kallinderis, Y.; Khawaja, A.; McMorris, H., Hybrid prismatic/tetrahedral grid generation for complex geometries, Aiaa j., 34, 291, (1996) · Zbl 0900.76488
[27] Kopriva, D.A., Multidomain spectral solutions of the Euler gas-dynamics equations, J. comput. phys., 96, 428, (1991) · Zbl 0726.76077
[28] 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
[29] Liu, M.-S., Mass flux schemes and connection to shock instability, J. comput. phys., 160, 623, (2000) · Zbl 0967.76062
[30] Liu, X.D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 2000, (1994)
[31] Luo, H.; Sharov, D.; Baum, J.D.; Lohner, R., on the computation of compressible turbulent flows on unstructured grids, (2000)
[32] Mavriplis, D.J.; Jameson, A., Multigrid solution of the navier – stokes equations on triangular meshes, Aiaa j., 28, 1415, (1990)
[33] Osher, S.; Chakravarthy, S., Upwind schemes and boundary conditions with applications to Euler equation in general geometries, J. comput. phys., 50, 447, (1983) · Zbl 0518.76060
[34] Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. numer. anal., 21, 217, (1984) · Zbl 0592.65069
[35] 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
[36] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357, (1981) · Zbl 0474.65066
[37] P. L. Roe, Optimum Upwind Advection on a Triangular Mesh, ICASE Report 90-75, 1990.
[38] Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. sci. statist. comput., 9, 1073, (1988) · Zbl 0662.65081
[39] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, edited by A. Quarteroni, Lecture Notes in Mathematics Springer-Verlag, Berlin/New York, 1998., Vol. 1697, p. 325.
[40] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. comput., 49, 105, (1987) · Zbl 0628.65075
[41] Sonar, T., On families of pointwise optimal finite volume ENO approximations, SIAM J. numer. anal., 35, 2350, (1998) · Zbl 0921.65069
[42] 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
[43] van Leer, B., Towards the ultimate conservative difference scheme II. monotonicity and conservation combined in a second-order scheme, J. comput. phys., 14, 361, (1974) · Zbl 0276.65055
[44] 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
[45] van Leer, B., Flux-vector splitting for the Euler equations, Lect. notes phys., 170, 507, (1982)
[46] Venkatakrishnan, V.; Mavriplis, D.J., Implicit solvers for unstructured meshes, J. comput. phys., 105, 83, (1993) · Zbl 0783.76065
[47] Wang, Z.J.; Richards, B.E., High-resolution schemes for steady flow computation, J. comput. phys., 97, 53, (1991) · Zbl 0738.76055
[48] Wang, Z.J., A fast flux-splitting for all speed flow, Proceedings of fifteen international conference on numerical methods in fluid dynamics, 141, (1997)
[49] Z. J. Wang, and, R. F. Chen, Anisotropic Cartesian Grid Method for Viscous Turbulent Flow, AIAA Paper No. 2000-0395, 2000.
[50] Yee, H.C., Construction of explicit and implicit symmetric TVD schemes and their applications, J. comput. phys., 68, 151, (1987) · Zbl 0621.76026
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