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Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method. II: Two dimensional case. (English) Zbl 1134.65358
Summary: A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional nonlinear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in Part I [J. Comput. Phys. 193, No. 1, 115–135 (2004; Zbl 1039.65068)]. In this paper, we extend the method to solve two dimensional nonlinear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
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[1] Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys., 160, 405-452, (2000) · Zbl 0961.65078
[2] Biswas, R.; Devine, K.D.; Flaherty, J., Parallel, adaptive finite element methods for conservation laws, Appl. numer. math., 14, 255-283, (1994) · Zbl 0826.65084
[3] Bouchut, F.; Bourdarias, C.; Perthame, B., A MUSCL method satisfying all the numerical entropy inequalities, Math. comput., 65, 1439-1461, (1996) · Zbl 0853.65091
[4] Burbeau, A.; Sagaut, P.; Bruneau, C.H., A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, J. comput. phys., 169, 111-150, (2001) · Zbl 0979.65081
[5] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. comput., 54, 545-581, (1990) · Zbl 0695.65066
[6] 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, 90-113, (1989) · Zbl 0677.65093
[7] Cockburn, B.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. comput., 52, 411-435, (1989) · Zbl 0662.65083
[8] Cockburn, B.; Shu, C.-W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. model. numer. anal. (M^{2}AN), 25, 337-361, (1991) · Zbl 0732.65094
[9] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. comput. phys., 141, 199-224, (1998) · Zbl 0920.65059
[10] Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin method for convection-dominated problems, J. scientific comput., 16, 173-261, (2001) · Zbl 1065.76135
[11] Dougherty, R.L.; Edelman, A.S.; Hyman, J.M., Nonnegativity-, monotonicity-, or convexity-preserving cubic and quintic Hermite interpolation, Math. comput., 52, 471-494, (1989) · Zbl 0693.41004
[12] Friedrichs, O., Weighted essentially non-oscillatory schemes for the interpolation of Mean values on unstructured grids, J. comput. phys., 144, 194-212, (1998) · Zbl 1392.76048
[13] Harten, A., On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. numer. anal., 21, 1-23, (1984) · Zbl 0547.65062
[14] Harten, A.; Engquist, B.; Osher, S.; Chakravathy, S., Uniformly high order accurate essentially non-oscillatory schemes, III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067
[15] Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. comput. phys., 150, 97-127, (1999) · Zbl 0926.65090
[16] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065
[17] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Math. model. numer. anal., 33, 547-571, (1999) · Zbl 0938.65110
[18] Liu, X.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076
[19] Nakamura, T.; Tanaka, R.; Yabe, T.; Takizawa, K., Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique, J. comput. phys., 174, 171-207, (2001) · Zbl 0995.65094
[20] Qiu, J.; Shu, C.-W., On the construction, comparison, and local characteristic decomposition for high order central WENO schemes, J. comput. phys., 183, 187-209, (2002) · Zbl 1018.65106
[21] Qiu J, Shu CW. Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Scientific Comput, in press · Zbl 1077.65109
[22] Qiu, J.; Shu, C.-W., Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one dimensional case, J. comput. phys., 193, 115-135, (2003) · Zbl 1039.65068
[23] Reed WH, Hill TR. Triangular mesh methods for neutron transport equation, Tech Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973
[24] Shi, J.; Hu, C.; Shu, C.-W., A technique of treating negative weights in WENO schemes, J. comput. phys., 175, 108-127, (2002) · Zbl 0992.65094
[25] Shu, C.-W., TVB uniformly high-order schemes for conservation laws, Math. comput., 49, 105-121, (1987) · Zbl 0628.65075
[26] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, (), 325-432 · Zbl 0927.65111
[27] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[28] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. comput. phys., 83, 32-78, (1989) · Zbl 0674.65061
[29] Takewaki, H.; Nishiguchi, A.; Yabe, T., Cubic interpolated pseudoparticle method (CIP) for solving hyperbolic type equations, J. comput. phys., 61, 261-268, (1985) · Zbl 0607.65055
[30] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
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