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High-resolution semi-discrete Hermite central-upwind scheme for multidimensional Hamilton-Jacobi equations. (English) Zbl 1288.65134

Summary: We introduce a high resolution fifth-order semi-discrete Hermite central-upwind scheme for multidimensional Hamilton-Jacobi equations. The numerical fluxes of the scheme are constructed by Hermite polynomials which can be obtained by using the short-time assignment of the first derivatives. The extensions of the proposed semi-discrete Hermite central-upwind scheme to multidimensional cases are straightforward. The accuracy, efficiency and stability properties of our schemes are finally demonstrated via a variety of numerical examples.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35F21 Hamilton-Jacobi equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Bryson, S.; Levy, D., Central schemes for multi-dimensional Hamilton-Jacobi equations (2001), NAS Technical Report NAS-01-014 · Zbl 1043.65098
[2] Bryson, S.; Levy, D., High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equations, J. Comput. Phys., 189, 63-87 (2003) · Zbl 1027.65126
[3] Bryson, S.; Levy, D., High-order central WENO schemes for multidimensional Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41, 1339-1369 (2003) · Zbl 1050.65076
[4] Bryson, S.; Kurganov, A.; Levy, D.; Petrova, G., Compressed Semi-Discrete Central-Upwind Schemes for Hamilton-Jacobi Equations (2003), NASA Ames Research Center
[5] Cai, L., Studies on modeling and control of liquid sloshing in spacecraft (2008), Northwestern Polytechnical University: Northwestern Polytechnical University China, PHD thesis
[6] Cai, L.; Feng, J. H.; Xie, W. X., A CWENO-type central-upwind scheme for ideal MHD equations, Appl. Math. Comput., 168, 600-612 (2005) · Zbl 1109.76039
[7] Cai, L.; Feng, J. H.; Xie, W. X.; Zhou, J., Tracking discontinuities in shallow water equations and ideal magnetohydrodynamics equations via ghost fluid method, Appl. Numer. Math., 56, 1555-1569 (2006) · Zbl 1139.76326
[8] Cai, L.; Feng, J. H.; Xie, W. X.; Zhou, J., Computations of steady and unsteady transport of pollutant in shallow water, Math. Comput. Simul., 71, 31-43 (2006) · Zbl 1088.76030
[9] Cai, L.; Xie, W. X.; Feng, J. H.; Zhou, J., Computations of transport of pollutant in shallow water, Appl. Math. Model., 31, 490-498 (2007) · Zbl 1146.76033
[10] Cai, L.; Zhou, J.; Feng, J. H.; Xie, W. X., Fluid simulation using an adaptive semi-discrete central-upwind scheme, Int. J. Comput. Methods, 4, 283-297 (2007) · Zbl 1198.76073
[11] Cai, L.; Zhou, J.; Zhou, F. Q.; Xie, W. X., A hybrid scheme for three-dimensional incompressible two-phase flows, Int. J. Appl. Mech., 2, 4, 889-905 (2010)
[14] Fagnan, K.; LeVeque, R. J.; Matula, T. J., Computational models of material interfaces for the study of extracorporeal shock wave therapy (2012) · Zbl 1278.92018
[15] Feng, J. H.; Cai, L.; Xie, W. X., CWENO-type central-upwind schemes for multidimensional Saint-Venant system of shallow water equations, Appl. Numer. Math., 56, 1001-1017 (2006) · Zbl 1092.76039
[16] Gottlieb, S.; Shu, C. W.; Tadmor, E., Strong stability-preserving high order time discretization methods, SIAM Rev., 43, 89-112 (2001) · Zbl 0967.65098
[17] Hu, C.; Shu, C. W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 666-690 (1999) · Zbl 0946.65090
[18] Jiang, G. S.; Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 2126-2143 (2000) · Zbl 0957.35014
[19] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228 (1996) · Zbl 0877.65065
[20] Jin, S.; Xin, Z., Numerical passage from systems of conservation laws to Hamilton-Jacobi equations and relaxation schemes, SIAM J. Numer. Anal., 35, 2385-2404 (1998) · Zbl 0921.65063
[21] Karlsen, K. H.; Risebro, N. H., Unconditionally stable methods for Hamilton-Jacobi equations, J. Comput. Phys., 180, 710-735 (2002) · Zbl 1143.65365
[22] Kurganov, A.; Levy, D., A third-order semidiscrete central schemes for conservation laws and convection-diffusion equations, SIAM J. Sci. Comput., 22, 1461-1488 (2000) · Zbl 0979.65077
[23] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 241-282 (2000) · Zbl 0987.65085
[24] Kurganov, A.; Tadmor, E., New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23, 707-740 (2001)
[25] Kurganov, A.; Tadmor, E., Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differ. Equ., 18, 584-608 (2002) · Zbl 1058.76046
[26] Kurganov, A.; Noelle, S.; Petrova, G., Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., 23, 707-740 (2000) · Zbl 0998.65091
[27] Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun. Pure Appl. Math., 7, 159-193 (1954) · Zbl 0055.19404
[28] Lax, P. D.; Liu, X. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19, 319-340 (1998) · Zbl 0952.76060
[29] Lepsky, O.; Hu, C.; Shu, C. W., Analysis of the discontinuous Galerkin method for Hamilton-Jacobi equations, Appl. Numer. Math., 33, 423-434 (2000) · Zbl 0968.65073
[30] 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
[31] Levy, D.; Puppo, G.; Russo, G., Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22, 2, 656-672 (2000) · Zbl 0967.65089
[32] Levy, D.; Puppo, G.; Russo, G., A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 24, 2, 480-506 (2002) · Zbl 1014.65079
[33] Lin, C. L.; Tadmor, E., High-resolution non-oscillatory central schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 2163-2186 (2000) · Zbl 0964.65097
[34] Osher, S.; Shu, C. W., High-order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28, 907-922 (1991) · Zbl 0736.65066
[35] Qiu, J. X.; Shu, C. W., Hermite WENO schemes for Hamilton-Jacobi equations, J. Comput. Phys., 204, 82-99 (2005) · Zbl 1070.65078
[36] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 22, 1-31 (1978) · Zbl 0387.76063
[37] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), Spring-Verlag: Spring-Verlag Berlin · Zbl 0888.76001
[38] Zhang, T.; Zheng, Y., Conjecture on the structure of solutions of the Riemann problem for two-dimensional gas dynamics systems, SIAM J. Math. Anal., 21, 593-630 (1990) · Zbl 0726.35081
[39] Zhou, J.; Cai, L.; Zhou, F. Q., A hybrid scheme for computing incompressible two phase flows, Chin. Phys. B, 17, 1535-1544 (2008)
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