Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws

Wang, Sulin; Xu, Zhengfu

doi:10.1090/mcom/3364

Total variation bounded flux limiters for high order finite difference schemes solving one-dimensional scalar conservation laws
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Math. Comp. 88 (2019), 691-716 Request permission

Abstract:

In this paper, we focus on developing locally conservative high order finite difference methods with provable total variation stability for solving one-dimensional scalar conservation laws. We introduce a new criterion for designing high order finite difference schemes with provable total variation stability by measuring the total variation of an expanded vector. This expanded vector is created from grid values at $t^{n+1}$ and $t^n$ with ordering determined by upwinding information. Achievable local bounds for grid values at $t^{n+1}$ are obtained to provide a sufficient condition for the total variation of the expanded vector not to be greater than total variation of the initial data. We apply the Flux-Corrected Transport type of bound preserving flux limiters to ensure that numerical values at $t^{n+1}$ are within these local bounds. When compared with traditional total variation bounded high order methods, the new method does not depend on mesh-related parameters. Numerical results are produced to demonstrate: the total variation of the numerical solution is always bounded; the order of accuracy is not sacrificed. When the total variation bounded flux limiting method is applied to a third order finite difference scheme, we show that the third order of accuracy is maintained from the local truncation error point of view.

References

J. P. Boris and D. L. Book, Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works, Journal of computational physics 11 (1973), no. 1, 38–69.
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Phillip Colella, A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Statist. Comput. 6 (1985), no. 1, 104–117. MR 773284, DOI 10.1137/0906009
Ulrik S. Fjordholm, Siddhartha Mishra, and Eitan Tadmor, Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal. 50 (2012), no. 2, 544–573. MR 2914275, DOI 10.1137/110836961
James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.) 47 (89) (1959), 271–306 (Russian). MR 0119433
Jonathan B. Goodman and Randall J. LeVeque, On the accuracy of stable schemes for $2$D scalar conservation laws, Math. Comp. 45 (1985), no. 171, 15–21. MR 790641, DOI 10.1090/S0025-5718-1985-0790641-4
Ami Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), no. 3, 357–393. MR 701178, DOI 10.1016/0021-9991(83)90136-5
Ami Harten, Björn Engquist, Stanley Osher, and Sukumar R. Chakravarthy, Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys. 71 (1987), no. 2, 231–303. MR 897244, DOI 10.1016/0021-9991(87)90031-3
Amiram Harten, Peter D. Lax, and Bram van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), no. 1, 35–61. MR 693713, DOI 10.1137/1025002
G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes., Tech. report, DTIC Document, 1995.
Guang-Shan Jiang and Eitan Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput. 19 (1998), no. 6, 1892–1917. MR 1638064, DOI 10.1137/S106482759631041X
Peter Lax and Burton Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237. MR 120774, DOI 10.1002/cpa.3160130205
Peter D. Lax and Xu-Dong Liu, Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput. 19 (1998), no. 2, 319–340. MR 1618863, DOI 10.1137/S1064827595291819
Xu-Dong Liu and Stanley Osher, Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes. I, SIAM J. Numer. Anal. 33 (1996), no. 2, 760–779. MR 1388497, DOI 10.1137/0733038
Xu-Dong Liu, Stanley Osher, and Tony Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115 (1994), no. 1, 200–212. MR 1300340, DOI 10.1006/jcph.1994.1187
Xu-Dong Liu and Eitan Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws, Numer. Math. 79 (1998), no. 3, 397–425. MR 1626324, DOI 10.1007/s002110050345
Stanley Osher and Sukumar Chakravarthy, High resolution schemes and the entropy condition, SIAM J. Numer. Anal. 21 (1984), no. 5, 955–984. MR 760626, DOI 10.1137/0721060
Richard Sanders, A third-order accurate variation nonexpansive difference scheme for single nonlinear conservation laws, Math. Comp. 51 (1988), no. 184, 535–558. MR 935073, DOI 10.1090/S0025-5718-1988-0935073-3
Chi-Wang Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp. 49 (1987), no. 179, 105–121. MR 890256, DOI 10.1090/S0025-5718-1987-0890256-5
Chi-Wang Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev. 51 (2009), no. 1, 82–126. MR 2481112, DOI 10.1137/070679065
Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comput. Phys. 83 (1989), no. 1, 32–78. MR 1010162, DOI 10.1016/0021-9991(89)90222-2
Chi-Wang Shu and Stanley Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR 954915, DOI 10.1016/0021-9991(88)90177-5
P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984), no. 5, 995–1011. MR 760628, DOI 10.1137/0721062
B. Van Leer, Towards the ultimate conservative difference scheme. II. monotonicity and conservation combined in a second-order scheme, J. Comput. Phys. 14 (1974), no. 4, 361–370.
Bram van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method [J. Comput. Phys. 32 (1979), no. 1, 101–136], J. Comput. Phys. 135 (1997), no. 2, 227–248. With an introduction by Ch. Hirsch; Commemoration of the 30th anniversary {of J. Comput. Phys.}. MR 1486274, DOI 10.1006/jcph.1997.5757
Tao Xiong, Jing-Mei Qiu, and Zhengfu Xu, A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows, J. Comput. Phys. 252 (2013), 310–331. MR 3101509, DOI 10.1016/j.jcp.2013.06.026
Zhengfu Xu, Parametrized maximum principle preserving flux limiters for high order schemes solving hyperbolic conservation laws: one-dimensional scalar problem, Math. Comp. 83 (2014), no. 289, 2213–2238. MR 3223330, DOI 10.1090/S0025-5718-2013-02788-3
Steven T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys. 31 (1979), no. 3, 335–362. MR 534786, DOI 10.1016/0021-9991(79)90051-2
Xiangxiong Zhang and Chi-Wang Shu, A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM J. Numer. Anal. 48 (2010), no. 2, 772–795. MR 2670004, DOI 10.1137/090764384