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A note on \((2K+1)\)-point conservative monotone schemes. (English) Zbl 1075.65113
Summary: First-order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a \((2K+1)\)-point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.

MSC:
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
35L65 Hyperbolic conservation laws
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References:
[1] M.G. Crandall and A. Majda , Monotone difference approximations for scalar conservation laws . Math. Comput. 34 ( 1980 ) 1 - 21 . Zbl 0423.65052 · Zbl 0423.65052 · doi:10.2307/2006218
[2] A. Harten , High resolution schemes for hyperbolic conservation laws . J. Comput. Phys. 49 ( 1983 ) 357 - 393 . Zbl 0565.65050 · Zbl 0565.65050 · doi:10.1016/0021-9991(83)90136-5
[3] A. Harten and S. Osher , Uniformly high order accurate non-oscillatory schemes I . SIAM J. Numer. Anal. 24 ( 1987 ) 229 - 309 . Zbl 0627.65102 · Zbl 0627.65102 · doi:10.1137/0724022
[4] A. Harten , J.M. Hyman and P.D. Lax , On finite difference approximations and entropy conditions for shocks . Comm. Pure Appl. Math. 29 ( 1976 ) 297 - 322 . Zbl 0351.76070 · Zbl 0351.76070 · doi:10.1002/cpa.3160290305
[5] C. Helzel and G. Warnecke , Unconditionally stable explicit schemes for the approximation of conservation laws , in Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, B. Fiedler Ed., Springer ( 2001 ). Also available at http://www.math.fu-berlin.de/\(\widetilde{\;}\)danse/bookpapers/ MR 1850329 | Zbl 0999.65093 · Zbl 0999.65093
[6] N.N. Kuznetsov , Accuracy of some approximate methods for computing the weaks solutions of a first-order quasi-linear equation . USSR. Comput. Math. Phys. 16 ( 1976 ) 105 - 119 . Zbl 0381.35015 · Zbl 0381.35015 · doi:10.1016/0041-5553(76)90046-X
[7] X.D. Liu and E. Tadmor , Third order nonoscillatory central scheme for hyperbolic conservation laws . Numer. Math. 79 ( 1998 ) 397 - 425 . Zbl 0906.65093 · Zbl 0906.65093 · doi:10.1007/s002110050345
[8] F. Sabac , The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws . SIAM J. Numer. Anal. 34 ( 1997 ) 2306 - 2318 Zbl 0992.65099 · Zbl 0992.65099 · doi:10.1137/S003614299529347X
[9] R. Sanders , On the convergence of monotone finite difference schemes with variable spatial differencing . Math. Comput. 40 ( 1983 ) 91 - 106 . Zbl 0533.65061 · Zbl 0533.65061 · doi:10.2307/2007364
[10] E. Tadmor , The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs schemes . Math. Comput. 43 ( 1984 ) 353 - 368 . Zbl 0598.65067 · Zbl 0598.65067 · doi:10.2307/2008281
[11] T. Tang and Z.-H. Teng , The sharpness of Kuznetsov’s \(O(\sqrt{\Delta x})L^1\)-error estimate for monotone difference schemes . Math. Comput. 64 ( 1995 ) 581 - 589 . Zbl 0845.65053 · Zbl 0845.65053 · doi:10.2307/2153440
[12] T. Tang and Z.-H. Teng , Viscosity methods for piecewise smooth solutions to scalar conservation laws . Math. Comput. 66 ( 1997 ) 495 - 526 . Zbl 0864.65060 · Zbl 0864.65060 · doi:10.1090/S0025-5718-97-00822-3
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