An Investigation on Three Point Explicit Schemes and Induced Numerical Oscillations

Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. Comp. Phys. 14(2), 159–179 (1974)

Lax, P.D.: Gibbs phenomena. J. Sci. Comput. 28(2–3), 445–449 (2006)

Godunov, S.K.: A Difference method for numerical solution of discontinuous solution of hydrodynamic equations. Matematicheskii Sbornik 47, 271306 (1959)

Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure. Appl. Math. 13, 217–237 (1960)

LeVeque, R.J.: Numerical Methods for Conservation Laws, Lectures in Mathematics ETH Zurich, 2nd edn. Birkhäuser Verlag, Basel (1992)

Laney, C.B.: Computational Gas-Dynamics. Cambridge University Press, Cambridge (1998)

Lefloach, P.G., Liu, J.G.: Generalized monotone schemes, discrete paths of extrema and discrete entropy conditions. Math. Comp. 68, 1025–1055 (1999)

Breuß, M.: An analysis of the influence of data extrema on some first and second order central approximations of hyperbolic conservation laws. ESAIM. Math. Model. Numer. Anal. 39(5), 965–994 (2005)

Li, Jiequan, Tang, H., Warneke, G., Zhang, L.: Local oscillations in finite difference solutions of hyperbolic conservation laws. Math. Comput. 78(268), 1997–2018 (2009)

Li, J., Yang, Z.: Heuristic modified equation analysis of oscillations in numerical solutions of conservation laws. SIAM J. Numer. Anal. 49(6), 2386–2406 (2011)

Harten, A., Osher, S.: Uniformly high-order accurate non-oscillatory schemes I. SIAM J. Numer. Anal. 24, 279–309 (1987)

Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comp. Phys. 77(2), 439–471 (1988)

Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Adv. Numer. Approx. Nonlinear Hyperb. Equ. Lect. Notes Math. 1697, 325–432 (1998)

Nessyahu, H., Tadmor, E.: Non-Oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87(2), 408–436 (1990)

Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd edn. Springer, New York (2009)

van Leer, B.: Towards the ultimate conservative difference scheme, II. monotonicity and conservation combined in a second-order scheme. J. Comp. Phys 14(4), 361–370 (1974)

Zhang, D., Jiang, C., Liang, D., Cheng, L.: A review on TVD schemes and a refined flux-limiter for steady-state calculations. J. Comp. Phys. 302, 114–154 (2015)

Dubey, R.K., Biswas, B., Gupta, V.: Local maximum principle satisfying high-order non-oscillatory schemes. Int. J. Numer. Meth. Fluids 81(11), 689–715 (2016)

Dubey, R.K.: Data dependent stability of forward in time and centred in space (FTCS) scheme for scalar hyperbolic equations. Int. J. Numer. Anal. Model. Univ. Alberta 13(5), 689–704 (2016)

Dubey, R.K., Biswas, B.: An analysis on induced numerical oscillations by Lax-Friedrichs scheme. Differ. Equ. Dyn. Syst. (Springer) 25(2), 151–168 (2017)