1. Balsara, D.S. and Shu, C.W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, Journal of Computational Physics, 160 (2000), 405–452.
2. Burman, E. A monotonicity preserving, nonlinear, finite element upwind method for the transport equation, Applied Mathematics Letters, 49 (2015), 141–146.
3. Capdeville, G. A high-order monotonicity-preserving scheme for hyperbolic conservation laws, Computers & Fluids, 144 (2017), 86–116.
4. Crandall, M.G. and Andrew, M. Monotone difference approximations for scalar conservation laws, Mathematics of Computation, 149 (1980), 1–21.
5. Crandall, M. and Majda, A. The method of fractional steps for conservation laws, Numerische Mathematik, 34 (1980), 285–314.
6. Daru, V. and Tenaud, C. High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations, Journal of Computational Physics, 193 (2004), 563–594.
7. Du, Q., Huang, Z. and LeFloch, P.G. Nonlocal conservation laws. A new class of monotonicity-preserving models, SIAM Journal on Numerical Analysis, 55 (2017), 2465–2489.
8. Evans, L.C. Partial differential equations, American Mathematical Society, 1998.
9. Fang, J., Yao, Y., Li, Z. and Lu, L. Investigation of low-dissipation monotonicity-preserving scheme for direct numerical simulation of compressible turbulent flows, Computers & Fluids, 104 (2014), 55–72.
10. Farzi, J. Global error estimation of linear multistep methods through the Runge-Kutta methods, Iranian Journal of Numerical Analysis and Optimization, 6 (2016), 99–120.
11. Farzi, J., Hosseini, S.M. High order immersed interface method for acoustic wave equation with discontinuous coefficients, Iranian Journal of Numerical Analysis and Optimization, 4 (2014), 1–24.
12. Farzi, J. and Khodadosti, F. A total variation diminishing high resolution scheme for nonlinear conservation laws, Computational Methods for Differential Equations, 6 (2018), 456–470.
13. Gardner, L.R. and Dag, I. A numerical solution of the advection-diffusion equation using B-spline finite element, Proceedings International AMSE Conference, Lyon, France, (1994), 109–116.
14. Ha, C.T. and Lee, J.H. A modified monotonicity-preserving high-order scheme with application to computation of multi-phase flows, Computers & Fluids, 197 (2020), 1–29.
15. He, Z., Li, X., Fu, D. and Ma, Y. A 5th order monotonicity-preserving upwind compact difference scheme, Science China Physics Mechanics and Astronomy, 54 (2011), 511–522.
16. He, Z., Zhang, Y., Gao, F., Li, X. and Tian, B. An improved accurate monotonicity-preserving scheme for the Euler equations, Computers & Fluids, 140 (2016), 1–10.
17. Higueras, I., Ketcheson, D.I. and Kocsis, T.A. Optimal monotonicity preserving perturbations of a given Runge-Kutta method, Journal of Scientific Computing, 76 (2018), 1337–1369.
18. Holly, F.M. and Usseglio-Polatera, J.M. Dispersion simulation in two dimensional tidal flow, Journal of Hydraulic Engineering, 110 (1984), 905–926.
19. Hundsdorfer, W. and Verwer, J.G. A note on splitting errors for advection-reaction equations, Applied Numerical Mathematics, 18 (1997), 191–199.
20. Hundsdorfer, W. and Verwer, J.G. Numerical solution of time-dependent advection-diffusion-reaction equations, Springer Science & Business Media, 2013.
21. Huynh, H.T. A piecewise-parabolic dual-mesh method for the Euler equations, In 12th Computational Fluid Dynamics Conference, 1995.
22. Huynh, H.T. Schemes and constraints for advection, In Fifteenth Inter national Conference on Numerical Methods in Fluid Dynamics, Springer, Berlin, Heidelberg, (1997) 498–503.
23. Karlsen, K.H. and Risebro, N.H. Corrected operator splitting for nonlinear parabolic equations, SIAM Journal on Numerical Analysis, 37 (2000), 980–1003.
24. Khalsaraei, M.M. An improvement on the positivity results for 2-stage explicit Runge-Kutta methods, Journal of Computational and Applied Mathematics, 235 (2010) 137–143.
25. Khalsaraei, M.M. and Khodadosti, F. 2-stage explicit total variation diminishing preserving Runge-Kutta methods, Computational Methods forDifferential Equations, 1 (2013), 30–38.
26. Khalsaraei, M.M. and Khodadosti, F. Qualitatively stability of nonstandard 2-stage explicit Runge–Kutta methods of order two, Computational Mathematics and Mathematical Physics, 56 (2016), 235–242.
27. Koren, B. A robust upwind discretization method for advection, diffusion and source terms, In Numerical methods for advection-diffusion problems, Amsterdam: Centrum voor Wiskunde en Informatica, (1993), 117–138.
28. Langseth, J.O., Tveito, A. and Winther, R. On the convergence of operator splitting applied to conservation laws with source terms, SIAM Journal on Numerical Analysis, 33 (1996), 843-863.
29. LeVeque, R.J. Finite volume methods for hyperbolic problems, Cambridge University Press; 2002.
30. LeVeque, R.J. and Yee, H.C. A study of numerical methods for hyper bolic conservation laws with stiff source terms, Journal of Computational Physics, 86 (1990), 187–210.
31. Li, X.L., Fu, D.X., Ma, Y.W. and Liang, X. Direct numerical simulation of compressible turbulent flows, Acta Mechanica Sinica, 26 (2010), 795–806.
32. Matsushita, S. and Aoki, T. A weakly compressible scheme with a diffuse interface method for low Mach number two-phase flows, Journal of Computational Physics, 376 (2019), 838–862.
33. Mickens, R.E. Nonstandard finite difference models of differential equations, world scientific, 1994.
34. Oruc, O., Bulut, F. and Esen, A. A Haar wavelet-finite difference hybrid method for the numerical solution of the modified Burgers equation, Journal of Mathematical Chemistry, 53 (2015), 1592–1607.
35. Ropp, D.L. and Shadid, J.N. Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems, Journal of Computational Physics, 228 (2009), 3508–3516.
36. Shu, C.W. and Osher, S. Efficient implementation of essentially nonoscillatory shock-capturing schemes, Journal of Computational Physics, 77 (1988), 439–471.
37. Speth, R.L., Green, W.H., MacNamara, S. and Strang, G. Balanced splitting and rebalanced splitting, SIAM Journal on Numerical Analysis, 51(2013), 3084–3105.
38. Strang, G. On the construction and comparison of difference schemes, SIAM Journal on Numerical Analysis, 5 (1968), 506–517.
39. Suresh, A. and Huynh, H.T. Accurate monotonicity-preserving schemes with Runge-Kutta time stepping, Journal of Computational Physics, 136(1997), 83–99.
40. Ucar, Y., Yagmurlu, N.M. and Celikkaya, I. Operator splitting for numerical solution of the modified Burgers’ equation using finite element method, Numerical Methods for Partial Differential Equations, 35 (2019),
478–492.
41. Yu, Y., Tian, B. and Mo, Z. Hybrid monotonicity-preserving piecewise parabolic method for compressible Euler equations, Computers & Fluids, 159 (2017), 1–8.
Send comment about this article