Abstract
In the paper, the space second-order conservative characteristic finite difference method based on essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) interpolation for solving two-dimensional conservative convection–diffusion equations in divergence form is developed. Combining the splitting technique, characteristic difference and mass correction method, a two-dimensional convection–diffusion equations are changed into the two-dimensional parabolic equations, where the convection term and unsteady term are considered as one term. The solutions and fluxes on the staggered meshes are computed by the splitting implicit solution-flux coupled scheme. Numerical experiments are presented to illustrate mass conservation and convergence.
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References
Chaudhuri A, Hadjadj A, Chinnayya A, Palerm S (2011) Numerical study of compressible mixing layers using high-order WENO schemes. J Sci Comput 47:170–197
Chaudhuri A, Hadjadj A, Sadot O, Glazer E (2012) Computational study of shock-wave interaction with solid obstacles using immersed boundary methods. Int J Numer Meth Eng 89:975–990
Cheng J, Shu C (2008) A high order accurate conservative remapping method on staggered meshes. Appl Numer Math 58:1042–1060
Colella P, Woodward P (1984) The piecewise parabolic method (PPM) for gas-dynamical simulations. J Comput Phys 54:174–201
Crnjaric-Zic N, Vukovic S, Sopta L (2004) Extension of ENO and WENO schemes to one-dimensional sediment transport equations. Comput Fluids 33:31–56
Douglas J Jr, Russell T (1982) Numerical solution for convection-dominated diffusion problem based on combining the method of characteristics with finite element or differncen procedures. SIAM J Numer Anal 19:871–885
Douglas J Jr, Huang C, Pereira F (1999) The modified method of characteristics with adjust advection. Numer Math 83:353–369
Du C, Liang D (2010) An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media. J Comput Phys 229:4501–4521
Lei N, Cheng J, Shu C (2021) A high order positivity-preserving conservative WENO remapping method on 2D quadrilateral meshes. Comput Methods Appl Mech Eng 373:113497
Fu K, Liang D (2016) The conservative characteristic FD methods for atmospheric aerosol transport problems. J Comput Phys 305:494–520
Fu K, Liang D (2017) The time second order mass conservative characteristic FDM for advection–diffusion equations in high dimensions. J Sci Comput 73:26–49
Fu K, Liang D (2019) A mass-conservative temporal second order and spatial fourth order characteristic finite volume method for atmosphertic pollution advection diffusion problems. SIAM J Sci Comput 41:1178–1210
Jiang G, Shu C (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228
Harten A, Engquist B, Osher S, Chakravarthy S (1986) Uniformly high order accurate essentially non-oscillatory schemes. J Comput Phys 71:231–303
Liu X, Osher S, Chan T (1994) Weighted essentially non-oscillatory schemes. J Comput Phys 115:200–212
Li R, Zhou Z, Li L, etc (2020) The mass-preserving domain decomposition scheme for solving three-dimensional convection–diffusion equations. Math Comput Simul 177:527–555
Li C, Yuan Y (2009) A modified upwind difference domain decomposition method for convection–diffusion equations. Appl Numer Math 59:1584–1598
Liang D, Du C, Wang H (2007) A fractional step ELLAM approach to high-dimensional convection–diffusion problems with forward particle tracking. J Comput Phys 221:198–225
Liang D, Zhou Z (2020) The conservative splitting domain decomposition method for multicomponent contamination flows in porous media. J Comput Phys 400:108974
Piquet A, Zebiri B, Hadjadj A, Shadloo M (2019) A parallel high-order compressible flows solver with domain decomposition method in the generalized curvilinear coordinates system. Int J Numer Methods Heat Fluid flows. https://doi.org/10.1108/HFF-01-2019-0048
Rui H, Tabata M (2010) A mass-conservative characteristic finite element scheme for convection-diffusion problems. J Sci Comput 43:416–432
You T (2004) The ENO-MMOCAA finite difference method for convection–diffusion equation. Chin J Eng Math 21:377–381
You T (2009) The three-step ENO-MMOCAA difference method for convection diffusion equation. Math Appl 22:137–143
You T (2005) The three-step WENO-MMOCAA difference method for convection diffusion equation. Acta Math Appl Sin 28:713–722
You T (2004) The modifeid method of characteristic with adjusted advection based on WENO interpolation for nonlinear convection diffusion equation. Chin J Eng Math 21:931–935
Zhang J, Yang D, Shen S, Zhu J (2014) A new MMOCAA-MFE method for compressible miscible displacement in porous media. Appl Numer Math 80:65–80
Zhou Z, Liang D (2017) The mass-preserving and modified-upwind splitting DDM scheme for time-dependent convection–diffusion equations. J Comput Appl Math 317:247–273
Zhu J, Shu C (2018) A new type of multi-resolution WENO schemes with increasingly higher order of accuracy. J Comput Phys 375:659–683
Zhu J, Shu C (2019) A new type of multi-resolution WENO schemes with increasingly higher order of accuracy on triangular meshes. J Comput Phys 392:19–33
Zhou Z, Sun X, Pan H, Wang Y (2020) An efficient characteristic finite difference S-DDM scheme for convection–diffusion equations. Comput Math Appl 80:3044–3065
Zhou Z, Hang T, Jiang T (2021) etc, Mass conservative characteristic finite difference method for convection–diffusion equations. Int J Comput Math. https://doi.org/10.1080/00207160.2021.1876229
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Communicated by Abdellah Hadjadj.
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This work was supported partially by the National Natural Science Foundation of China (Grant No. 61703250) and Shandong Agricultural University (Grant No. xxxy201704). Tongtong Hang and Yuxiao Zhai are co-first authors of the article .
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Hang, T., Zhai, Y., Zhou, Z. et al. Conservative characteristic finite difference method based on ENO and WENO interpolation for 2D convection–diffusion equations. Comp. Appl. Math. 40, 202 (2021). https://doi.org/10.1007/s40314-021-01594-4
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DOI: https://doi.org/10.1007/s40314-021-01594-4