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**High order ADI method for solving unsteady convection-diffusion problems.**
*(English)*
Zbl 1053.65067

Summary: We propose a high order alternating direction implicit (ADI) solution method for solving unsteady convection-diffusion problems. The method is fourth order in space and second order in time. It permits multiple use of the one-dimensional tridiagonal algorithm with a considerable saving in computing time, and produces a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable for 2D problems. Numerical experiments are conducted to test its high accuracy and to compare it with the standard second-order Peaceman-Rachford ADI method and the spatial third-order compact scheme of B. J. Noye and H. H. Tan [Int. J. Numer. Methods Eng. 26, No. 7, 1615–1629 (1988; Zbl 0638.76104)].

### MSC:

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K15 | Initial value problems for second-order parabolic equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

### Keywords:

Unsteady convection-diffusion equation; High order compact scheme; ADI method; Stability; comparison of methods; alternating direction implicit method; algorithm; Numerical experiments; Peaceman-Rachford method### Citations:

Zbl 0638.76104
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\textit{S. Karaa} and \textit{J. Zhang}, J. Comput. Phys. 198, No. 1, 1--9 (2004; Zbl 1053.65067)

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### References:

[1] | D’yakonov, E. G., Difference schemes of second-order accuracy with a splitting operator for parabolic equations without mixed derivatives, Zh. Vychisl. Mat. I Mat. Fiz., 4, 935-941 (1964), (in Russian) |

[2] | Gupta, M. M.; Manohar, R. P.; Stephenson, J. W., A single cell high-order scheme for the convection-diffusion equation with variable coefficients, Int. J. Numer. Methods Fluids, 4, 641-651 (1984) · Zbl 0545.76096 |

[3] | Kalita, J. C.; Dalal, D. C.; Dass, A. K., A class of higher order compact schemes for the unsteady two-dimensional convection-diffusion equation with variable convection coefficients, Int. J. Numer. Methods Fluids, 38, 1111-1131 (2002) · Zbl 1094.76546 |

[4] | Karaa, S.; Zhang, J., Analysis of stationary iterative methods for discrete convection-diffusion equation with a nine-point compact scheme, J. Comput. Appl. Math., 154, 447-476 (2003) · Zbl 1029.65119 |

[5] | Noye, B. J.; Tan, H. H., A third-order semi-implicit finite difference method for solving the one-dimensional convection-diffusion equation, Int. J. Numer. Methods Eng., 26, 1615-1629 (1988) · Zbl 0638.76104 |

[6] | Noye, B. J.; Tan, H. H., Finite difference methods for solving the two-dimensional advection-diffusion equation, Int. J. Numer. Methods Fluids, 26, 1615-1629 (1988) · Zbl 0638.76104 |

[7] | Peaceman, D. W.; Rachford, H. H., The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math., 3, 28-41 (1959) · Zbl 0067.35801 |

[8] | Rigal, A., Schémas compacts d’ordre élevé: application aux problémes bidimensionnels de diffusion-convection instationnaire I, C.R. Acad. Sci. Paris Sr. I Math., 328, 535-538 (1999) · Zbl 0936.65099 |

[9] | Rigal, A., High order difference schemes for unsteady one-dimensional diffusion-convection problems, J. Comput. Phys., 114, 59-76 (1994) · Zbl 0807.65096 |

[10] | P.J. Roach, Computational Fluid Dynamics, Hermosa, Albuquerque, NM, 1976; P.J. Roach, Computational Fluid Dynamics, Hermosa, Albuquerque, NM, 1976 |

[11] | Spotz, W. F.; Carey, G. F., Extension of high-order compact schemes to time-dependent problems, Numer. Methods Partial Differential Equations, 17, 657-672 (2001) · Zbl 0998.65101 |

[12] | W.F. Spotz, High-order compact finite difference schemes for computational mechanics, Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1995; W.F. Spotz, High-order compact finite difference schemes for computational mechanics, Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1995 |

[13] | J. Zhang, Multigrid acceleration techniques and applications to the numerical solution of partial differential equations, Ph.D. Thesis, The George Washington University, Washington, DC, 1997; J. Zhang, Multigrid acceleration techniques and applications to the numerical solution of partial differential equations, Ph.D. Thesis, The George Washington University, Washington, DC, 1997 |

[14] | Zhang, J., Multigrid method and fourth order compact difference scheme for 2D Poisson equation with unequal meshsize discretization, J. Comput. Phys., 179, 170-179 (2002) · Zbl 1005.65137 |

[15] | Zhang, J.; Sun, H.; Zhao, J. J., High order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems, Comput. Methods Appl. Mech. Eng., 191, 4661-4674 (2002) · Zbl 1068.76066 |

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