High order ADI method for solving unsteady convection-diffusion problems.

*(English)*Zbl 1053.65067Summary: 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
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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:

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