A fourth-order compact ADI method for solving two-dimensional unsteady convection–diffusion problems.

*(English)*Zbl 1104.65086Summary: An exponential high-order compact (EHOC) alternating direction implicit (ADI) method, in which the Crank-Nicolson scheme is used for the time discretization and an exponential fourth-order compact difference formula for the steady-state 1D convection-diffusion problem is used for the spatial discretization, is presented for the solution of the unsteady 2D convection-diffusion problems. The method is temporally second-order accurate and spatially fourth order accurate, which requires only a regular five-point 2D stencil similar to that in the standard second-order methods.

The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of S. Karaa and J. Zhang [J. Comput. Phys. 198, No. 1, 1–9 (2004; Zbl 1053.65067)] and the spatial third-order compact scheme of B. J. Note and H. H. Tan [Int. J. Numer. Methods Eng. 26, No. 7, 1615–1629 (1988; Zbl 0638.76104)].

The resulting EHOC ADI scheme in each ADI solution step corresponds to a strictly diagonally dominant tridiagonal matrix equation which can be inverted by simple tridiagonal Gaussian decomposition and may also be solved by application of the one-dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. The unconditionally stable character of the method was verified by means of the discrete Fourier (or von Neumann) analysis. Numerical examples are given to demonstrate the performance of the method proposed and to compare mostly it with the high order ADI method of S. Karaa and J. Zhang [J. Comput. Phys. 198, No. 1, 1–9 (2004; Zbl 1053.65067)] and the spatial third-order compact scheme of B. J. Note 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 |

##### Keywords:

higher-order compact scheme; alternating direction implicit method; finite difference method; comparison of methods; stability; error estimates; Crank-Nicolson scheme; unsteady 2D convection-diffusion problems; tridiagonal Gaussian decomposition; tridiagonal Thomas algorithm; numerical examples
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\textit{Z. F. Tian} and \textit{Y. B. Ge}, J. Comput. Appl. Math. 198, No. 1, 268--286 (2007; Zbl 1104.65086)

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