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

• Published in: Journal of computational and applied mathematics 2007, Vol.198 (1), p.268-286
• Main Authors: Tian, Z.F., Ge, Y.B.
• Format: Article
• Language: English
• Subjects: Algebra; Alternating direction implicit method; Convection–diffusion; Exact sciences and technology; Finite differences and functional equations; Fourier analysis; Higher-order compact scheme; Linear and multilinear algebra, matrix theory; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Unsteady
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2005.12.005

In this article, 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 Karaa and Zhang and the spatial third-order compact scheme of Note and Tan.