An extension ofA-stability to alternating direction implicit methods

Completely implicit, noniterative, finite-difference schemes have recently been developed by several authors for nonlinear, multidimensional systems of hyperbolic and mixed hyperbolic-parabolic partial differential equations. The method of Douglas and Gunn or the method of approximate factorization can be used to reduce the computational problem to a sequence of one-dimensional or alternating direction implicit (ADI) steps. Since the eigenvalues of partial differential equations (for example, the equations of compressible fluid dynamics) are often widely distributed with large imaginary parts,A-stable integration formulas provide ideal time-differencing approximations. In this paper it is shown that if anA-stable linear multistep method is used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation, then one can always construct an ADI scheme by the method of approximate factorization which is alsoA-stable, i.e., unconditionally stable. A more restrictive result is given for three spatial dimensions. Since necessary and sufficient conditions forA-stability can easily be determined by using the theory of positive real functions, the stability analysis of the factored partial difference equations is reduced to a simple algebraic test.The main results of this paper were presented at the SIAM National Meeting, Madison, Wis., May 24 to 26, 1978, and section 9 was part of a presentation at the 751st Meeting of the American Mathematical Society, San Luis Obispo, California, Nov. 11 to 12, 1977.