On the structure and use of linearized block implicit schemes

The recent use of methods which may be termed “linearized block ADI methods” or more generally “consistently split linearized block implicit” methods has been a significant development in the efficient and noniterative solution of certain systems of coupled nonlinear multidimensional partial differential equations. Some observations on their structure, derivation, and use are given. Consistently split linearized block implicit (LBI) methods are unified here and are related to the earlier scalar ADI schemes, as well as to existing iterative and noniterative methods for solving both systems of nonlinear algebraic equations, and systems of nonlinear ordinary differential equations (including those having multipoint boundary conditions). It is shown that the method used by Lindemuth and Killeen and that of Briley and McDonald (utilizing a two-dimensional Crank-Nicolson formulation) are both consistently split block implicit schemes which differ in principle only with regard to implementation of the linearization technique. It is also observed that the first approximate factorization scheme of Beam and Warming utilizes a splitting due to D'Yakanov whose intermediate steps are inconsistent in the sense that they do not approximate the governing equations to within a truncation error which vanishes to some order for small Δt. Methods based on splittings which have inconsistent intermediate steps are placed in a separate category and are shown to present serious difficulties, which apparently have escaped notice, in treating derivative boundary conditions accurately. Although similar difficulties can arise in the transient with consistently split schemes, the consistent splitting normally provides one order of accuracy improvement. It is further demonstrated that the two-level version of the second and more recent “delta” form approximate factorization scheme of Warming and Beam and the earlier method of Briley and McDonald have identical linearized block implicit structures. Finally, further substantial gains in efficiency resulting from reducible block submatrices and the use of multiple time steps are described.

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