Algebraic approximants and the numerical solution of parabolic equations

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The heat equation is but one example of problems which involve multiple scales. There is a lot of transient behavior which for many problems is of no particular interest. What is of concern is the long time-scale behavior. However the presence of the short time-scale behavior would seem to require numerical integration methods to take very short time steps to follow the behavior accurately. For these problems, what is desired is a numerical method which is accurate for the long time-scale behavior, and causes the transients to die out quickly. That their rate of decay is not quite right is not important for this class of problems. A formalism is developed which allows the straightforward derivation of finite-difference schemes which involve several prior times from algebraic approximants. The algebraic approximants turn out to be, in a quite natural way, approximations to the function exp{−4μ[ arcsin( w/4 )] 2} where μ= κΔt/( Δx) 2 is the Courant number. Several of the simpler cases are investigated, and, of the implicit schemes, a couple are found to be, not only of higher order accuracy than most currently popular schemes, but also unconditionally stable and in fact unconditionally stiff stable! The higher order accuracy and stiff stability properties are just what is required for this sort of problem.