Optimal Parameter Choice in the ?-Method for Parabolic Equations

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Abstract
The general first-order method, known as the θ-method, is applied to the semi-discrete form of a parabolic equation. It is shown that to every required local accuracy ɛ there corresponds a value of the parameter θ that is optimal in the sense of allowing the largest step for which the error remains bounded below ɛ. An asymptotic formula for θ in terms of ɛ⅓ is obtained, showing that the maximum step-size for the optimal θ-method is more than twice as large as that for the Crank-Nicolson method. A numerical example is given, showing good agreement between theory and practice.

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