S-stability of piecewise-linearized and linearized θ-methods

摘要

The S-stability of piecewise-linearized and linearized θ-methods is analyzed, and it is shown that piecewise-linearized methods are not S-stable, although they are A-stable, while linearized θ-methods are S-stable if and only if 1/2⩽θ⩽1. It is also shown that, for Prothero and Robinson's problems, the piecewise-linearized method yields accurate results even for large values of the stepsize, while, for stiff differential equations, the piecewise-linearized method yields more accurate solutions than S-stable techniques like linearized and standard, implicit θ-methods with θ=1/2 even for large values of the stepsize.