Superstable two-step methods for the numerical integration of general second order initial value problems

Abstract For the numerical integration of general second order initial value problems y ″ = f ( t , y , y ′), y ( t 0 ) = y 0 , y ′( t 0 ) = y ′ 0 , we report a new family of two-step fourth order methods which are superstable for the test equation: y ″ + 2 αy ′ + β 2 y = 0, α , β ⩾ 0, α + β > 0. We also note a modification of the trapezoidal method which results in a superstable method.