AbstractWhen solving the equations of structural dynamics using direct time integration methods, algorithmic damping is useful to control spurious high‐frequency oscillations. Ideally, an algorithm should possess asymptotic annihilation of the high‐frequency response, i.e. spurious oscillations are eliminated after one time step. Numerous one‐step algorithms, spectrally equivalent to linear multistep (LMS) methods, have been developed which include controlled numerical dissipation. This paper proves that the only unconditionally stable, second‐order accurate, 3‐step LMS method possessing asymptotic annihilation is Houbolt's method, which is known to be overly dissipative in the low‐frequency regime. Thus, using LMS methods, obtaining asymptotic annihilation with little low‐frequency dissipation requires at least a
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