Abstract: The two L-stable algorithms are presented for solving structural dynamics response. One is the Rosenbrock method and the other is a new single-step Houbolt direct integral scheme. A finite difference analysis shows that both are capable of asymptotically annihilating the high-frequency modes and are two-order accurate, unconditionally stable and devoid of overshoot phenomenon. The comparison between the new algorithms and the classical ones is made. The Rosenbrock method has a good property of dissipation and dispersion, and is easily implemented when it is used in solving nonlinear structural dynamic problems. The performance of the proposed algorithms are numerically validated by analyzing some examples including a simulated two degree-of-freedom system representing a large structure, elastic bar impact system and nonlinear spring pendulum system.