Time Integration Algorithms with Optimized Controllable Numerical Dissipation for Dynamic Systems

• New single-step and linear multi-step time integration methods are developed. • The methods achieve second-order accuracy with controllable numerical dissipation. • The framework bridges the Park method and the Crank-Nicolson/midpoint rule. • Superior numerical performance is shown by comparisons with widely used methods. This paper presents a new family of implicit time integration algorithms with controllable numerical dissipation for first- and second-order initial value problems. The generalized time-weighted residual methodology provides the fundamental design framework, which is further investigated through cubic approximations to construct a new class of single-step algorithms with optimized numerical properties. Equivalent linear multistep formulations are also derived to enclose the design space. A non-standard stability analysis via the second-order equation of motion and a novel initialization strategy enable the proposed algorithms to achieve second-order accuracy in time, unconditional stability, zero-order overshooting, and controllable numerical dissipation and dispersion through a user-defined parameter, ρ ∞ , with optimal spectral accuracy. For the first time, the proposed computational framework unifies the L-stable Park method ( ρ ∞ = 0 ) and the A-stable midpoint rule ( ρ ∞ = 1 ), while introducing a spectrum of novel schemes for 0 < ρ ∞ < 1, which do not require a distinct starting procedure as with their equivalent linear multistep counterparts. The enhanced dissipation control is validated through detailed comparisons with existing single-step and sub-step/multi-stage algorithms. Numerical examples involving both first- and second-order dynamic problems further demonstrate the improved numerical performance. The MATLAB code for the two-dimensional nonlinear heat transfer and three-dimensional solid mechanics problems is open-source to demonstrate the numerical implementations and compare the performance of various time integration algorithms in practical applications.