The single-step explicit time integration methods have long been valuable for solving large-scale nonlinear dynamic problems, classified into single-solve and multi-sub-step approaches. However, no existing conventional single-solve methods achieve third-order accuracy in any primary variables (displacement, velocity or acceleration). The paper addresses this gap by the framework of conventional self-starting single-solve time integration algorithms, which incorporates eleven algorithmic parameters. The study reveals that self-starting single-solve explicit methods cannot reach third-order accuracy in any primary variables for solving damped problems. Consequently, two novel algorithms are proposed: Algorithm 1 is an explicit scheme that achieves third-order accuracy in displacement and velocity undamped problems; Algorithm 2 is an implicit scheme which degenerates to an explicit scheme when solving the undamped problems, and it achieves third-order accuracy in displacement and velocity for any damped problems. Across a suite of both linear and nonlinear benchmarks, the new algorithms consistently outperform existing conventional explicit methods in accuracy. Their built-in numerical dissipation effectively filters out spurious high-frequency components, as demonstrated by two wave propagation problems. Finally, when applied to the realistic engineering problem, both of them deliver superior numerical precision at a reasonable computational cost.
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