Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

Time-dependent parabolic and hyperbolic equations are widely encountered in practical engineering. Accurate and fast solution methods for such equations have always attracted attention. To reach this aim, this article proposes a strategy to construct highly precise and efficient time integration methods (TIMs) for linear heat conduction and structural dynamic systems. In the proposed strategy, an integrated amplification matrix of a TIM is created to precisely transfer the free responses of the previous step, and the Gauss–Legendre quadrature is employed to precisely compute the forced responses of the current step. To reduce computational cost and rounding error, a 2^m algorithm and a method of storing incremental matrix are applied in the construction of the integrated matrix. Moreover, the L-stable backward difference formula for heat conduction and the generalized trapezoidal rule for structural dynamics, which is A-stable and has controllable dissipation, are utilized to conduct this strategy. Numerical experiments validate that compared with some existing TIMs, the TIMs generated by the proposed strategy enjoy advantages both in precision and efficiency.