A unified dispersion error framework for direct time integration methods in isogeometric and finite element analyses

AbstractTo advance research on second-order implicit time integration methods within the framework of isogeometric spatial discretization, the paper proposes a novel second-order mass matrix formulated through a weighted combination of the consistent and lumped mass matrices in the context of quadratic isogeometric analysis. The newly constructed mass matrix enhances compatibility with second-order implicit time integration schemes while preserving the modeling capabilities of quadratic isogeometric analysis. Furthermore, a unified dispersion error framework is proposed, extending the fully discrete error analysis to time integration algorithms based on arbitrary spatial discretization. Applied to second-order implicit algorithms combined with isogeometric discretization, this unified dispersion error framework enables the determination of the leading-term dispersion error. Based on the error analysis, the corresponding optimal Courant–Friedrichs–Lewy (CFL) number is derived analytically. For comparison, the leading-term dispersion error and optimal CFL number are also provided for second-order implicit time integration methods using conventional finite element spatial discretization. The validity of the optimal CFL number obtained via the proposed dispersion error analysis is demonstrated through two scalar wave propagation problems. A performance comparison between second-order implicit time integration algorithms based on isogeometric and finite element discretizations is also presented.