My research develops higher-order and structure-preserving numerical methods for nonlinear continuum mechanics, with particular emphasis on long-time stability and the faithful reproduction of fundamental physical laws at the discrete level. Rather than designing numerical schemes solely from the viewpoint of local approximation accuracy, we construct fully discrete formulations that inherit the energy balance and conservation properties of the underlying continuum models. For finite elastodynamics and viscoelastodynamics, this leads to energy-momentum consistent schemes that conserve linear and angular momenta while either preserving the total energy in conservative systems or reproducing physically meaningful dissipation in inelastic materials. A key aspect of this work is the consistent design of algorithmic stresses and internal-variable updates so that constitutive integration and global balance laws remain compatible after discretization.

Another direction of my research focuses on higher-order numerical methods for incompressible flows and turbulence modeling. We develop half-explicit Runge–Kutta (HERK) schemes within the residual-based variational multiscale (VMS) framework, with the goal of achieving higher-order accuracy in both space and time while maintaining a mathematically consistent treatment of the incompressibility constraint. Guided by the Rothe method, the temporal discretization is designed directly for the differential-algebraic structure of the incompressible Navier–Stokes equations, avoiding the order-reduction issues that can arise in conventional multi-stage schemes. The explicit treatment of nonlinear convection further enables a systematic VMS analysis with fewer modeling assumptions. Through stability and Fourier analyses, together with turbulent-flow and hydrodynamic-instability benchmarks, these methods demonstrate improved dissipation and dispersion characteristics and enhanced ability to resolve sensitive multiscale flow dynamics.

Related Publications

Y. Sun, C. Ding, and J. Liu*, "Half-explicit Runge-Kutta integrators for variational multiscale turbulence modeling: Toward higher-order accuracy in space and time", Computer Methods in Applied Mechanics and Engineering, 456:118930, 2026. [link]

J. Liu* and J. Guan, "A continuum and computational framework for viscoelastodynamics: II. Strain-driven and energy-momentum consistent schemes", Computer Methods in Applied Mechanics and Engineering, 417:116308, 2023. [link]

J. Liu*, "On the design of non-singular, energy-momentum consistent integrators for nonlinear dynamics using energy splitting and perturbation techniques", Journal of Computational Physics, 487:112177, 2023. [link]

J. Guan, H. Yuan, and J. Liu*, "A structure-preserving integrator for incompressible finite elastodynamics based on a grad-div stabilized mixed formulation with particular emphasis on stretch-based material models", Computer Methods in Applied Mechanics and Engineering, 414:116145, 2023. [link]