We discuss a novel class of “Hessian-free force-gradient integrators” in the context of gauge field simulations in lattice QCD.

The force-gradient approach offers substantial performance enhancements, particularly for larger lattice volumes where higher-order integrators exhibit greater efficiency. However, conventional force-gradient integrators require the evaluation of the force-gradient term involving the Hessian of the potential, which may not be available in software implementations or could be significantly more expensive to evaluate than a standard force evaluation. In contrast, Hessian-free variants approximate the force-gradient updates, effectively replacing the force-gradient term with another force evaluation. This modification makes it straightforward to apply the integrators in existing software packages.

We will discuss a refined error analysis and the geometric properties of the new class of integrators. Additionally, a stability analysis will be conducted to assess the robustness of these integrators. By investigating the accuracy and stability of the integrators, promising variants among the family of self-adjoint Hessian-free force-gradient integrators with up to eleven stages will be identified. Numerical results for four-dimensional gauge field simulations in lattice QCD with two heavy Wilson fermions demonstrate the superior efficiency of the proposed integrators compared to commonly employed non-gradient schemes.