## Young Researchers Workshop: Ki-Net 2012-2019

### Toward sharp error analysis of extended Langrangian molecular dynamics for polarizable force field simulation

Michael Lindsey

New York University

Abstract:

The inertial extended Langrangian self-consistent field iteration-free (iEL/0-SCF) method was recently introduced as an efficient method for molecular dynamics simulations of polarizable force field models. These simulations involve ODEs where the dynamics depend on the solution of a linear system $Ax=b$, where $A=A(r)$ and $b=b(r)$ themselves depend on the current state $r$. Mathematically, the iEL/0-SCF method introduces a singularly perturbed Hamiltonian system to solve these ODEs approximately without performing explicit matrix inversion at each time step. We prove the convergence of the iEL/0-SCF method provided accurate initial condition and study the asymptotics of the error in the parameter \epsilon which controls the perturbation. Sharp asymptotics depend on the careful analysis of some oscillatory integrals. Interestingly, it is possible to achieve $O(\epsilon)$ error for the state r, even though the error for the linear system solution $x$ is only $O(\sqrt(\epsilon))$ for certain initializations.