Integrators of higher order Aysam Gürler

Overview Molecular Dynamics Simulations Methods Explicit/Implicit/Symplectic Euler, Midpoint Rule, Strömer/Verlet, Gear 4th, Runge-Kutta 4th Results ATP-Video

Molecular Dynamics Computing the equilibrium Classical physics ignoring quantum effects

Hamiltonian

Molecular Dynamics Computation of force is very time consuming Consequently Methods involving more force calculations per step could be critical Increase the step size Reduce the number of pairs e.g. pharmacologically important atoms only or by a cut off value

Harmonic oscillator Oscillation with period = 2π Symplectic No energy drift / long term stable Accuracy Calculating the distance to exact solution

Test runs 0.001 0.01 0.1 0.3 Euler (explicit) Verlet Gear4th Method / Step size 0.001 0.01 0.1 0.3 Euler (explicit) 10 80 Verlet Gear4th Runge-Kutta4th PERIODS

Euler’s method Simplest approach by a short Taylor series Explicit Euler (error of 2nd order) No use of force derivatives

Euler’s explicit method Not symplectic Not reversible Not area preserving Extreme energy drift Note Method is not recommend

Verlet-Störmer Taylor expansion in both directions Note Reversible, because of symmetry

Verlet-Störmer Summing both equations yields the verlet integrator Local error of 4th order Disadvantages Bad conditioned Velocity for energy calculation through simple approx.

Verlet-Störmer Symplectic Reversible Little long term drift Moderate short term energy conservation Note Accurate for long term runs

Gear algorithms Open or predictor methods Predicting q(n+1) directly. Closed or predictor-corrector methods 1) Predicting a value y(n+1) 2) Use f(y(n+1)) to make a better prediction of q(n+1) Repeatable (more force calculations per step) Only one force per step called Gear algorithms

Gear algorithms N-Representation Nordsieck (4,1)

Gear algorithms Predictor A in N-rep. (Taylor) Predictor matrix A by Taylor expansion

Gear algorithms Predictor step Corrector

Gear algorithms Correction vector a Index 1 2 3 4 1/6 5/6 1/3 5 19/120 1 2 3 4 1/6 5/6 1/3 5 19/120 3/4 1/2 Numerical Initial Value Problems in Ordinary Differential Equations (C.W.Gear)

Gear 4th algorithms Highly accurate Not Symplectic Not Reversible “Does not seem to improve for higher order” Note Very good for short term runs with high precision

Runge-Kutta Solving analytical Approximation implicit trapezoidal rule

Runge-Kutta implicit trapezoidal midpoint rule

Runge-Kutta Main formula

Runge-Kutta Represention by coefficients

Runge-Kutta Implementation of 4th order explicit method Error of 5th order

Runge-Kutta 4th order k2 k4 k3 k1 qi qi + h/2 qi + h

Analyzing Runge-Kutta Rule for symplectic Runge-Kutta Algorithms Result Not symplectic / explicit

Analyzing Runge-Kutta Rule for symmetry Runge-Kutta algorithms Result Not reversible

Runge-Kutta Not Symplectic Not Reversible Extremely good for moderate step size Very stable up to large step sizes But error is either permanently growing Note Very good for short term runs

Notes Symplectic algorithms of higher order are time consuming Non symplectic algorithms of higher order drift Different approach by optimizing the coefficients numerically possible

Video ATP Verlet-Störmer 100.000 steps at 1.3 fs without solvence

References Hairer, Lubich, Wanner Berendsen, Gunsteren Geometric Numerical Integration Berendsen, Gunsteren Practical Algorithms for Dynamic Simulations Dullweber, Leimkuhler, McLachlan Split-Hamiltonian Methods for Rigid Body Molecular Dynamics Schmidt, Schütte Hamilton’sche Systeme und klassische Moleküldynamik Allen, Tildesley Computer simulation of liquids Frenkel, Smit Understanding Molecular Simulation Ratanapisit, Isbister, Ely Symplectic integrators and their usefulness McLachlan On the numerical integration of ordinary dierential equations by symmetric composition methods. SIAM J. Sci. Comput. Ordinary Differential Equations – IVP (Lecture 21) http://www.personal.psu.edu (CSE455-NumericalAnalysis)