1. Integrators of higher order Aysam Gürler

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

3. Molecular Dynamics Computing the equilibrium Classical physics ignoring quantum effects

4. Hamiltonian

5. 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

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

7. Test runs PERIODS

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

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

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

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

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

17. 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

18. Gear algorithms N-Representation Nordsieck (4,1)

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

20. Gear algorithms Predictor step Corrector

21. Gear algorithms Correction vector a Numerical Initial Value Problems in Ordinary Differential Equations (C.W.Gear)

25. Gear 4th algorithms • Highly accurate • Not Symplectic • Not Reversible • “Does not seem to improve for higher order” Note Very good for short term runswithhigh precision

26. Runge-Kutta Solving analytical Approximation implicit trapezoidal rule

27. Runge-Kutta implicit trapezoidal midpoint rule

28. Runge-Kutta Main formula

29. Runge-Kutta Represention by coefficients

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

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

32. Analyzing Runge-Kutta Rule for symplectic Runge-Kutta Algorithms Result Notsymplectic / explicit

33. Analyzing Runge-Kutta Rule for symmetry Runge-Kutta algorithms Result Notreversible

36. 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

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

41. Video ATPVerlet-Störmer 100.000 steps at 1.3 fs without solvence

42. References • Hairer, Lubich, Wanner 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)