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 to exact solution

7. 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

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
Bad conditioned
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 A in N-rep. (Taylor)
Predictor matrix A by Taylor expansion

20. Gear algorithms
Predictor step
Corrector

21. Gear algorithms Correction vector a Index 1 2 3 4 1/6 5/6 1/3 5 19/1201 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)

25. 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

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
Not symplectic / explicit

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

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
ATP Verlet-Störmer steps at 1.3 fs without solvence

42. 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)
(CSE455-NumericalAnalysis)