1. Topics in Molecular Modeling: II
Topics in Molecular Modeling: II. Numerical Methods for the Time-integration
Xiantao Li
Department of Mathematics, Pennsylvania State University

2. Outline Hamiltonian system (NVE ensemble) Operator-splitting methods
Energy conservation properties
Stochastic models
Strong and weak convergence

3. Hamiltonian systems Hamilton’s principle 𝑞 = 𝜕𝐻 𝜕𝑝 𝑝 =− 𝜕𝐻 𝜕𝑞
Examples: mass spring model, many-body problems, pendulum model, Lotka-Volterra model etc.
The mapping 𝑞 0 , 𝑝 0 →(𝑞 𝑡 ,𝑝 𝑡 ) is symplectic
A numerical method is symplectic if it defines a symplectic mapping

4. Many-Particle Systems
Coordinate and Momentum 𝑞 𝑖 𝑡 , 𝑝 𝑖 𝑡 , 𝑖=1,2,⋯,𝑁.
The equations: 𝑞 𝑖 = 𝑝 𝑖 𝑚 𝑖
𝑝 𝑖 =− 𝑗≠𝑖 𝛻𝑉( 𝑞 𝑖 − 𝑞 𝑗 )
Linear momentum:𝑃= 𝑖 𝑝 𝑖
Angular momentum: 𝐿= 𝑖 𝑞 𝑖 × 𝑝 𝑖
Total energy: 𝐻 𝑞,𝑝 = 𝑖 𝑝 𝑖 2 2 𝑚 𝑖 𝑗≠𝑖 𝑉( 𝑞 𝑖 − 𝑞 𝑗 )
Conservation: 𝑑 𝑑𝑡 𝑃=0, 𝑑 𝑑𝑡 𝐿=0, 𝑑 𝑑𝑡 𝐻=0.
(q_i, p_i), i=1, 2, \cdots, N.
\dot{q}_i=\frac{p_i}{m_i}
\dot{p}_i= -\sum_{j\ne i}V'(|q_i-q_j|)\frac{q_i-q_j}{|q_i-q_j|}
L=\sum_i q_i \times p_i
H= \sum_i \frac{p_i^2}{2m_i} + \frac12\sum_{i\ne j}V(|q_i - q_j|)

5. First Integrals (Invariants)
In general, for an ODE system, 𝑥 =𝑓 𝑥 , 𝜙 𝑥 is a first integral (or invariants) if 𝑑 𝑑𝑡 𝜙 𝑥 𝑡 =0.
Most numerical methods preserve linear invariant
Only some methods preserve quadratic invariants

6. Symmetry If 𝜌∘𝑓=−𝑓∘𝜌, the function is symmetric wrt 𝜌.
The solutions of the corresponding ODEs will be ρ- reversible 𝜌∘Φ= − Φ −1 ∘𝜌.
For example, 𝜌 𝑞,𝑝 = 𝑞,−𝑝 .
A numerical method is ρ- reversible if the solution satisfies the same properties.
Explicit RK methods are not symmetric.

7. Symplectic Structure
Let 𝐽= 0 𝐼 −𝐼 0 . If y=(𝑞,𝑝), then the Hamiltonian ODEs can be written as
𝑦 =𝐽∇𝐻(𝑦).
A linear mapping A is symplectic if 𝐴 𝑇 𝐽𝐴=𝐽.
A nonlinear mapping g(x) is symplectic if ∇𝑔= 𝜕𝑔 𝜕𝑥 is symplectic matrix.
For Hamiltonian system, the Jacobian matrix 𝜕(𝑞 𝑡 ,𝑝 𝑡 ) 𝜕( 𝑞 0 , 𝑝 0 ) is symplectic. Therefore, the mapping 𝑞 0 , 𝑝 0 →(𝑞 𝑡 ,𝑝 𝑡 ) is symplectic

8. Symmetric and symplectic methods
Symmetric methods
Order of accuracy is always even
Very convenient for an extrapolation procedure
Symplectic methods
Very good energy conservation properties
Promising accuracy over long time integration
Provide good statistics.
Explicit RK methods are not symplectic.

9. A special class of symplectic methods
Nonlinear dynamical system: 𝑥 =𝑓 𝑥 ,𝑥 0 = 𝑥 0 . dim 𝑥 =𝑁.
Wronskian: 𝑊 𝑡 = 𝜕𝑥(𝑡) 𝜕 𝑥 0 .
Variational equation:
𝑊 𝑡 =𝐴 𝑡 𝑊 𝑡 , 𝐴 𝑡 :=∇𝑓 𝑥 𝑡 , 𝑊 0 =𝐼.
Fundamental solution: 𝑦 =𝐴 𝑡 𝑦, 𝑦 0 = 𝑦 0 𝑦 𝑡 =𝑊 𝑡 𝑦 0 .
In particular, If 𝑦 𝑡 =𝑓 𝑥 𝑡 , then 𝑦 =𝐴 𝑡 𝑦, 𝑦 0 =𝑓 𝑥 0 .
Therefore, 𝑓 𝑥 𝑡 =𝑊 𝑡 𝑓 𝑥 0 = 𝜕𝑥 𝑡 𝜕 𝑥 0 𝑓( 𝑥 0 ).

10. Semi-group notation
For any quantity of interest: 𝐴 𝑡, 𝑥 0 ≔𝜙 𝑥 𝑡; 𝑥 0 .
Time derivative:
𝐴 =𝑓 𝑥 𝑡 𝑇 ∇𝜙 𝑥 𝑡 =𝑓 𝑥 0 ⋅ ∇ 𝑥 0 𝐴=:𝐿𝐴.
L is a differential operator w.r.t. 𝑥 0
Time evolution: 𝐴 =𝐿𝐴 𝐴 𝑡 = 𝑒 𝑡𝐿 𝐴 0 .
This is the foundation for operator-splitting methods.

11. Newtonian dynamics Equations of motion: 𝑥 =𝑓(𝑥)
Operator:𝐿= 𝑣 0 ⋅ 𝜕 𝑥 0 +𝑓 𝑥 0 ⋅ 𝜕 𝑣 0
Operator splitting: 𝐿= 𝐿 1 + 𝐿 2
corresponds to
x'=v, \quad v'=f(x)

12. Operator splitting
Both equations are solvable
Do we have
No, unless

13. BCH (Baker-Campbell-Hausdorff) formulas
Commutator:
BCH formula I:
First-order splitting:
Implementation:
Symplectic Euler:

14. Symmetric Splitting BCH formula II: Symmetric splitting:
Implementation:

15. Properties of the second splitting method
Second order accuracy in time
Symplectic
Symmetric
Energy conservation:
𝐻 𝑞 Δ𝑡 𝑡 , 𝑝 Δ𝑡 𝑡 −𝐻 𝑞 𝑡 ,𝑝 𝑡 =𝑂(Δ 𝑡 2 )
Conserve the linear and angular momentum

16. General splitting methods
Introduce multiple fractional steps
The coefficients are determined by maximizing the order of accuracy.
Formulas are available up to eighth order.

17. Extended Hamiltonian systems
In practice, the Hamiltonian system is often connected a heat bath or pressure bath, modeled by additional equations.
Nose-Hoover thermostat
𝑞 𝑖 = 𝑝 𝑖 𝑚 𝑖
𝑝 𝑖 =− ∇ 𝑞 𝑖 𝑉−𝜉 𝑝 𝑖
𝜉 = 𝑔 𝑄 [ 1 dN 𝑖 𝑝 𝑖 2 𝑚 𝑖 −𝑇]
Symmetric splitting: 𝐿= 𝐿 𝑥 + 𝐿 𝑝 + 𝐿 𝑁𝐻
𝑒 Δ𝑡𝐿 ≈ 𝑒 𝐿 𝑁𝐻 Δ𝑡 𝑒 𝐿 𝑣 Δ𝑡 2 𝑒 𝐿 𝑥 Δ𝑡 𝑒 𝐿 𝑣 Δ𝑡 2 𝑒 𝐿 𝑁𝐻 Δ𝑡 2

18. Problem with multiple scales
Hamiltonian system driven by two forces
𝑥 = 1 𝜖 2 𝑓 𝑥 +𝑔(𝑥)
𝑓 𝑥 : short-range fast force
𝑔 𝑥 : long-range slow force; More difficult to compute.
Multiple time-stepping: evaluate 𝑓 𝑥 on the fast time scale with step size 𝛿𝑡, and evaluate g 𝑥 on the slow time scale with step size Δ𝑡: Δ𝑡≫𝛿𝑡.

19. Multiple time-stepping
Let 𝑒 𝑡 𝐿 1 be the evolution operator for the fast dynamics:
𝑥 =𝑣, 𝑣 = 1 𝜖 2 𝑓 𝑥
Let 𝑒 𝑡 𝐿 2 be the evolution operator for the slow dynamics
𝑥 =0 𝑣 =𝑔 𝑥
Overall algorithm: 𝑒 Δ𝑡𝐿 ≈ 𝑒 Δ𝑡 2 𝐿 2 𝑒 Δ𝑡 𝐿 1 𝑒 Δ𝑡 2 𝐿 It is symplectic.
Operations: Kick + Evolve + Kick.
Local accuracy: 𝑂 Δ 𝑡 3 𝜖 2 .

20. More accurate splitting
To eliminate the leading term in the error, we constructed the following splitting method
𝑒 Δ𝑡𝐿 ≈ 𝑒 Δ𝑡 4 𝐿 2 𝑒 2Δ𝑡 3 𝐿 1 𝑒 3Δ𝑡 4 𝐿 2 𝑒 Δ𝑡 3 𝐿 1
This method also have better energy conservation property.

21. Example: dynamics of butane molecule

22. 𝑑𝑥=𝑣𝑑𝑡 𝑑𝑣= 𝑓 𝑥 −𝛾𝑣 𝑑𝑡+𝜎𝑑𝑊(𝑡)
Langevin dynamics
Langevin stochastic differential equation:
𝑑𝑥=𝑣𝑑𝑡 𝑑𝑣= 𝑓 𝑥 −𝛾𝑣 𝑑𝑡+𝜎𝑑𝑊(𝑡)
𝑥,𝑣∈ 𝑅 𝑑 : position, velocity
𝑓=𝑓 𝑥 : 𝑅 𝑑 → 𝑅 𝑑 : non-linear conservative potential, 𝑓 𝑥 = − ∇ 𝑥 𝑉(𝑥)
𝛾∈ 𝑅 𝑑×𝑑 ,𝜎∈ 𝑅 𝑑×𝑚 : friction matrix constant and diffusion matrix constant, related by the fluctuation dissipation formula
𝜎 𝜎 𝑇 =2 𝑘 𝐵 𝑇𝛾
𝑊=( 𝑊 1 , 𝑊 2 ,…, 𝑊 𝑚 )={𝑊 𝑡 :𝑡≥0}: standard 𝑚-dimensional Wiener process / Brownian motion
Assume additive noise: 𝜎 is constant

23. Strong convergence
Definition: A one-step method 𝑌= 𝑌 Δ𝑡 with step size Δ𝑡>0 is said to converge in the strong sense to a stochastic process 𝑋= 𝑋 𝑡 at time 𝑇=𝑁Δ𝑡, 𝑁=𝑁 Δ𝑡 ∈ {1,2,…} provided
𝐸 𝑋 𝑇 − 𝑌 Δ𝑡 𝑇 →0 as Δ𝑡↓0.
Further, we say the convergence is of order 𝛾∈{0.5,1,1.5,2,…} provided there exists constants 𝐶=𝐶 𝑇 ,Δ>0 such that
𝐸 𝑋 𝑇 − 𝑌 Δ𝑡 𝑇 <𝐶Δ 𝑡 𝛾 for all 0<Δ𝑡<Δ.
Reduces to usual def. of convergence when 𝜎=0
To test convergence rates numerically, we take M many samples (M>>1) of X and Y and approximate the error:
𝐸 𝑋 𝑇 − 𝑌 Δ𝑡 𝑇 ≈ 1 𝑀 𝑘=1 𝑀 𝑋 𝑇 𝜔 𝑘 − 𝑌 Δ𝑡 𝑇, 𝜔 𝑘 =:𝐸 Δ𝑡 .

24. Strong Itô-Taylor expansion
Analysis will involve comparison with the Itô-Taylor formula.
Notation: step size Δ𝑡>0, independent Brownian increments Δ 𝑊 𝑗 ≔ 𝑊 𝑗 Δ𝑡 , 𝐸 Δ 𝑊 𝑖 Δ 𝑊 𝑗 =Δ𝑡 𝛿 𝑖𝑗 , and independent stochastic integrals
𝐼 𝑗,0 ≔ 0 Δ𝑡 𝑊 𝑗 𝑡 𝑑𝑡 , 𝐸 𝐼 𝑖,0 𝐼 𝑗,0 = Δ 𝑡 𝛿 𝑖𝑗
Order 1 and :
𝑥 Δ𝑡 𝑣 Δ𝑡 = 𝑥 𝑣 + 𝑣 𝑓 𝑥 −𝛾𝑣 Δt+ 𝑗=1 𝑚 0 𝜎 𝑗 Δ 𝑊 𝑗 +𝐻.𝑂.𝑇 𝑥 Δ𝑡 𝑣 Δ𝑡 = 𝑥 𝑣 + 𝑣 𝑓 𝑥 −𝛾𝑣 Δ𝑡+ 𝑗=1 𝑚 0 𝜎 𝑗 Δ 𝑊 𝑗 + 𝑓 𝑥 −𝛾𝑣 ∇ 𝑥 𝑓 𝑥 𝑣−𝛾𝑓 𝑥 + 𝛾 2 𝑣 Δ 𝑡 𝑗=1 𝑚 𝜎 𝑗 −𝛾 𝜎 𝑗 𝐼 𝑗,0 +𝐻.𝑂.𝑇.
We derived an order 3. Higher order stochastic integrals are involved:
0 Δ𝑡 0 𝑡 𝑑𝑠𝑑 𝑊 𝑗 (𝑡) , 0 Δ𝑡 0 𝑡 0 𝑠 𝑑 𝑊 𝑗 𝑢 𝑑𝑠𝑑𝑡

25. 𝑥 𝑛+1 𝑣 𝑛+1 = 𝑥 𝑛 𝑣 𝑛 + 𝑣 𝑛 𝑓 𝑥 𝑛 −𝛾 𝑣 𝑛 Δt+ 𝑗=1 𝑚 0 𝜎 𝑗 Δ 𝑊 𝑛 𝑗
Existing methods
References: for example, (1) Leimkuhler & Matthews, Molecular Dynamics (2014) and (2) Kloeden & Platen, Num. Sol. Of SDE’s (1991)
Euler Maruyama (order 1):
𝑥 𝑛+1 𝑣 𝑛+1 = 𝑥 𝑛 𝑣 𝑛 𝑣 𝑛 𝑓 𝑥 𝑛 −𝛾 𝑣 𝑛 Δt+ 𝑗=1 𝑚 0 𝜎 𝑗 Δ 𝑊 𝑛 𝑗
Milstein (order 1) and Itô-Taylor method (order 1): reduce to the same method as Euler Maruyama for Langevin dynamics with additive noise (i.e. constant diffusion 𝜎)
Itô-Taylor method (order 2): requires computing ∇ 𝑥 𝑓(𝑥)
Stochastic velocity Verlet (SVV): no ∇ 𝑥 𝑓(𝑥) and only one 𝑓 evaluation. No clear generalization to a third order method. Refer to Swope, Anderson, Berens & Wilson (1982).
Theorem: (Telatovich and Li) SVV has strong order 2.
Itô-Taylor method (order 3): requires computing ∇ 𝑥 𝑓(𝑥) and ∇ 𝑥 2 𝑓(𝑥)

26. Existing methods (continued)
Direct splitting methods: split 𝑑𝑥=𝑣𝑑𝑡 𝑑𝑣= 𝑓 𝑥 −𝛾𝑣 𝑑𝑡+𝜎𝑑𝑊(𝑡) into
𝑑𝑥=𝑣𝑑𝑡 𝑑𝑣= and 𝑑𝑥= 𝑑𝑣= 𝑓 𝑥 −𝛾𝑣 𝑑𝑡+𝜎𝑑𝑊(𝑡)
Solve these simpler linear systems exactly, back-to-back
Solution expansion involves Lie brackets
High order convergence only in the weak sense
Can do more sophisticated direct splittings
Direct splitting methods lack stochastic integrals necessary for higher order convergence in strong sense
Theorem: (Telatovich & Li) The splitting methods have strong order at most 1.
Challenge: Higher order schemes require not only more smoothness but also more information about W!

27. Higher order operator-splitting methods
Starting point: Kunita’s solution operator 𝑌 𝑡 (Kunita, 1980).
𝑆 𝑡 = exp 𝑌 𝑡 𝑆 0 , 𝑆 0 = 𝑥,𝑣 𝑇
𝑌 𝑡 is given by:
𝑌 𝑡 =𝑡 𝑋 0 + 𝑗=1 𝑚 𝑊 𝑗 𝑡 𝑋 𝑗 𝑗=1 𝑚 𝑡, 𝑊 𝑗 𝑡 [ 𝑋 0 , 𝑋 𝑗 ] 𝑗=1 𝑚 𝑡, 𝑊 𝑗 𝑡 ,𝑡 𝑋 0 , 𝑋 𝑗 , 𝑋 0 +𝐻.𝑂.𝑇.
Vector fields 𝑋 0 ≔𝑣 𝜕 𝑥 + 𝑓 𝑥 −𝛾𝑣 𝜕 𝑣 , 𝑋 𝑗 ≔ 𝜎 𝑗 𝜕 𝑣 ,
Lie bracket 𝑋 0 , 𝑋 𝑗 ≔ 𝑋 0 𝑋 𝑗 − 𝑋 𝑗 𝑋 0 , stochastic integrals 𝑡, 𝑊 𝑗 𝑡 ≔ 0 𝑡 𝑠𝑑 𝑊 𝑗 𝑠 − 0 𝑡 𝑊 𝑗 𝑠 𝑑𝑠
Third order brackets and integrals defined similarly
ODE special case (no noise 𝜎=0): 𝑌 𝑡 =𝑡 𝑋 0 is a simple finite sum (comprehensive exam)
We can make truncations of 𝑌 𝑡

28. Truncations of the operator 𝑌 𝑡
Truncation I:
𝑌 𝑡 ≈ 𝑌 𝑡 𝐼 =𝑡 𝑋 0 + 𝑗=1 𝑚 𝑊 𝑗 𝑡 𝑋 𝑗
Truncation II:
𝑌 𝑡 ≈ 𝑌 𝑡 𝐼𝐼 =𝑡 𝑋 0 + 𝑗=1 𝑚 𝑊 𝑗 𝑡 𝑋 𝑗 𝑗=1 𝑚 𝑡, 𝑊 𝑗 𝑡 [ 𝑋 0 , 𝑋 𝑗 ]
Truncation III:
𝑌 𝑡 ≈ 𝑌 𝑡 𝐼𝐼𝐼 =𝑡 𝑋 0 + 𝑗=1 𝑚 𝑊 𝑗 𝑡 𝑋 𝑗 𝑗=1 𝑚 𝑡, 𝑊 𝑗 𝑡 [ 𝑋 0 , 𝑋 𝑗 ]
𝑗=1 𝑚 𝑡, 𝑊 𝑗 𝑡 ,𝑡 𝑋 0 , 𝑋 𝑗 , 𝑋 0

29. Multiple stochastic integrals
For truncations I, II, and III (Misawa, 2000):
Δ 𝑊 𝑗 = 𝑊 𝑗 Δ𝑡 =∫1𝑑 𝑊 𝑗 (𝑡), Δ 𝑈 𝑗 ≔ 1 2 Δ𝑡,Δ 𝑊 𝑗 = 𝑑𝑠𝑑 𝑊 𝑗 𝑡 − 1𝑑 𝑊 𝑗 𝑠 𝑑𝑡 , Δ 𝑉 𝑗 ≔ Δ𝑡,Δ 𝑊 𝑗 ,Δ𝑡 = 𝑑𝑢𝑑 𝑊 𝑗 𝑠 𝑑𝑡 −2 1𝑑 𝑊 𝑗 𝑢 𝑑𝑠𝑑𝑡 +Δ𝑡Δ 𝑈 𝑗
Joint symmetric covariances (T.): 𝐸 Δ 𝑊 𝑖 Δ 𝑊 𝑗 𝐸 Δ 𝑊 𝑖 Δ 𝑈 𝑗 𝐸 Δ 𝑊 𝑖 Δ 𝑉 𝑗 𝐸 Δ 𝑈 𝑖 Δ 𝑈 𝑗 𝐸 Δ 𝑈 𝑖 Δ 𝑉 𝑗 𝐸 Δ 𝑉 𝑖 Δ 𝑉 𝑗 3𝑚×3𝑚 = Δ𝑡𝐼 Δ 𝑡 𝐼 − Δ 𝑡 𝐼 Δ 𝑡 𝐼
Computed using Itô’s isometry
𝐸 0 Δ𝑡 𝑓 𝑡 𝑑 𝑊 𝑗 𝑡 =𝐸 0 Δ𝑡 𝑓 2 𝑡 𝑑𝑡 and 𝐸 𝑊 𝑡 𝑊 𝑠 = min (𝑡,𝑠)

30. Reduced Differential Equations
𝑌 Δ𝑡 ≈ 𝑌 Δ𝑡 𝐼 ↔ 𝑥 =𝑣Δ𝑡 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊
𝑌 Δ𝑡 ≈ 𝑌 Δ𝑡 𝐼𝐼 ↔ 𝑥 =𝑣𝛥𝑡−𝜎𝛥𝑈 𝑣 = 𝑓 𝑥 −𝛾𝑣 𝛥𝑡+𝜎𝛥𝑊+𝛾𝜎𝛥𝑈
𝑌 Δ𝑡 ≈ 𝑌 Δ𝑡 𝐼𝐼𝐼 ↔ 𝑥 =𝑣Δ𝑡−𝜎Δ𝑈+𝛾𝜎Δ𝑉 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈− 𝛻 𝑥 𝑓 𝑥 𝜎+ 𝛾 2 𝜎 𝛥𝑉
Once sampled, we can treat Δ𝑊,Δ𝑈,Δ𝑉 as constants
Rescale 𝑡→𝑡Δ𝑡 and solve from 0 to 𝑇Δ𝑡 instead of 0 to 𝑇
Approximate 𝑆 Δ𝑡 = exp 𝑌 Δ𝑡 𝑆(0)≈ 𝑆 𝐼,𝐼𝐼,𝐼𝐼𝐼 Δ𝑡 ≔ exp 𝑌 Δ𝑡 I,II,III 𝑆 0 , still non-linear due to f
Approximate 𝑆 𝐼,𝐼𝐼,𝐼𝐼𝐼 𝛥𝑡 numerically by standard splitting techniques

31. Operator-splitting methods
Useful integration method for ODEs.
Order 1 naïve splitting:
exp 𝐴+𝐵 ≈ exp 𝐴 exp 𝐵
Order 2 symmetric/Strang splitting:
exp 𝐴+𝐵 ≈ exp 𝐴 2 exp 𝐵 exp 𝐴 2
Order 4 Neri splitting:
exp 𝐴+𝐵 ≈ exp 𝑐 1 𝐴 exp 𝑑 1 𝐵 exp 𝑐 2 𝐴 exp 𝑑 2 𝐵 exp 𝑐 3 𝐴 exp 𝑑 3 𝐵 exp 𝑐 4 𝐴
where 𝑐 1 = 1 2 2− , 𝑐 2 = 1 2 − 𝑐 1 , 𝑐 3 = 𝑐 2 , 𝑐 4 = 𝑐 1 and
𝑑 1 = 1 2− 3 2 , 𝑑 2 =1−2 𝑑 1 , 𝑑 3 = 𝑑 1

32. Baker-Campbell-Hausdorff (BCH) formula
Useful for obtaining a “local” time estimate for the strong error.
Theorem: If A and B are (non-commuting) matrices and 𝐴,𝐵 ≔𝐴𝐵− 𝐵𝐴≠0, then
exp 𝑡𝐴 exp 𝑡𝐵 = exp (𝑡 𝐶 1 + 𝑡 2 𝐶 2 + 𝑡 3 𝐶 3 +𝑂( 𝑡 4 )) ,
where exp 𝐴 ≔ 𝑛=0 ∞ 𝐴 𝑛 𝑛! is the matrix exponential, and the coefficients are given by,
𝐶 1 =𝐴+𝐵, 𝐶 2 = 1 2 𝐴,𝐵 , and 𝐶 3 = 𝐴, 𝐴,𝐵 𝐵, 𝐵,𝐴 .
Proof: Follow the definitions above. For the formula in full generality, with a description of every coefficient, see Hairer, Lubich & Wanner (2005).

33. Doob’s inequality (special case)
Useful for obtaining a uniform “global” time estimate on the strong error.
Theorem (Doob): If 𝑋 𝑡 is a martingale then
𝐸 sup 𝑋 𝑡 2 :0≤𝑡≤𝑇 ≤4𝐸 𝑋 𝑇

34. Truncation I Recall: 𝑌 Δ𝑡 𝐼 ↔ 𝑥 =𝑣Δ𝑡 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊
Split 𝑌 Δ𝑡 𝐼 =𝐴+𝐵 where 𝐴={𝑣Δ𝑡} 𝜕 𝑥 𝐵={ 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊} 𝜕 𝑣
Notice: 𝐴↔ 𝑥 =𝑣Δ𝑡 𝑣 = and 𝐵↔ 𝑥 = 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊
These equations can be solved exactly.
Theorem: The naïve, symmetric, and Neri splittings give consistent methods with order 1 convergence.
Proof (sketch): Use BCH formula, compare with Itô-Taylor expansions, and use Doob.

35. Truncation II Recall: 𝑌 Δ𝑡 𝐼𝐼 ↔ 𝑥 =𝑣Δ𝑡−𝜎Δ𝑈 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈
Split 𝑌 Δ𝑡 𝐼𝐼 =𝐴+𝐵 where 𝐴={𝑣Δ𝑡−𝜎Δ𝑈} 𝜕 𝑥 𝐵={ 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈} 𝜕 𝑣
Notice: 𝐴↔ 𝑥 =𝑣Δ𝑡−𝜎Δ𝑈 𝑣 = and 𝐵↔ 𝑥 = 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈
These equations can be solved exactly.
Theorem: (T. & Li) The naïve, symmetric, and Neri splittings give consistent methods with convergence orders 1, 2, and 2, respectively.
Proof (sketch): Use the same strategy as for truncation I.

36. Truncation III Recall:
𝑌 Δ𝑡 𝐼𝐼𝐼 ↔ 𝑥 =𝑣Δ𝑡−𝜎Δ𝑈+𝛾𝜎Δ𝑉 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈− ∇ 𝑥 𝑓 𝑥 𝜎+ 𝛾 2 𝜎 Δ𝑉
Split 𝑌 Δ𝑡 𝐼𝐼𝐼 =𝐴+𝐵 where 𝐴={𝑣Δ𝑡−𝜎Δ𝑈+𝛾𝜎Δ𝑉} 𝜕 𝑥 𝐵={ 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈− ∇ 𝑥 𝑓 𝑥 𝜎+ 𝛾 2 𝜎 Δ𝑉} 𝜕 𝑣
Notice: 𝐴↔ 𝑥 =𝑣Δ𝑡−𝜎Δ𝑈+𝛾𝜎Δ𝑉 𝑣 = and
𝐵↔ 𝑥 = 𝑣 = 𝑓 𝑥 −𝛾𝑣 Δ𝑡+𝜎Δ𝑊+𝛾𝜎Δ𝑈− ∇ 𝑥 𝑓 𝑥 𝜎+ 𝛾 2 𝜎 Δ𝑉
Numerical evidence: truncation III with the Neri splitting gives strong order 3
Analysis is complicated

37. Numerical tests
One dimensional pendulum: 𝑓 𝑥 =− sin 𝑥 ∈ 𝑅 1 , 𝛾=𝜎=2, 𝑥 0 = 𝑣 0 =1
Lennard-Jones cluster (LJ7), function of distance:
𝑓 𝑖 𝑥 = 𝑗≠𝑖,𝑗= 𝑥 𝑖𝑗 − 𝑥 𝑖𝑗 𝑥 𝑖𝑗 / 𝑥 𝑖𝑗 ∈ ℝ 3
𝑥 𝑖𝑗 = 𝑥 𝑖 − 𝑥 𝑗 ∈ ℝ 3 , Euclidean distance |𝑥 𝑖𝑗 |:= 𝑥 𝑖𝑗 2
𝑥= 𝑥 1 , 𝑥 2 ,…, 𝑥 7 ∈ ℝ 21 , 𝑓= 𝑓 1 , 𝑓 2 ,…, 𝑓 7 : ℝ 21 → ℝ 21
𝛾=10 𝐼 21×21 ∈ ℝ 21×21 , 𝜎= 2 𝑘 𝐵 𝑇 𝛾 = 2×0.3 𝛾 ∈ ℝ 21×21

38. Trajectory comparison
One trajectory using three different operator splitting methods.
Constants 𝛾=𝜎=2; initial conditions 𝑥 0 = 𝑣 0 =1.
Same driving Wiener process with small 𝛿𝑡= 2 −10 .
Exact solution: Euler Maruyama with step size 𝛿𝑡.
Approximate solutions: use larger step size Δ𝑡= 2 −2 .

39. Numerical tests