Enhanced Integration Algorithms: Leapfrog Method & Hard Sphere Repulsion, Papers of Physics

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Lec

• dt errors. Leapfrog algorithm
• Many particles - balls in boxes

Recap week 1

• Discussed (simplest) method for solving New- ton’s laws for motion of single particle un- der external force
• Write as two first order equations:

dr dt

= v dv dt

m

F

• Use Euler algorithm to evolve (update) in discrete time
• Harmonic oscillators (1,2 dims), projectiles
• Discussed python code to implement this algorithm

More accurate integrators

Using Taylor ....

x(t + dt) = x(t) + v(t)dt + a(t)dt^2 /2 + O(dt^3 )

leading to

xn+1 = xn + vndt + andt^2 / 2

Subtracting x(t − dt) from x(t + dt) find

v(t) = x(t + dt) − x(t − dt) + O(dt^3 )

leading to

vn+1 = xn+2 − xn 2 dt Substiting for xn+2 using previous equation:

vn+1 = vn + dt 2

(an + an+1)

leapfrog algorithm Accurate to O(dt^2 )

Leapfrog errors

Almost as simple to implement in code.

• Compute and store new variable oldforce to keep initial acceleration.
• force() function same as before.
• See much smaller errors in energy. Consis- tent with ∆ EE ∼ dt^2

Newton’s 2nd law

If many mutually interacting particles:

Fa =

∑ b

Fab

where label b runs over all particles in system

dxa dt

= va dva dt

= Fa/ma

Get a pair of (vector) equations for each parti- cle. Simulating system requires summing over all particles.

Example - balls in box

• Consider a force of form

Fab = Ae −

(ra−r r 0 b

) 2

n

• n is unit vector from ballb to balla.
• Take r0 to be 1/10 radius of ball say. Sim- ulates hard sphere repulsion.
• Take dt <

√ b/A. Stable integration.

More for loops

To carry out a sum over all balls use for com- mand eg.

for ball in system: ball.pos=ball.pos+ball.vel*dt+...

• Note indentation – all statements at same level are executed before passing to next element of list
• Note give each ball a position, velocity and (2) accelerations.
• Use leapfrog to update.
• Add elastic force for walls of container - reverse velocity if hits wall.

Integrate module

• Change force function to newforce. Latter is passed the labels to two individual balls and computes mutual force vector.
• Function totalforce gets total force on a single ball.

Summary

• Discussed improved interation algorithms
• Developed code to study multiparticle sys- tems – here hard spheres
• Hard sphere replusion leads to chaotic tra- jectories