Preserving the Structure of Motion

 

Erik Jansson, DAMTP

Who am I?

I am mathematician, postdoc at the Department of Applied Mathematics and Theoretical Physics

I work with telling computers how to solve equations approximately

And, more importantly, proving that these approximations are correct

Equations?? 

  • Equations appear early in a mathematical curriculum.
  • Each year: harder equations and advanced techniques to write down a  solution with pen and paper.

However, most equations cannot be solved  with pen and paper! 

Let's explain what I mean by some examples....

~ x+3 = 5 \implies x = 5-3 = 2~
~ x^2 + 3x +2 = 0 \implies x = \frac{-3 \pm \sqrt{9-8}}{2} = \frac{-3 \pm 1}{2}~\\

Equations?? 

~\dot y = y^2+2, y(0) = 0 \implies y(t) = \sqrt{2}\tan(\sqrt{2}t)~
~u_t−u_{xx}-2u_x=0 \implies
~\begin{cases}C_1xe^{-x-t}+C_2e^{-x-t}+\sum\limits_sC_3(s)e^{-x-t((f(s))^2+1)}\sin(xf(s))+\sum\limits_sC_4(s)e^{-x-t((f(s))^2+1)}\cos(xf(s))&\text{when}~\text{Re}(t)\geq0\\C_1xe^{-x-t}+C_2e^{-x-t}+\sum\limits_sC_3(s)e^{-x+t((f(s))^2-1)}\sinh(xf(s))+\sum\limits_sC_4(s)e^{-x+t((f(s))^2-1)}\cosh(xf(s))&\text{when}~\text{Re}(t)\leq0\end{cases}~\\
~\ddot r_i = -G \sum_{j = 1, \\j \neq i }^N \frac{m_j}{|r_j-r_i|^3} \implies r=?~

Intro to approximation

Coarse

Fine

Smaller steps seems better! Are we done? 

Let's look at an example.

Equations for our solar system

Classical mechanics problem. 

Dates back to Newton

Physics demands: energy is conserved; planets stay on their orbits. 

Which simulations of the N-body problem can we trust?

~\ddot r_i = -G \sum_{j = 1, \\j \neq i }^N \frac{m_j}{|r_j-r_i|^3}~\\

Two ways of simulating our solar system

Which one is the best? Both are accurate and good methods and the steps are small! 

Why does the left Mercury crash into the sun?

Violation of physics!

The non-geometric algorithm loses energy, leading to the crash and non-physical behavior

The fix is to build physics into the method: this is geometric integration, or, methods that 

preserve the structure of motion

This goes way back...

...and is still an important tool for scientific discovery

Uses of structure-preserving numerics

Used every day in for instance materials science, plasma physics, and fluid simulations

Additional maths work: derive which structures from geometry and physics we want to 

preserve, construct methods that provably does this.

My research

Personally, I have derived geometric methods for topics such as:

Turbulence modelling

Stochastic fluid dynamics

Spin chains

Punchline: trustworthy approximations by respecting physics