Introduction to symplectic methods

Standard approaches trade-off

• - Explicit methods: (+) Simple to compute, (-) limited stability
• - Implicit methods: (-) Hard to compute (especially on non linear functions), (+) very stable
• - Oscillatory systems are not easy to model
• • (-) Numerical solution either diverge or converge.

Symplectic approach

• - Remark: Mechanical systems have position and velocity variables
• • - Derivative of position is linear w/r velocity
  • - Derivative of velocity is more complex (forces - non linear)

• \(\Rightarrow\) General idea: separate treatment of velocity and position

• Semi-implicit:
• • - Implicit scheme for position \(p^{k+1}\) (linear part)
  • - Explicit scheme for velocity \(v^{k+1}\) (non linear part)

• \(\Rightarrow\) In practice: use velocity \(v^{k+1}\) to evaluate \(p^{k+1}\).

Pro

• (+) As simple as explicit method to implement
• (+) Improved stability
• (+) Well adapted to oscillatory systems