Spring Mass Systems: Implicit Methods for Solving ODE

PREVIOUS < - - - - > CS 184 HOME < - - - - > CURRENT < - - - - > NEXT

Implicit Euler Method

Key Slides from David Baraff's lecture on

Implicit Methods for Differential Equations

In the forward Euler method one has to carefully control the size of the time-step h
The larger k is, the stiffer the ODE becomes and the smaller the time step h has to be.
Below is an example for a simple one-particle system with a restoring derivative proportional to the deviation from 0.
The charts showing time running downwards in discrete steps of h compare 5 systems with different k:

k = 0.5/h; k = 1/h; k = 1.5/h; k = 2/h; k = 3/h;

If you must deal with a system with stiff ODEs, use an implicit method;

For our simple one-particle system with a restoring force, this results in:

This involves more computation,
but in this case we can take arbitrarily large time steps:

Some intuitive insight why this works:

To Learn More:

Andy Witkin and David Baraff: "Physically Based Modeling: Principles and Practice"
http://www-2.cs.cmu.edu/~baraff/sigcourse/index.html

Grinspun, Hirani, Desbrun, and Peter Schroder: "Discrete Shells," SCA 2003