Overview
Class #2 (Jan 16)
• Brief introduction to time-stepping ODEs
• Application: Particle systems
• Numerical stiffness issues
No class next Tuesday
• Instead:
– assignment
– readings
• http://www.cs.cmu.edu/~djames/pbmis
What is stiffness?
• From [AscherPetzold97]
• “Loosely speaking, the initial value
problem is referred to as being stiff if the
absolute stability requirement dictates a
much smaller time step than is needed to
satisfy approximation requirements alone.”
• Definition 3.1 [p48]: An IVP is stiff in some interval
[0,b] if the step size needed to maintain stability of
the forward Euler method is much smaller than the
step size required to represent the solution
accurately.
What is stiffness?
• Stiffness depends on your problem and…
– Accuracy criterion
– Length of the interval of integration, [0,b]
– Region of absolute stability of the method
• Related to local Jacobian eigenvalues {j}
– Stiff on [0,b] if
• “Stiff decay”
Some higher-order explicit methods
Some higher-order implicit methods
Complications
• Smoothness assumption in Taylor
expansion