Overview Class #2 (Jan 16) Brief introduction to time-stepping ODEs Application: Particle systems Numerical stiffness issues

No class next Tuesday Instead: http://www.cs.cmu.edu/~djames/pbmis 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