Time-Stepping Methods

Was introduced to this in CS370.

Time-stepping applies a Recurrence Relation to approximate the function values at later and later times.

  • Given , approximate . Given , approximate . etc.

There are several varieties/categories of time-stepping methods:

  • Single-step vs. Multistep: do we use information only from the current time, or from previous timesteps too?
  • Explicit vs. Implicit: Is given as an explicit function to evaluate, or do we need to solve an implicit equation?
  • Timestep size: Do we use a constant timestep , or allow it to vary?


Implicit vs. Explicit Function

e.g., given and , determine . Explicit function ( only on one side):

  • Easy – just evaluate

Implicit function ( on both sides):

  • Harder – requires solving

Explicit vs. Implicit schemes

Explicit vs. Implicit Schemes


  • Simpler, and fast to compute per step.
  • Less stable – require smaller timesteps to avoid β€œblowing up”. (More later!)


  • Often more complex and expensive to solve per step.
  • More stable – can safely use larger timesteps.