# 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?

Methods:

### Implicit vs. Explicit Function

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

• Easy β just evaluate

Implicit function ( on both sides):

• Harder β requires solving

### Explicit vs. Implicit schemes

Explicit vs. Implicit Schemes

Explicit:

• Simpler, and fast to compute per step.
• Less stable β require smaller timesteps to avoid βblowing upβ. (More later!)

Implicit:

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