15.1. Overview of Multistep Methods

Try these first

1. 1.

   Define multistep methods in your own words.

   Hint

   Think about how these methods leverage past computations.

2. 2.

   What is the main benefit of using multistep methods?

   Hint

   Consider efficiency in solving ODEs over long periods.

3. 3.

   What is the main purpose of multistep methods?

   • To calculate a single value
   • To use multiple previous values for predictions
   • To simplify the ODE

   Hint

   Think about the efficiency in solving differential equations.

4. 4.

   True or false: The Adams–Bashforth method is an example of an implicit method.

   • True
   • False

   Hint

   Recall the definitions of explicit and implicit methods.

5. 5.

   Given the differential equation dy/dx = y + x and the initial condition y(0) = 1, calculate y(0.5) using a 2-step Adams-Bashforth method after getting the first two values through a single-step method.

   Hint

   Ensure accuracy in your function evaluations.

6. 6.

   How would you demonstrate the stability of chosen step sizes across multiple runs of a numerical solution problem using the Adams-Bashforth method?

   Hint

   Focus on error analysis between varying step sizes.