153.3. 3-Step Adams–Bashforth Method

Try these first

1. 1.

   Define the 3-step Adams-Bashforth method in your own words.

   Hint

   Think about how it employs past values.

2. 2.

   What is the formula for the 3-step Adams-Bashforth method?

   Hint

   Recall how the coefficients come from interpolation.

3. 3.

   What does the 3-step Adams-Bashforth method rely on?

   • Past two function values
   • Three past function values
   • All past function values

   Hint

   Refer back to the definition of this multistep method.

4. 4.

   The global error of the 3-step Adams-Bashforth method is:

   • True
   • False

   Hint

   Think about the relationship between local and global errors.

5. 5.

   Given the equation dy/dx = sin(x) + cos(y), y(0)=0, compute y(0.5) using the Adams-Bashforth method with h = 0.1. Include all intermediate calculations.

   Hint

   Ensure you derive the initial values correctly before applying the multistep method.

6. 6.

   Analyze how changing the step size h affects the accuracy and stability of the 3-step Adams-Bashforth method when solving dy/dx = e^{-x} - y.

   Hint

   Consider how lower step sizes generally improve accuracy but at the cost of computational load.