15.3.4. 4-Step Adams–Bashforth Method

Try these first

1. 1.

   What is the purpose of the Adams–Bashforth method?

   Hint

   Think of what we aim for in numerical methods.

2. 2.

   What does 'step size' refer to in numerical methods?

   Hint

   Consider how we define intervals between values.

3. 3.

   What is the primary goal of the Adams-Bashforth method?

   • Calculate integrals
   • Solve ordinary differential equations
   • Optimize computational speed

   Hint

   Focus on the specific application of this method.

4. 4.

   True or False: The local truncation error for the 4-step Adams-Bashforth method is $O(h^4)$.

   • True
   • False

   Hint

   Remember, local truncation error and global error are not always the same.

5. 5.

   Given the initial value problem $\frac{dy}{dx} = x + y$ with $y(0)=1$, estimate $y(0.1)$ using the 4-Step Adams-Bashforth method. Calculate required initial values using a basic method first.

   Hint

   Make sure to determine initial function evaluations needed for the formula before applying the 4-step method.

6. 6.

   Evaluate how a larger step size would affect the accuracy of the Adams–Bashforth method in a long-term integration scenario. Provide examples to illustrate your response.

   Hint

   Consider real-life scenarios where accuracy over time is crucial, such as weather predictions.