15. Numerical Solutions of ODEs

Try these first

1. 1.

   What does the acronym IVP stand for?

   Hint

   Think about how we begin solving differential equations.

2. 2.

   What is meant by step size in numerical methods?

   Hint

   It's related to how quickly we move along the x-axis.

3. 3.

   What type of method is the Adams-Bashforth method?

   • Implicit
   • Explicit
   • Neither

   Hint

   Recall the key features of Adams-Bashforth.

4. 4.

   True or False: The Local Truncation Error for a 2-step Adams-Bashforth method is O(h^2).

   • True
   • False

   Hint

   Consider the definition of Local Truncation Error.

5. 5.

   Given the ODE $\frac{dy}{dx} = y + 3x^2$ with initial condition $y(0)=1$, apply the 4-step Adams-Bashforth method to compute an approximation to $y(0.6)$ using a step size of $h=0.2$. Start with values obtained from a single-step method.

   Hint

   Don’t forget to derive initial values before applying the 4-step method.

6. 6.

   Critically analyze how increasing the number of steps in the Adams-Bashforth method affects accuracy and computational load. Provide a practical scenario where one might choose a higher or lower step count.

   Hint

   Reflect on typical use-cases for rapid versus precise results.