18.11. Numerical and Computational Aspects

Try these first

1. 1.

   What is the Gibbs Phenomenon?

   Hint

   Think about what happens at jumps in functions.

2. 2.

   What does truncation mean in numerical methods?

   Hint

   It relates to how you might round something off.

3. 3.

   What do numerical methods help engineers achieve?

   • Precise analytical solutions
   • Approximations for complex problems
   • No solutions at all

   Hint

   Recall the discussion about complications in shapes.

4. 4.

   True or False: The Gibbs Phenomenon indicates that using more terms always prevents oscillations.

   • True
   • False

   Hint

   Consider what happens at a sudden jump.

5. 5.

   Consider a complex boundary problem in fluid dynamics. Describe how you would set up a finite element solution to analyze the flow.

   Hint

   Think about how breaking a complex shape into smaller shapes can make the problem easier to handle.

6. 6.

   Using the Fourier series, derive the first three terms for the function given by f(x) = sin(3πx/10) on the interval [0,10]. Discuss what happens when you add more terms.

   Hint

   Start with the Fourier sine coefficients and recall what happens with complex functions.