19.10.1. Finite Difference Method (FDM)

Try these first

1. 1.

   What does the Finite Difference Method (FDM) help solve?

   Hint

   Think about where analytical solutions might be challenging.

2. 2.

   Define the CFL Condition in your own words.

   Hint

   Consider what happens if that condition is violated.

3. 3.

   What is the primary purpose of the Finite Difference Method?

   • To solve linear equations.
   • To approximate solutions to differential equations.
   • To create graphical representations of data.

   Hint

   Think about what FDM is often applied to.

4. 4.

   True or False: The CFL condition is used to ensure the validity of numerical solutions.

   • True
   • False

   Hint

   Consider stability aspects of numerical methods.

5. 5.

   Given a grid where Δx = 0.05 m, Δy = 0.05 m, and Δt = 0.01 s, derive the CFL condition parameters for a wave speed of 300 m/s. Determine if the condition is satisfied.

   Hint

   Focus on calculating each part, ensuring you respect the parameters set.

6. 6.

   Apply the explicit finite difference method to find the displacement at the next time step for a given set of initial values un_i,j=2, un-1_i,j=1, un_i,j+1=3, and un_i,j-1=0. Define the neighboring contributions as well.

   Hint

   Make sure to plug in all grid point values correctly and perform the appropriate calculations.