Assignments
Introduction to Finite Differences
Summer Session 2016

1. 10 May:
2. 17 May:
   Notebook #1: Linear convection in 1D.
3. 24 May:
   
   1. Notebook #2: Nonlinear convection in 1D.
      
   2. Assuming that x1, x2, and x3 are equally spaced, show how the second derivative of a function f(x) defined at x2 can be approximated by a finite difference. Explain the steps involved, starting with a Taylor series. Explain the remainder terms.
4. 31 May:
   
   1. Notebook #3; Diffusion in 1D.
   2. Assume that x1, x2 and x3 are not equally spaced. Let h be the spacing x2-x1, and k be the spacing x3-x2. The finite difference approximation for the second derivative of the function u(x) at x2 is

                      2 * ( k * u1 - ( h + k ) * u2 + h * u3 ) / ( h * k * ( h + k )
                    

      Show that, if h = k, this formula agrees with the central difference formula.
   3. In the formula, replace u1 by the Taylor series with remainder for u(x1), continuing up to the O(h^4) term. Do the same for u3. Work out the algebra and show why you arrive at an approximation for u''(x2). What is the order of the error?
5. 7 June:
   
   1. Notebook #4: The Burgers equation in 1D.
   2. Finish the unequally-spaced second derivative approximation.
   3. Redo Notebook #3 using an implicit time stepping method.
6. 14 June:
   Notebook #5: Linear convection in 2D.
7. 21 June:
   Notebook #6: Nonlinear convection in 2D.
8. 28 June:
   Notebook #7: Diffusion in 2D.
9. 5 July:
   Notebook #8: The Burgers equation in 2D.
10. 12 July:
    Notebook #9: Laplace equation in 2D.
11. 19 July:
    Notebook #10: Poisson equation in 2D.
12. 26 July:
    Notebook #11: Navier-Stokes cavity flow in 2D.
13. 2 August:
    Notebook #12: Navier-Stokes channel flow in 2D.

You can return to the FEM 2016 web page.