Questions and Answers on Finite Difference Methods

• What is a two-point boundary value problem?
  
• What is a finite difference formula?
• What are three commonly used FD formulas for approximating u'(x)? How accurate of those methods?
• What's the best 'h' that gives the best possible accuracy for FD formulas for approximating u'(x) in terms of the machine precision? 
• What is a convergence order? How do we find them numerically?
  
• When we solve a 1D Sturm-Liouville problem using a FD method, what kind of conditions should we have for the coefficients to guarantee the solution exist and unique?
• How do we get finite difference formulas for u''(x)? How many different possibilities? What's the best one?
• What are two most important parts in deriving the conservative FD scheme for S-L BV problems?
• What is a finite difference stencil and a master grid point?
• What is a compact FD scheme?
  
• Why should we use a conservative FD scheme if available?
• What are local truncation errors and global errors?
  
• What conditions that can guarantee the convergence of a FD method.
• What is the stability of a FD method?
• What is the ghost point method? Suppose we have   q u(a) + r u'(a) = s at x=a? How do we use the ghost point method?  * What is the order of the local truncation error at x=a?
• What is the matrix-structure of the linear system terms of equations from the FDM applied to S-L problem? Is it an M-matrix?
  

• Given u''(x) = f(x), u'(a)=u'(b), does the problem exist? Or under what kind of condition,  the solution exists? If the solution does exist, is it unique?
• Given u''(x) = f(x), u(a)=u(b), does the problem exist? Or under what kind of condition,  the solution exists? If the solution does exist, is it unique? It is called a periodic boundary condition.
  
• What is a S-L eigenvalue problem? Can you use a FDM to solve it? How?

• What are the continuous and discrete maximum principle?
• Can you prove the convergence of the finite difference method using the maximum principle?
• What are the matrix structure of the finite difference equations for 2D Sturm_Liouville elliptic PDEs using natural ordering and red-black ordering, respectively?
• How's 9-point compact 4th order finite difference scheme derived? When we wish to use 4-th order schemes instead of 2nd order?
  

Exercises:

• Modify the 1D two point BVP codes to use a tri-diagonal solver.
  
• Extra Credit: Derive and implement a FDM for a general 1D linear two-point BVP, using numerical integration if necessary. Hint: Find an integration factor to convert the 1D problem to a standard Sturm-Liouville problem.
  
• Extra Credit: Use the Green function to show that the 3-point FD scheme for S-L two-point BVP is second order convergent in the infinite norm. Hint: Refer to Dr. R. LeVeques's FD book. 

Suggested Projects:

1.    Develop and test a finite difference method in spherical coordinates for Poisson equations.
2.    Explore the possibility of a finite difference scheme for a 2D diffusion and advection elliptic PDEs.
   
3.    Develop and test a finite difference method in Matlab for Helmholtz equations in Matlab using FFT (fast Fourier Transforms). It should allow three linear boundary conditions.
   
4.    Develop and test a finite difference method in Matlab for Helmholtz equations in in the entire 2D plane with some numerical boundary conditions.
   
5.    Develop and test a multigrid solver for 2D Sturm-Liouville elliptic PDEs. 
   
6.    Develop MOL for 2D heat equations and compare with the ADI method. 
7.    Implement the projection method for Navier-Stokes equations on a rectangular domain. Test your code using u(x,y,t) = sin(pi*x)*sin(pi*y)*h(t). 
8.   Simulate the lid-driven flow using Helmholtz/Poisson solves (the projection method)
9.   Use an extrapolation technique to get fourth order finite difference schemes for 1D, 2D Poisson and/or elliptic PDEs of BVP