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EGN 5456 Homework 7 Fall 1996

1.   The backward-time central-space (BTCS) scheme for unsteady heat conduction in a uniform bar has the form

   

   Find the growth factor g of this scheme. Use it to show that each Fourier mode of the numerical solution converges to the exact Fourier mode. Discuss whether or not this proves that the numerical solution always converges to the exact solution. Discuss fully!

2. You developed a new finite difference scheme for fluid flows and you used it to compute the flow for a case in which an exact solution is known. Your results are in good agreement with the exact solution. Will your solution get more and more accurate when you keep refining the mesh (using more points)? Discuss fully!
3. For the hypothetical finite difference method of the previous question, you find that on refining your mesh your numerical solution clearly converges to the exact solution. Does this mean that your method will also work for similar flows, even if no exact solution is available? Discuss fully!
4. Examine whether the backward-time, central-space scheme of question 1is stable. What conclusion do you draw from your result? Discuss fully!
5. A certain very fast finite difference scheme is unstable. To be able to use this fast scheme, you get the idea of taking the spatial and time steps as powers of two. With a bit of work, this allows you to perform all computations exactly. Since you no longer have to worry about round-off errors spoiling the solution, will the computations converge to the correct solution? Discuss fully!
6. Use the fast method to show that the one-way-wave, or convection, equation

   

   is properly posed.

7.   Use the fast method to determine whether the following equation is properly posed:

   

8. For the equation of question 7, what can you say about the stability condition to use for corresponding finite difference schemes?
9. Use the fast method to determine when the BTCS scheme of question 1 is stable.
10.   Use the fast method to determine when the forward-time, backward space (upwind) scheme for the convection equation,

    

    is stable. Here a>0. Use the strong version of the stability condition.

11.   Use the fast method to determine whether the FTCS scheme for the convection equation,

    

    is stable. Be careful (compare section 2.1ff in the book).

12. For the scheme of question 10, would we not get a less restrictive condition by using the weak version of the stability condition? Comment on the desirability of doing so.
13. Use the Taylor series approach to determine the consistency and order of accuracy of the upwind scheme of question 10.
14. Use the Taylor series approach to determine the consistency and order of accuracy of the central space scheme of question 11.

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