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EGN 5456 Homework 9 Fall 1996

1. Solve the unsteady heat conduction in a bar with homogeneous Dirichlet boundary conditions and an initial condition in which the temperature is maximum in the center of the bar and decays linearly to zero at each end. Use the Crank-Nicholson method to do so.
2.   Use approximate factorization to develop an ADI scheme for the three-dimensional heat equation

   

3. Show stability for the scheme of question 2.
4. Derive the truncation error for the scheme of question 2
5. Explain how the scheme of question 2 would be implemented numerically,
6. Discuss how you would implement Dirichlet boundary conditions on a block-shaped domain for the scheme of question 2.
7. What is the maximum stable Courant number if the fourth order Runge-Kutta method is applied to the one-way wave equation, differenced centrally in space?
8.   What is the maximum stable time step if the fourth order Runge-Kutta method is applied to the two-dimensional unsteady heat equation? Compare with the maximum time-step for the forward-time (Euler) approximation.
9. Discuss which of the two schemes of question 8 is preferable.
10. Project (25% of your grade): Solve the unsteady heat conduction in a plate of length 2 and height 4, and a heat conduction coefficient 3. Assume the sides are at zero temperature for all time, and that the initial temperature is

    

    Compute up to time t=0.5. Implement and compare the Peaceman-Rachford and Mitchell-Fairweather methods. Assume that the desired maximum error at time t=0.5 is 0.1. What is the minimum number of points that needs to be computed using the Peaceman-Rachford method to achieve this accuracy? And what is the minimum number of points that needs to be computed using the Mitchell-Fairweather method? Use neat and good coding procedures.

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