Partial Differential Equations and High Performance Computing - Computer Science Assignment Help

Assignment Task

Tasks
 

1. Write a program which solves the above problem with the initial conditions and boundary conditions
f
0 and f
1 on a uniform grid containing 100 points in x, using the following schemes:
 

• FTCS
• Upwind
• Lax–Friedrichs
• Lax-Wendroff
• MacCormack
 

2. Solve the problem and output the numerical and exact solutions at time t = 5 and t = 10.
 

3. Explain the behaviour of numerical solutions in terms of expected properties of the schemes involved in terms of stability and accuracy - hint use numerical viscosity and/or other ways of assessing the stability of a numerical schemes in an a priori fashion.
 

4. Compute the error of the computed solutions and discuss the evolution of error in time, use L1, L2 and Linfinity norms.
 

5. Compare analytical and numerical solutions. How do the numerical solutions vary with the Courant number? How the solution involves in time? Comparisons should be both qualitative and quanti- tative.

6. Increase the grid size to 200, 800, 1600, 6400 points in x. Present and examine the grid convergence of numerical solutions (use the error) obtained with the numerical scheme at t = 10.
 

7. Measure the execution time of the sequential code.
 

8. Devise a way to parallelise this problem for 2, 4, 8, 16, 32 and 64 processors using MPI and measure the execution time of the parallel code for these parameters - Hint use high number of points (at least several thousands) to measure the computational time.
 

9. Ascertain that the results of the execution are exactly the same by measuring the difference between solution obtained with parallel and sequential codes at t = 5 and t = 10.

10. Measure the cost of communications in the boundary exchange and the cost of computing a time- step for each process.

11. Measure the performance of your serial and parallel codes and discuss. How does the actual parallel code performance compare to the theoretical one? Is the performance of your parallel code expected?

12. Based on your results, comment on the problem size which is necessary in order for MPI paralleli- sation to become efficient for this case - hint the problem will increase with size of grid.

You should perform your parallel computations on the HPC facility. To make use of the the worker nodes of the HPC facility you must submit your simulation job to the queue of the scheduler.

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