Numerical Methods for Partial Di?erential Equations and High Performance Computing - IT Assignment Help

Assignment Task:

Tasks:
1. Write a program which solves the above problem with the initial conditions and boundary conditions f0 and f1 on a uniform grid containing 100 points in x, using the following schemes: 
• FTCS 
• Upwind 
• Lax–Friedrichs 
• Lax-Wendro? 
• 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 scheme 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 Lin?nity 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 quantitative.

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 uses the high number of points (at least several thousand) to measure the computational time.

9. Ascertain that the results of the execution are exactly the same by measuring the di?erence between the 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 timestep 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 parallelisation to become e?cient for this case - hint the problem will increase with the size of the grid.

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

Report and Requirements:

Summarise your result in a report. The serial code should be written without external dependencies/libraries, a low-level programming language should be used e.g. FORTRAN, c, no Matlab or python. The parallelisation of the serial code should be done with MPI directives. Write a report summarising the numerical discretisation schemes (theory), your code structure (use ?owchart and/or pseudo-code if necessary), your parallelisation strategy, your ?ndings with respect to accuracy and computational e?ciency. Use the books in the bibliography section and lecture notes/slides to facilitate your work. Literature survey with books and papers should be performed and used to also enriched your discussion and analysis section. The report should be not less than 1000 words and should not exceed 5000 words.

The report can contain any number of ?gures and tables. All ?gures and tables in the report should be numbered and discussed. The report should be structured like paper i.e. abstract, nomenclature, introduction (literature survey), methodology and theoretical background, results and discussion, conclusions, references and appendix. 

This IT Assignment has been solved by our IT Experts at My Uni Paper. Our Assignment Writing Experts are efficient to provide a fresh solution to this question. We are serving more than 10000+ Students in Australia, UK & US by helping them to score HD in their academics. Our experts are well trained to follow all marking rubrics & referencing style.

Be it a used or new solution, the quality of the work submitted by our assignment experts remains unhampered. You may continue to expect the same or even better quality with the used and new assignment solution files respectively. There’s one thing to be noticed that you could choose one between the two and acquire an HD either way. You could choose a new assignment solution file to get yourself an exclusive, plagiarism (with free Turnitin file), expert quality assignment or order an old solution file that was considered worthy of the highest distinction.