Course Description

This course is intended to serve as a sequel to an introductory finite element or computational mechanics courses. It is designed to deepen student’s understanding of the characteristics of elliptic, parabolic, and hyperbolic partial differential equations (PDE) and get familiar with solution techniques for dynamic problems.

Course Objectives

  1. Provide sufficient mathematical background to read the current literature and understand new developments in the field
  2. Familiarize the students with various numerical schemes for continuum dynamics
  3. Relate theory to practical applications in computational science and engineering.
  4. Develop the student’s capabilities for technical communication and independent research in computational science and engineering.


Class Information

  • Hours: Mondays and Wednesdays  12:40-1:55 pm EST (11:40 am-12:55 pm CST)
  • Location: UTK:  Doherty 406                    UTSI: Main Academic Building E110

Office Hours

By appointment (email or phone call)  

Office: B203, Main academic building, UTSI
Phone: (931) 393-7334
Skype username: rpabedi


  • HW1: link. Due 02/10/2016
  • HW2: link. Due 03/07/2016                link to supplementary plots (they can also be found in the dropbox folder)
  • HW3: link. Due 03/21/2016                link to supplementary plots (they can also be found in the dropbox folder)
  • HW4: link. Due 04/11/2016
  • HW5: link. Due 04/29/2016
  • HW6: link. Due 05/09/2016               Note: The return of this assignment is optional. It will help you to better understand the concepts of dispersion and dissipation from the last chapter (chapter 7) and well-posedness, dynamic stability, and robustness from chapter 7.
    If you return this assignment half of your total grade (that is a maximum of 100 points) will be added to your total points as extra credit points.
  • Final presentation: Friday 5/6 11am – 1 pm EST (10 am – 12 pm CST) in the usual classrooms.

Presentation file

I am changing the format of the presentation file. Until the transition is complete, please use part1 for previous sections and part2 for new sections. part1     part2

Class timeline

  1. 01/13/2016 Lecture: notes          Topics: Discussion of material to be covered in the course
  2. 01/20/2016 Lecture: notes          Topics: Classification of PDEs: 1.a and 1.b
  3. 01/25/2016 Lecture: notes          Topics: Classification of PDEs: 1.c  (i. characteristics for 1st order PDEs, classification of 2nd order PDEs).
  4. 01/27/2016 Lecture: notes          Topics: 1.c.iii D’Alembert solution of the wave equation; 1.c.iv systems of PDEs (part 1l).
  5. 02/01/2016 Lecture: notes          Topics: Systems of first order PDEs 1.c.iii.
  6. 02/03/2016 Lecture: notes          Topics: Quasi-linear systems, shocks and expansion waves (brief discussion) 1.c.iv; Riemann solution and 2 methods for linear systems 1.c.v.
  7. 02/08/2016 Lecture: notes          Topics: 2.c Finite Element formulation for dynamic problems (part 1).
  8. 02/10/2016 Lecture: notes          Topics: 2.c Finite Element formulation for dynamic problems (part 2); 2.a Finite Difference method (part 1).
  9. 02/15/2016 Lecture: notes          Topics: 2.a Finite Difference method (part 2).
  10. 02/17/2016 Lecture: notes          Topics: 2.a Finite Difference method (part 3); 2.b Finite Volume method (part 1).
  11. 02/22/2016 Lecture: notes          Topics: 2.b Finite Volume method (part 2).
  12. 02/24/2016 Lecture: notes          Topics: 3.a Exact temporal integration; 3.b time marching schemes; 3.c Modal superposition (part 1).
  13. 02/29/2016 Lecture: notes          Topics: 3.c Modal superposition (part 2).
  14. 03/02/2016 Lecture: notes          Topics: 3.c Modal superposition (part 3).
  15. 03/07/2016 Lecture: notes          Topics: 3.c Modal superposition (part 4).
  16. 03/09/2016 Lecture: notes          Topics: 4.a Time marching schemes (part 1): Linear Multi-step methods.
  17. 03/21/2016 Lecture: notes          Topics: 4.a Time marching schemes (part 2): Multivariate single-step methods.
  18. 03/23/2016 Lecture: notes          Topics: 4.a Time marching schemes (part 3): Runge-Kutta methods.
  19. 03/28/2016 Lecture: notes          Topics: Runge-Kutta methods; 5. Analysis of MDOF/SDOF for generalized alpha method (part 1).
  20. 03/30/2016 Lecture: notes          Topics: 5. Stability, consistency, and convergence analysis of MDOF/SDOF for generalized alpha method (part 2).
  21. 04/04/2016 Lecture: notes          Topics: 5. Stability analysis of Linear Multi-step (LMS) methods (e.g. Central difference, Houbolt) and multivariate single step methods (Mewmark, θ-Wilson).
  22. 04/06/2016 Lecture: notes          Topics: 5. Absolute stability, practical consideration in using numerical methods (part 1).
  23. 04/11/2016 Lecture: notes          Topics: 5. practical consideration (part 2); Topic 6: FD analysis (convergence and consistency).
  24. 04/13/2016 Lecture: notes          Topic 6: FD analysis consistency; stability (part 1).
  25. 04/18/2016 Lecture: notes          Topic 6: FD stability, Introduction to von Neumann analysis (part 2).
  26. 04/20/2016 Lecture: notes          Topic 6: FD stability, von Neumann analysis: FTBS, Lax-Friedrichs, and leapfrog schemes (part 3).
  27. 04/22/2016 Lecture: notes          Topic 6: FD stability, von Neumann analysis: temporally second order PDEs, solution to recursive relations, polynomial root condition (part 4); Dispersion analysis (part 1).
  28. 04/25/2016 Lecture: notes          Topic 6: Dispersion analysis (part 2); Dispersion relation, phase velocity, and physical dispersion and dissipation.
  29. 04/27/2016 Lecture: notes          Topic 6: Dispersion analysis (part 3); Examples of dispersive media; Numerical dispersion and dissipation errors with examples from FD methods.
  30. Frequency, dissipation error for ODEs: link
    Based on a discussion in the classroom the following hand-written analysis shows how frequency, related to period elongation, and dissipation, related to amplitude decay, errors can be computed for an ODE.
    The first half of the document works on a first order ODE that corresponds to spatial discretization of a parabolic diffusion equation, while the latter performs the same analysis for a first order ODE that corresponds to spatial discretization of the wave equation. In either case the analysis is carried over for generalized alpha method.

Selected Bibliography

  • [Strikwerda, 2004] Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.
  • [Hughes, 2012] Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element analysis. Courier Corporation.
  • [Bathe, 2006] Bathe, K.-J. (2006). Finite element procedures. Klaus-Jurgen Bathe.
  • [Farlow, 2012] Farlow, S. J. (2012). Partial differential equations for scientists and engineers. Courier Corporation.
  • [LeVeque, 2002] LeVeque, R. L. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
  • [Chapra and Canale, 2010] Chapra, S. C. and Canale, R. P. (2010). Numerical methods for engineers, volume 2. McGraw-Hill. 6th edition.