ENGRD 241: Engineering Computation
Fall 2002
LEARNING OBJECTIVES FOR FINAL EXAM
Friday, December 13, 9:00 – 11:30 AM; Upson 111, 207, 211, and 215
Makeup exam: TBA
The Final Exam is comprehensive, that is, it covers material from the entire course. You are
responsible for all of the learning objectives distributed for Preliminary Exams I and II. You are
especially responsible for the new objectives below, and there will be some extra emphasis on the
material covered since the Second Prelim, i.e., ODE’s and PDE’s.
Numerical Solution of Ordinary Differential Equations (Chapters 25-27)
1. Be able to recognize linear and non-linear differential equations, be able to categorize equations
properly (e.g., initial value problems, boundary value problems, 1st order, 2nd order, etc.), and
recognize the context in which each arises.
2. Be able to define and illustrate the differences between local truncation error and global error.
3. For initial value problems, be able to use the appropriate Runge-Kutta formula (Euler, Heun,
midpoint, classical 4th order, Runge-Kutta-Fehlberg: 2-3rd and 4-5th) to integrate single ODE’s
and systems of ODE’s.
4. For initial value problems, be able to use different multi-step or predictor-corrector (NewtonCotes and Adams) formulas to integrate single ODE’s and systems of ODE’s.
5. Be able to list the relative advantages of Runge-Kutta versus multi-step methods; and of NewtonCotes versus Adams multistep formulas.
6. Know how to decompose higher order ODE’s into systems of 1st order equations and set up and
solve the system of equations.
7. Know what methods are used, how to estimate truncation error, and how to adjust step size in
adaptive procedures for integrating ODE’s with Runge-Kutta methods.
8. Be able to analyze the stability of different formulas for integrating simple linear ordinary
differential equations, and know the meaning and importance of stability. Students should be
able to provide an example of when a reasonable solution method is unstable, and when it is
stable.
9. Know how to define stiff differential equations and describe the problems they pose when
solved numerically. Provide examples. Reformulate ODE problems to generate non-stiff
systems.
10. Be able to use the shooting method to solve boundary values problems
11. Know how to use finite differences to solve boundary value problems, including when values
of the function or derivatives are specified.
12. Be able to discuss the relative advantages and disadvantages of the shooting method and finite
differences for solving boundary values problems.
13. [We did not have time to cover in any detail eigenvalues and eigenvectors (C&C 27.2) so you will not be
responsible for this topic on the exam. However, the learning objective for the topic would be that you could
provide examples of how eigenvalue problems arise in the solution of systems of partial differential equations using
separation of variables, mass-spring systems (C&C example 27.4), or the analysis of the buckling loads of structural
elements (see example C&C 27.5). In addition you should know how eigenvalues arise in the computation of matrix
norms and the analysis of the convergence of iterative solution of systems of linear equations. Finally, I would like
you to be able to estimate the largest and smallest eigenvalues of a matrix by the power and inverse power methods
(C&C 27.2.5), and know what properties symmetric matrices have which guarantee the success of these two
methods. C&C section 27.2.6 discusses methods that work for real problems.]
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ENGRD 241: Engineering Computation
Fall 2002
Partial Differential Equations (Chapters 29-31)
1. Be able to recognize and distinguish among the 3 classes of 2nd-order linear PDE’s. Students
should know physical problems each class represents, and the physical and mathematical
characteristics of each.
2. Know how to distinguish between linear & nonlinear equations.
3. Be able to describe differences between finite-difference and finite-element methods for solving
PDE’s.
4. Be able to formulate finite difference molecules for the solution of partial differential equations.
5. Know how to set up and solve Elliptic (Laplace/Poisson) PDE’s using the finite difference
method.
6. Know how to set up and solve Parabolic (Heat/Diffusion) PDE’s using finite differences with
both explicit and implicit methods. Students should know the relative advantages of the two
methods, and when and why the explicit method is unstable.
7. Be able to explain the idea of stability in the solution of ODE’s and PDE’s, what it means for a
numerical method to be stable or unstable, and tell why stability is important.
8. Be able to identify appropriate methods for solving the systems of equations generated for the
solution of PDE’s (Gauss-Seidel, LU decomposition, Thomas algorithm, etc.).
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