Questions

Below you will find some sample question that may help you better digest the material for this course. In terms of their scope and depth, these questions are similar to what you are likely to encounter at the quizzes.

• Use the Taylor series expansion to estimate the truncation error of the central finite difference formula approximating the second derivative f''(x) of the function f(x).
• Derive the central difference approximation of the first derivative f'(x) using interpolating polynomials
• What are the compact finite difference formulas? Provide an example.
• Are differentiation matrices invertible? Justify your answer.
• Show how to approximate the derivative f'(x) using the complex step derivative.
• How to discretize in time initial-value problems of order higher than one?
• Would you recommend using an implicit technique to solve a nonlinear ODE?
• Define what it means for an ODE integration technique to be conservative. Provide examples.
• What is the difference between the algebraic and spectral rares of convergence of an approximation?
• Give a Weighted Residual Method with the set of basis functions e_i, i=1,...,N and the set of test (trial) functions h_k, k=1,...,N, indicate what additional assumptions are necessary to derive from it
a) the spectral Galerkin method
a) the spectral collocation method
• Given a boundary-value problem of the general form Lu=f, where u is the solutions and L a second-order differential operator, and a set of orthonormal basis functions e_i, i=1,...,N who how this equation can be solved using a spectral Galerkin method
• You are given a function u and a set of basis functions e_i, i=1,...,N. Comment on how solution of the approximation problem will differ depending on whether or not the basis functions are orthonormal.
• What is the relationship between the smoothness of a function f : R -> R and the decay of its Fourier transform for increasing wavenumbers?
• What are the properties required of the function f in order for its approximation to exhibit spectral convergence?