Review Questions
Truncation vs. round-off errors: origins and characterization.
Similarities and differences between the fixed-point iteration,
Newton's method and the secant method for solution is nonlinear
equations.
Convergence analysis of different root-finding methods; the role of
the fixed-point theorem.
What are the origins of the Runge phenomenon in polynomial
interpolation? What does the interpolation error depend on?
Given a set of points (x_i,y_i), i=1,...,N and an approximating function
in the form y= Ax + B, use the least-squares method to derive a system
of equations characterizing the unknown parameters A and B; how do solutions
of approximation and interpolation problem differ?
Numerical differentiation: one-sided (forward / backward) and central
difference formulas for the first and second derivative. Estimates of
numerical differentiation errors.
Derive an estimate for the order of accuracy of the trapezoidal rule for
numerical integration.
Why are the Gaussian quadratures more accurate than the Newton-Cotes
quadratures when the integrand function is a polynomial of a given
order? What are the disadvantages of the Gaussian quadratures?
Compare the Runge-Kutta and multistep methods for numerical integration
of ODEs. Address accuracy and computational efficiency.
Numerical solution of initial-value problems for ODEs of order higher than one.
What determines stability of explicit methods in such cases?
Analyze implementation of Dirichlet and Neumann boundary conditions in finite-difference
solutions of boundary value problems (both for ODEs and elliptic PDEs).
How to solve efficiently sparse (banded) algebraic systems depending on the number of their
diagonals
Discuss the criteria determining the choice of time step for explicit
and implicit integration schemes for parabolic and hyperbolic
PDEs. What are the trade-offs between stability and computational
performance in these two approaches?
Derive and compare explicit and implicit (in time) schemes for numerical
solution of parabolic PDEs
What is the relationship between spatial and temporal (explicit)
discretization of parabolic and hyperbolic PDEs. How does this
relationship depend on the order of derivatives in space and in time?
Does it depend on the spatial dimension of the problem?
Compare the different variants of the Finite Element Method for solution
of a given ODE or PDE.