Ph.D. Preliminary Examination: Analysis of Numerical Methods, II Fall 2025

(20 pts) On an equidistant mesh \(x_j = jh\), define the operator \(\widetilde {D}_0\) as, \[ \widetilde {D}_0u(x_j) = \frac {u(x_j+h/2)-u(x_j-h/2)}{h}, \] For a given smooth \(\kappa \) and \(f\), consider the scheme, \[ -\widetilde {D}_0\left (\kappa (x_j)\widetilde {D}_0u_j\right ) = f(x_j), \] for the ODE \[ -\frac {d}{dx}\left (\kappa (x)\frac {d}{dx}u(x)\right ) = f(x). \] What order is the local truncation error?

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the time-integration scheme \[ u_{n+1} = u_n + \frac {k}{4}f(t_n,u_n) + \frac {3k}{4}f\left (t_n+\frac {2}{3}k,\ u_n+\frac {2}{3}kf(t_n,u_n)\right ). \]

1. (5 pts) Identify the Butcher tableau for this scheme.
2. (8 pts) Compute the order of consistency of this scheme.
3. (7 pts) Compute the stability/amplification factor of the scheme.

(20 pts) Consider the ODE, \[ u'(t) = Au + N(t,u), \]

where \(A\) is a fixed matrix and \(N\) is an arbitrary, e.g., nonlinear, function.

1. (6 pts) With initial data \(u(0) = u_0\), show that the solution to this IVP at time \(t > 0\) is given by, \[ u(t) = e^{tA}u_0 + \int _0^t e^{(t-s)A}N(s,u(s))\,ds, \] where \(e^{tA}\) is the matrix exponential of \(tA\).
2. (14 pts) Exponential integrators form a scheme to compute \(u_{n+1}\) in terms of \(u_n\) by integrating the above expression over the interval \([t_n,t_{n+1}]\) (instead of \([0,t]\)), by approximating \(N(u(s))\) with a quadrature rule/polynomial approximation, and by exactly integrating the matrix exponential term. A Forward Euler exponential integrator makes the approximation \(N(u(s)) \approx N(u_n)\). Write out the Forward Euler exponential integrator scheme explicitly (i.e., by eliminating integrals).

(20 pts) For the ODE \(u'(t) = f(t,u)\), consider the scheme, \[ u_{n+1}+\alpha _1u_n = k\beta _0 f_{n+1}+k\beta _1 f_n, \] where \(u_n \approx u(t_n)\), \(f_n = f(t_n,u_n)\), \(k = t_{n+1}-t_n\), and \(u' = f(t,u)\).

1. (10 pts) Identify the coefficients \(\alpha , \beta \) that yield a scheme of optimal \(k\)-order of accuracy, and identify this order of accuracy.
2. (10 pts) Determine whether or not this scheme is 0-stable and/or A-stable.

(20 pts)

1. (10 pts) Use von Neumann stability analysis to determine a stability condition for, \[ D^+u_j^n = D_0u_j^n, \] where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\), \(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and \(D_\pm u_j^n = \pm (u_{j\pm 1}^n-u_j^n)/h\).
2. (10 pts) Use von Neumann stability analysis to determine a stability condition for \[ D^+u_j^n = D_+D_-u_j^n. \]