Math 6620 • Analysis of Numerical Methods II

(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.
\begin{enumerate}

\item
(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 \(tA\).


\item
(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).

\end{enumerate}