Semi-Lagrangian Methods and Exponential Integrators

(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).

Padé Approximants Define what is meant by the Padé approximant of a function \(f\) and compute the \([1/1]\) Padé approximant of the exponential.

The \(n\) data points \((x_i,y_i)\), \(i=1,2,\ldots ,n\) are believed to lie on an exponential curve \[ y_i=A\exp (\lambda x_i). \] Estimate \(\lambda \).

The \(n\) data points \((x_i,y_i)\), \(i=1,2,\ldots ,n\), are believed to lie on an exponential curve \(y_i=A\exp (\lambda x_i)\). Formulate a linear regression problem to estimate \(\lambda \).

The \(n\) data points \((x_i,y_i)\), \(i=1,2,\ldots ,n\) are believed to lie on an exponential curve \(y_i=A\exp (\lambda x_i)\). Estimate \(\lambda \).