Dense output for highly oscillatory numerical solutions

Understand

We present a method to construct a continuous extension (otherwise known as dense output) for a numerical routine in the special case of the numerical solution being a scalar-valued function exhibiting rapid oscillations.

• Such cases call for numerical routines that make use of the known global behaviour of the solution, one example being methods using asymptotic expansions to forecast the solution at each step of the independent variable.
• An example is oscode, numerical routine which uses the Wentzel-Kramers-Brillouin (WKB) approximation when the solution oscillates rapidly and otherwise behaves as a Runge-Kutta (RK) solver.
• Polynomial interpolation is not suitable for producing the solution at an arbitrary point mid-step, since efficient numerical methods based on the WKB approximation will step through multiple oscillations in a single step.