High-order stroboscopic averaging methods for highly oscillatory delay problems (Q2301446)

The problem of numerically integrating highly oscillatory delay differential systems of the form \(\dot{x}(t) =f(x(t), x(t-\tau), t, \Omega t; \Omega), 0<t<t_{\max}, x(t) = \varphi(t), -\tau\leq t\leq 0\) is considered. Here \(\tau>0\) is the constant delay, the angular frequency \(\Omega \gg 1\) is a large parameter and \(f\) is smooth, takes values in \(\mathbb{R}^D\) and is \(2\pi\)-periodic in its fourth argument. A family of heterogeneous multiscale methods for differential equations as above are introduced and discussed. The suggested methodology provides algorithms with arbitrarily high accuracy.