Minimal residual linear multistep methods

Barys Faleichyk (Belarusian State University)

Consider an initial value problem for the system of ODEs \(y'=f(t,y)\) and suppose that we have \(k\) starting values \(y_0,\ldots,y_{k-1}\) at points \(\{t_j\}\) which are not necessarily equidistant. To compute \(y_k\approx y(t_{k-1}+\tau)\) take an explicit linear multistep method with unknown coefficients: \[y_k=\sum_{j=0}^{k-1}(\tau \beta_j f_j - \alpha_j y_j).\] On the other hand consider the corresponding classic \(p\)-step implicit BDF formula \[c_{k-p} y_{k-p} + \ldots + c_k y_k = \tau f_k,\quad p\leq k.\] In the talk we discuss what happens if on each step of numerical integration the coefficients \(\{\alpha_j, \beta_j\}\) of (1) are chosen to minimize the norm of the residual of method (2). The main focus will lie on the most tractable case of linear problems with \(f(t,y)=A(t) y+b(t)\).