Convergence of LMM when the solution is not smooth (Q1824999)

For \(y'=f(x,y),\quad a\leq x\leq b,\quad y(a)=y_ 0,\) a stable linear multistep method (LMM) which is of order p for a smooth problem is applied supposing that the solution y(x) only has q continuous derivatives instead of the usual \(p+1\) [see: \textit{P. Henrici}, Discrete variable methods in ordinary differential equations (1962; Zbl 0112.349)]. By a modification of Henrici's proof the order of convergence is reduced to min(q-1,p). If f(x,y) and y(x) are supposed piecewise smooth, the order is \(\min (q+1,p)\).