Weighting parameters for time-step integration algorithms with predetermined coefficients (Q2701970)

The effect of the predetermined coefficients in constructing time-step integration algorithms is studied by the weighting parameter method. First and second order differential equations are considered. It is proved that if there are \(m\) predetermined coefficients in the polynomial form of the approximate solution, additional to the initial conditions, and \(r\) unknown coefficients, then the followings hold. The order of accuracy could be from \(m+r\) to \(m+2r\). For first-order equations, unconditionally stable algorithms are possible if \(m\leq r\) with a maximum order of accuracy \(2r\). For second-order equations, unconditionally stable algorithms are possible if \(m+1\leq r\) with a maximum order of accuracy \(2r\).