A special linear multi-step method for special second-order differential equations (Q2914794)

This paper deals with the numerical solution of initial value problems for special second-order equations \( y'' = f(x, y)\) by means of some combinations of linear multistep methods. Assuming that on a uniform grid \( x_j = x_0 + j h\), \( j=0,1, \ldots \) with step size \(h\) the numerical solution \( y_j \simeq y(x_j)\) has been computed up to the grid point \( x_{n-k+1}\) to obtain the new \( y_{n+k} \simeq y(x_{n+k})\), the authors propose the following three-stage algorithm: First, an initial approximation \( y_{n+k}^{(1)}\) is determined from the linear \(k\)-step formula of order \((k-1)\) with the form NEWLINE\[NEWLINE \sum_{j=0}^k \alpha_j y_{n+j} = h^2 \beta_k f(x_{n+k}, y_{n+k}).\tag{1}NEWLINE\]NEWLINE Then, by using this value, an advanced value \( y_{n+k+1}^{(1)}\) is determined with the same formula for the next step. Finally, the value \( y_{n+k}\) is corrected from the \(k\)-th order formula of type NEWLINE\[NEWLINE \sum_{j=0}^k \widehat{\alpha}_j y_{n+j} = h^2 \left( \widehat{\beta}_k f(x_{n+k}, y_{n+k}) + \widehat{\beta}_{k+1} f(x_{n+k-1}, y_{n+k-1}^{(1)}) \right).NEWLINE\]NEWLINE With such a forward-backward combination of multistep formulas the authors show that the proposed methods have larger stability regions than the methods (1).