Peer methods for the numerical solution of general second order IVPs
Luis Randez (University of Zaragoza), Manuel Calvo, Juan Ignacio Montijano
In this work, we solve numerically general second order initial vale problems \(y''=f(t, y, y')\) by means of explicit two-step Peer methods, given by \[\begin{array}{l} Y_{m+1}= B Y_m + h A Z_m + h^2 Q F_{m-1} + h^2 R F_m, \\[6pt] Z_{m+1}= \widehat{B} Z_m + h \widehat{Q} F_{m-1} + h \widehat{R} F_m, \end{array}\] where the stage vectors evaluated at \(t_{mi}=t_m + c_i h\) are \[\begin{array}{l} Y_{m}= \left( Y_{mi}\right), \textrm{ where } Y_{mi} \simeq y(t_{mi}), \\[6pt] Z_{m}= \left( Z_{mi}\right), \textrm{ where } Z_{mi} \simeq y'(t_{mi}), \\[6pt] F_{m}= \left( f(t_{mi}, Y_{mi}, Z_{mi}) \right). \end{array}\]
We propose explicit Peer methods with minimum number of effective function evaluations per step. We analyze the 0-stability, consistency and convergence of these schemes.
S. Jebens, R. Weiner, H. Podhaisky, B.A. Schmitt. Explicit multi-step peer methods for special second-order differential equations. Applied Mathematics and Computation 202 803–813, 2008