NUMDIFF17

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 $\begin{array}{l} Y_{m + 1} = B Y_{m} + h A Z_{m} + h^{2} Q F_{m - 1} + h^{2} R F_{m}, \\ Z_{m + 1} = \hat{B} Z_{m} + h \hat{Q} F_{m - 1} + h \hat{R} F_{m}, \end{array}$ where the stage vectors evaluated at $t_{m i} = t_{m} + c_{i} h$ are $\begin{array}{l} Y_{m} = (Y_{m i}), where Y_{m i} ≃ y (t_{m i}), \\ Z_{m} = (Z_{m i}), where Z_{m i} ≃ y^{'} (t_{m i}), \\ F_{m} = (f (t_{m i}, Y_{m i}, Z_{m i})) . \end{array}$

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.

References

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

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