A family of implicit Chebyshev methods for the numerical integration of second-order differential equations

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A family of implicit methods based on intra-step Chebyshev interpolation is developed for the solution of initial-value problems whose differential equations are of the special second-order form y ″ = f ( y ( x); x). The general procedure allows stepsizes which are considerably larger than commonly used in conventional methods. Computation overhead is comparable to that required by high-order single or multistep procedures. In addition, the iterative nature of the method substantially reduces local errors while maintaining a low rate of global error growth.