Numerical Methods for ODEs

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We have developed a numerical method for solving initial-value problems (IVPs) in ordinary differential equations (ODEs). The method is a simple mix of a classical explicit Runge-Kutta (RK) method, and a Gauss-Legendre (GL) quadrature multistep. This new method, designated RKGL, has the interesting property that its global error is of order r+1, where r is the global order of the underlying RK method. Consequently, RKGL is generally found to be more efficient than RK. We have shown that RKGL is consistent, convergent and strongly stable. Current work concerns the interval of stability and local error control.