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A Jacobi waveform relaxation method for solving initial value problems for ordinary di erential equations (ODEs) is presented. In each window the method uses a technique called dynamic tting and a pair of continuous Runge-Kutta formulas to produce the initial waveform, after which a xed number of waveform iterates are computed. The reliability and e cacy of the method is demonstrated numerically by applying it to qualitatively di erent problems for linear tridiagonal ODEs. Key words. initialdoi:10.1137/s1064827596306562 fatcat:hg4qc5b5mffwtasscvxwv2jrpa