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The book covers key foundation topics:      o Taylor series methods     o Runge--Kutta methods     o Linear multistep methods     o Convergence     o Stability    and a range of modern themes:    o Adaptive stepsize selection     o Long term dynamics     o Modified equations     o Geometric integration     o Stochastic differential equations    The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. 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