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Runge Kutta methods achieve better results than Euler by using intermediate computations at intermediate time values
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The fourth-order rule is the favorite method as it achieves good accuracy with modest computational complexity -- the algorithm is in words:
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Use derivative of first time step to get trial midpoint
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Use its derivative at first time step to get second trial midpoint
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Use its derivative to get a trial end point
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Integrate by Simpon's Rule, using average of two midpoint estimates
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Global error is fourth order
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