Numerical methods for solving first-order IVPs often fall into one of two large categories : linear multistep methods, or Runge-Kutta methods.
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The Taylor expansion is used below to analyze the error committed by the Euler method, and it can be extended to produce Runge Kutta methods.
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And solved using, for example, the Livermore predictor-corrector method, the Rosenbrock methods or the 4th / 5th-order Runge Kutta method.
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What is the highest possible order of an explicit Runge-Kutta method ?-- talk ) 20 : 27, 27 September 2011 ( UTC)
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What is the highest order of a known explicit Runge-Kutta method ?-- talk ) 12 : 58, 28 September 2011 ( UTC)
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If you're lucky, there's a closed form solution, otherwise use your favorite numerical approximation such as the Runge-Kutta method.
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However, the simplest adaptive Runge Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1.
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For this reason, people usually employ alternative, higher-order methods such as Runge Kutta methods or linear multistep methods, especially if a high accuracy is desired.
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The modified midpoint method by itself is a second-order method, and therefore generally inferior to fourth-order methods like the fourth-order Runge Kutta method.
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He was co-developer and co-eponym of the Runge Kutta method ( German pronunciation : ), in the field of what is today known as numerical analysis.
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