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Unformatted text preview: expansions, but they take intermediate (extrapolated) steps in which the function is evaluated appropriately chosen times BETWEEN the time time steps (dt). The derivation is rather long and not too instructive, we’ll just quote results and use them. The 4 th order RungeKutta is the standard method for solving ODEs – not necessarily the fastest or most accurate, but it is quite stable. 2 nd order RK is basically just a type of Euler method and not very commonly used. 4 th order RK is the standard Assume we have an ODE of the form Truncation error for 4 th order RK is dx dt = f ( x ( t ) , t ) O ( τ 5 ) Requires 4 intermediate evaluations of the function f(x): F 1 = f ( x, t ) F 2 = f ( x + 1 2 dtF 1 , t + 1 2 dt ) F 3 = f ( x + 1 2 dtF 2 , t + 1 2 dt ) F 4 = f ( x + dtF 3 , t + dt ) x ( t + dt ) = x ( t ) + dt 6 [ F 1 + 2 F 2 + 2 F 3 + F 4 ] Equation to solve: dx dt = − x...
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 Spring '09
 Gladden
 Derivative, Taylor Series, RK, Numerical ordinary differential equations, Lotka–Volterra equation

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