If you chose the step size appropriately to ensure the exact same number of

If you chose the step size appropriately to ensure

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If you chose the step-size appropriately to ensure the exact same number of function evaluations, RK4 will quite likely give a far more accurate solution than Euler’s method for the same amount of “work”. So yes, high-order methods are worth it. Another important fact is that, for all methods, additional effort will tend to lead to dimin- ishing returns. This means that at some point performing additional function evaluations (making the step-size smaller) will not give a significant improvement in accuracy. Ian Jeffrey 20-7
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Lecture 20: Higher-Order Runge-Kutta Methods March 26, 2019 20.5 Systems of Ordinary Differential Equations Many engineering problems involve solving systems of ordinary differential equations. We will consider systems of first-order ODEs. A general system of first-order ODEs is: dy 1 dx = f 1 ( x, y 1 , y 2 , . . . , y n ) dy 2 dx = f 2 ( x, y 1 , y 2 , . . . , y n ) . . . dy n dx = f n ( x, y 1 , y 2 , . . . , y n ) Here we must stress that the y j are not different sample values but are instead different dependent variables (we could have used u, v, w etc., but would have quickly run out of letters). Notice also that the right-hand-side function changes as denoted by the subscript. Key Concept : What is so important about systems of ODEs? Many engineering appli- cations require them directly (think of a circuit with multiple capacitors/inductors). Also, high-order ODEs can be reduced to systems of first-order ODEs which makes their numer- ical solution quite convenient. Example : Consider Newton’s Second Law in one dimension F = ma This is equivalent to the second-order linear ODE: d 2 x dt 2 = F m which can be decomposed into two first order ODEs: dv dt = F m dx dt = v where v is the velocity and x is the position of the mass. Solving for the positions of the planets in our solar system would extend these equations to two dimensions (assuming planets revolve in a common plane). Ian Jeffrey 20-8
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Lecture 20: Higher-Order Runge-Kutta Methods March 26, 2019 Example : Consider the following third-order ODE d 3 y dx 3 = 2 x - 3 y + 4 dy dx + x d 2 y dx 2 To solve this ODE IVP we need 3 initial conditions, for example: y (0) = 3, dy dx x =0 = 2 and d 2 y dx 2 x =0 = 7 (the actual values aren’t important for our purposes). One way of solving this third-order ODE is to reduce it to a system of 3 first-order ODEs. We let: w 1 = dy dx , w 1 (0) = 2 w 2 = dw 1 dx = , d 2 y dx 2 w 2 (0) = 7 Substituting these relationships into the original ODE gives: dy dx = w 1 dw 1 dx = w 2 dw 2 dx = 2 x - 3 y + 4 w 1 + xw 2 where, in the last equation we have used the fact that dw 2 dx = d 3 y dx 3 . The result is just a system of first-order ODE IVPs which can be solved using standard RK methods. Key Concept : Runge-Kutta methods can be applied to systems of first-order ODEs in a straightforward manner that simply extends our usual application of RK methods for a single ODE.
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  • Winter '16
  • Ian Jeffery
  • Numerical Analysis, Yi, Midpoint method, Runge–Kutta methods

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