Ch4-3 (1) - Chapter 4 Numerical Solution of Initial-Value Problems 4.3 Runge-Kutta Method The Runge-Kutta Method is an explicit method that achieves the

Ch4-3 (1) - Chapter 4 Numerical Solution of Initial-Value...

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Chapter 4 Numerical Solution of Initial-Value Problems 4.3 Runge-Kutta Method The Runge-Kutta Method is an explicit method that achieves the accuracy of a Taylor series approach without requiring the calculation of higher derivatives. The general form of Runge- Kutta method can be written as y n+1 = y n + ( x n , y n , h ) h (4.3-1) where ( x n , y n , h ) is the increment function that represents an average slope over the interval h . ( x n , y n , h ) h = a 1 k 1 + a 2 k 2 + + a m k m (4.3-2) In the above expression k 1 / h is the slope at x 1 , k 2 / h is the slope at x 2 , and k m / h is the slope at x m for which x 1 < x 2 < x m x 1 + h . With this definition, the first order Runge-Kutta method is just Euler’s method. ( x n , y n , h ) h = a 1 k 1 = (1) h f ( x n , y n ) where a 1 = 1 and k 1 = h f ( x n , y n ) The second order Runge-Kutta method is written as y n+1 = y n + ak 1 + bk 2 (4.3-3) where k 1 = h f ( x n , y n ) and k 2 = h f ( x n + h , y n + k 1 ). The four constants a , b , , and can be determined by comparing equation (4.3-3) with Taylor’s method of second order. y n+1 = y n + f ( x n , y n ) h + ! 2 1 f ’( x n , y n ) h 2 where f ’( x n , y n ) = dx df = x f dx dx + y f dx dy = x f + y f dx dy Therefore y n+1 = y n + f ( x n , y n ) h + ! 2 1 [ x f + y f dx dy ] h 2 (4.3-4) From Taylor series expansion of a function with two variables g ( x + r , y + s ) = g ( x , y ) + r x g + s y g + 4
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The constant k 2 = h f ( x n + h , y n + h ) can be expressed as k 2 = h { f ( x n , y n ) + h x f + k 1 y f + O ( h 2 ) } Substitutions of k 1 and k 2 into y n+1 = y n + ak 1 + bk 2 yields y n+1 = y n + a hf ( x n , y n ) + b hf ( x n , y n ) + b h 2 x f + b h 2 f ( x n , y n ) y f + O ( h 3 ) The equation can be rearranged to y n+1 = y n + [ a + b ] hf ( x n , y n ) + [ b x f + b f ( x n , y n ) y f ] h 2 + O ( h 3 ) (4.3-5) Compare equation (4.3-5) with equation (4.3-4) y n+1 = y n + f ( x n , y n ) h + 2 1 [ x f + y
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  • Spring '06
  • Staff
  • Economics, Numerical differential equations, Runge–Kutta methods, Numerical ordinary differential equations, 2k2

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