L53 - Fall 2003 Math 308/501502 Numerical Methods 3.6, 3.7,...

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Fall 2003 Math 308/501–502 Numerical Methods 3.6, 3.7, 5.3 Runge-Kutta Methods Mon, 27/Oct c ± 2003, Art Belmonte Summary Geometrical idea Runge-Kutta methods numerically approximate the solution of y 0 = f ( t , y ), y ( a ) = y 0 by using weighted averages of slopes near a point instead of the single slope involved by following the tangent line at a point. These methods are much more accurate than Euler’s method. The second-order Runge-Kutta (RK2) method Let [ a , b ] be the interval over which an approximation to the solution is desired. (Thus t = a and t = b are the initial and final values of the independent variable, respectively.) Partition this interval into N subintervals each of length h = ( b - a )/ N , called the step size .Le t t 0 = a and define t k + 1 = t k + h , for k = 0 , 1 ,..., N - 1 . Notice that t N = b and the other t k so-defined are the interior endpoints of the subintervals. These collectively are the discrete values of the independent variable. The initial value of the dependent variable is given by the initial condition, y ( a ) = y ( t 0 ) = y 0 . The other discrete dependent variable values are computed iteratively as follows. for k = 0to N - 1 s 1 = f ( t k , y k ) s 2 = f ( t k + h , y k + hs 1 ) t k + 1 = t k + h y k + 1 = y k + h s 1 + s 2 2 This method is a single-step numerical solver since it depends only on data obtained from the preceding step. It is a fixed-step solver since the lengths of the subintervals of [ a , b ] are all equal. The maximum error in the approximation over the interval [ a , b ] satisfies the following upper bound. maximum error M L ± e L ( b - a ) - 1 ² h 2 Here L = max ( t , y ) R ³ ³ ³ ³ f y ³ ³ ³ ³ where R is a rectangle containing the solution curve. This constant is the same as in Euler’s method. The constant M is different than that in Euler’s method, but it also depends only on the values of f and some of its derivatives over R . The fact that h occurs to the second power in the error bound is the reason that we say that this method is a second-order method. The fourth-order Runge-Kutta (RK4) method Let the independent variable values t k be as above. The initial value of the dependent variable is given by the initial condition, y ( a ) = y ( t 0 ) = y 0 . The other discrete dependent variable values are computed iteratively as follows. for k = N - 1 s 1 = f ( t k , y k ) s 2 = f ´ t k + h 2 , y k + h 2 s 1 µ s 3 = f ´ t k + h 2 , y k + h 2 s 2 µ s 4 = f ( t k + h , y k + 3 ) t k + 1 = t k + h y k + 1 = y k + h s 1 + 2 s 2 + 2 s 3 + s 4 6 This method is also a single-step numerical solver since it depends only on data obtained from the preceding step. It is a fixed-step solver since the lengths of the subintervals of [ a , b ]are all equal. The maximum error in the approximation over the interval [ a , b ] satisfies the following upper bound. maximum error M L ± e L ( b - a ) - 1 ² h 4 Here L = max ( t , y ) R ³ ³ ³ ³ f y ³ ³ ³ ³ where R is a rectangle containing the solution curve. This constant is the same as in Euler’s method.
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L53 - Fall 2003 Math 308/501502 Numerical Methods 3.6, 3.7,...

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