solhw11

solhw11 - + f yy f 2 h 2 + 2 f ty fh 2 ) + O ( h 3 )...

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Homework 11 Foundations of Computational Math 2 Spring 2011 Problem 11.1 Consider the Runge Kutta method called the implicit midpoint rule given by: ˆ y 1 = y n 1 + h 2 f 1 f 1 = f ( t n 1 + h 2 , ˆ y 1 ) y n = y n 1 + hf 1 An alternate form of the the method is given by: y n = y n 1 + hf ( t n + t n 1 2 , y n + y n 1 2 ) Show that the two forms are identical. Solution: t n 1 + h 2 = t n 1 + t n - t n 1 2 = t n + t n 1 2 y n = y n 1 + hf 1 f 1 = y n - y n 1 h ˆ y 1 = y n 1 + h 2 f 1 = y n 1 + h 2 y n - y n 1 h = y n + y n 1 2 y n = y n 1 + hf ( t n 1 + h 2 , ˆ y 1 ) = y n 1 + hf ( t n + t n 1 2 , y n + y n 1 2 ) Problem 11.2 Consider the Runge Kutta method called the explicit trapezoidal rule given by: ˆ y 1 = y n 1 + hf ( t n 1 , y n 1 ) y n = y n 1 + h 2 ( f ( t n 1 , y n 1 ) + f ( t n , ˆ y 1 ) ) Show that the method has truncation error O ( h 2 ). Solution: 1
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d n = y ( t n ) - y ( t n 1 ) h - 1 2 ( f ( t n 1 , y n 1 ) + f ( t n , y ( t n 1 ) + hf ( t n 1 , y ( t n 1 )) ) y ( t n ) - y ( t n 1 ) h = y ( t n 1 ) + h 2 y ′′ ( t n 1 ) + h 2 6 y ′′′ ( t n 1 ) + O ( h 3 ) = y + h 2 y ′′ + h 2 6 y ′′′ + O ( h 3 ) where the argument is dropped any time it is at t n 1 . Similarly dropping the argument and letting subscripts indicate partial diferentiation, we have f ( t n , y + hf ) = f + f t h + f y fh + 1 2 ( f tt h 2
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Unformatted text preview: + f yy f 2 h 2 + 2 f ty fh 2 ) + O ( h 3 ) Combining the two expressions yields d n = y ′ + h 2 y ′′ + h 2 6 y ′′′-b f + f t h + f y fh + 1 2 ( f tt h 2 + f yy f 2 h 2 + 2 f ty fh 2 )B + O ( h 3 ) = y ′ + h 2 y ′′ + h 2 6 y ′′′-1 2 b 2 f + h ( f t + f y f ) + h 2 2 ( f tt + f yy f 2 + 2 f ty f )B + O ( h 3 ) = ( y ′-f ) + h 2 ( y ′′-f t-f y f ) + h 2 ( 1 6 y ′′′-1 4 f tt-1 4 f yy f 2-1 2 f ty f ) + O ( h 3 ) = h 2 ( 1 6 y ′′′-1 4 f tt-1 4 f yy f 2-1 2 f ty f ) + O ( h 3 ) = O ( h 2 ) We have used the identities y ′ = f y ′′ = f t + f y f y ′′′ = f tt + 2 f ty f + f y f t + f yy f 2 + f 2 y f 2...
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This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.

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solhw11 - + f yy f 2 h 2 + 2 f ty fh 2 ) + O ( h 3 )...

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