mws_gen_ode_ppt_finitedifference

mws_gen_ode_ppt_finitedifference - FiniteDifferenceMethod

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11/04/11 http://numericalmethods.eng.usf.edu 1 Finite Difference Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM  Undergraduates
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Finite Difference Method      http://numericalmethods.eng.usf.edu
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                                            http://numericalmethods.eng.usf.edu 3 Finite Difference Method An example of a boundary value ordinary differential equation is  " 0030769 . 0 ) 8 ( , " 008731 . 0 ) 5 ( , 0 1 2 2 2 = = = - + u u r u dr du r dr u d The derivatives in such ordinary differential equation are substituted by finite  divided differences approximations, such as  x y y dx dy i i - + 1 ( 29 2 1 1 2 2 2 x y y y dx y d i i i + - - +
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                                            http://numericalmethods.eng.usf.edu 4 Example Take the case of a pressure vessel that is being tested in the laboratory to  check its ability to withstand pressure. For a thick pressure vessel of inner  radius  a  and outer radius  b , the differential equation for the radial  displacement  u  of a point along the thickness is given by  0 1 2 2 2 = - + r u dr du r dr u d The pressure vessel can be modeled as,  ( 29 2 1 1 2 2 2 r u u u dr u d i i i + - - + r u u dr du i i - + 1 Substituting these approximations gives you,  ( 29 0 1 2 2 1 2 1 1 = - - + + - + - + i i i i i i i i r u r u u r r u u u ( 29 ( 29 ( 29 0 1 1 1 2 1 1 1 2 2 2 1 2 = + - - - + + - + i i i i i i u r u r r r r u r r r
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                                            http://numericalmethods.eng.usf.edu 5 Solution Step 1 At node  " 5 , 0 0 = = = a r i " 0038731 . 0 0 = u Step 2   At node  " 6 . 5 6 . 0 5 , 1 0 1 = + = + = = r
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This document was uploaded on 11/04/2011 for the course MME 512 at Miami University.

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mws_gen_ode_ppt_finitedifference - FiniteDifferenceMethod

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