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hmwk_6_2010

# hmwk_6_2010 - UNIVERSITY OF NOTRE DAME Department of Civil...

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UNIVERSITY OF NOTRE DAME Department of Civil Engineering and Geological Sciences CE 30125 November 18, 2010 J.J. Westerink Due: November 30, 2010 Homework Set # 6 Corrections Nov. 23, 2010 Background: Partial differentiation in a discrete context is implemented simply by holding the indices in the directions other than those being differentiated constant. We can then apply all the one-dimensional discrete differentiation formulae. Note that for a three dimensional spatial coordinate system ( x,y,z ), the discrete indices would be ( i,j,k ). Problem 1 Write down a second order accurate approximation to the following three dimensional partial differential equa- tion at node ( i,j,k ) Background: Initial value problems are solved one step at a time (going from one time level to the next time level is called time stepping ). It involves writing a discrete algebraic form of the differential equation to connect two time levels. Starting with the functional value specified as the initial condition, you advance from one level to the next solving for one unknown at a time using the algebraic equation that you have developed. Problem 2 Solve the problem Using a forward Euler method for with time steps equal to , , and . Compare, by plotting the three solutions and comment. Develop your own MATLAB or Fortran code to do this problem. 2 f x 2 ------- 2 f y 2 ------- 2 f z 2 ------- 2 f x y ----------- 2 f y z ----------- + + + + B x y z , , ( ) = dy dt ----- y 2 t 5 y ( ) 5 --------------- = y 0 ( ) 1 = 0 t 4 Δ t 0.2 = Δ t 0.02 = Δ t 0.002 =

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