UNIVERSITY OF NOTRE DAMEDepartment of Civil Engineeringand Geological SciencesCE 30125November 18, 2010J.J. WesterinkDue: November 30, 2010Homework Set # 6 Corrections Nov. 23, 2010Background:Partial differentiation in a discrete context is implemented simply by holding the indices in thedirections other than those being differentiated constant. We can then apply all the one-dimensional discretedifferentiation formulae. Note that for a three dimensional spatial coordinate system (x,y,z), the discrete indiceswould be (i,j,k).Problem 1Write 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 timelevel is called time stepping). It involves writing a discrete algebraic form of the differential equation toconnect two time levels. Starting with the functional value specified as the initial condition, you advance fromone level to the next solving for one unknown at a time using the algebraic equation that you have developed. Problem 2Solve the problemUsing 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 thisproblem.∂2f∂x2-------∂2f∂y2-------∂2f∂z2-------∂2f∂x∂y-----------∂2f∂y∂z-----------++++B x y z,,()=dydt-----y2t5y–()5---------------=y0()1=0t4≤≤Δt0.2=Δt0.02=Δt0.002=
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