HW 5 Problem 1: Perform Von Neumann stability analysis of Crank Nicolson Method on 1D diffusion equation. Problem 2: For the convection-diffusion equation, the steady case reads !c∇$ − &Δ$ = 0$(0) = $,$(-) = 0 The dimensionless Peclet number can be defined ./ = 012 . If we normalize $ by $, and 4 by -, the nondimensional equation becomes ⎩⎨ ⎧∇$ − 1./ Δ$ = 0$(0) = 1$(1) = 0 1) Derive the analytical solution of the nondimensional equation and express it in terms of 4 and ./ 2) Plot the analytical solution with ./ = 1, 10 and 100. Comment on the plot. 3) Use central difference for both convective and diffusive terms and get the numerical solutions for the nondimensional equation. Based on the mesh length Δ4 (uniform mesh is assumed), the local Peclet is defined as .:/ = 0;<2 = =;<=/?@ =Δ4./. Solve the non-dimensional equation with ./ = 10 and 100 with different .:/ from very large to very small. Plot the numerical solutions against the analytical solutions. What do you see when .:/ is very big? What do you see when .:/ is very small? 4) Try your best to establish a rigorous criterion on how small .:/ needs to be in order to get rid of unphysical oscillatory solution. 5) Now use upwind for convection and central difference for diffusion. Repeat the analysis for the non-dimensional equation. Does it relax the requirement of .:/? 6) Let’s make the equation dynamical. Pick any combination of time integration, spatial discretization, and linear solvers you learned so far to solve the following PDEs numerically. Plot the solution at three time instances (you decide when). Do you like your method choice and your results? ⎩⎪⎨ ⎪⎧B$BC + ∇$ − 1./ Δ$ = 0$(C, 0) = 1$(C, E) = −1$(0, 4) = cos (4) o iO o 肛柴 TT ⼆千⼏
