Introduction to PDEs and Numerical Methods: Exam 1

Transcription

1 Prof Dr Thomas Sonar, Institute of Analysis Winter Semester 2003/ Introduction to PDEs and Numerical Methods: Exam 1 To obtain full points explain your solutions thoroughly and self-consistently with all necessary intermediate conclusions and calculation steps as to leave no doubt about the correctness and your understanding Use ball pen or similar tools, no removable ink There are in total 7 questions Manage to fit your solutions into the space within this document Name: Matrikel-Number: Points:

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5 Question 1 10 points A membrane with midpoint 0 and radius 1 satisfies the equation 2 u t 2 = c2 u with x 2 + y 2 < 1 a) Transform the equation to polar coordinates (r, φ) b) Use a separation ansatz of the form to reduce the PDE to three ODE for R, Φ and T u(r, φ, t) = R(r) Φ(φ) T (t)

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9 Question 2 11 points Find the regions in the xy-plane in which the following PDEs are elliptic, parabolic or hyperbolic or of no type a) 2 u y 2 y 2 u t 2 = 0 b) c) d) 2y 2 u x 2 2 u y 2 = x2 y 2 y 3 2 u x u y x + 2 u y 2 = x3 2 u x 2 u 2 u y 2 = K

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13 Question 3 Consider the following (eigenvalue) equation and boundary conditions 10 points d 2 u dx 2 = a2 u ; u(0) = 0 ; u(l) = 0 Assume a is positive and also use the short cut λ = a 2 a) Write down the general solution of this equation Use the boundary conditions to derive an expression for all possible values of λ and with this an expression for the eigenfunctions u(x) which directly follows b) Show that the eigenfunctions u n (x) form the orthogonal basis for a function space by (for example) demonstrating the orthogonal property Help: You might find the identity sin α sin β = 1/2(cos(α β) cos(α + β)) useful

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17 Question 4 13 points θ-method and the heat equation a) Give or derive the matrix form of the iteration scheme as introduced for the θ-method Show that the iteration step can be written as u n+1 = B 1 1 B 2u n = Bu n with the B 1 and B 2 as suitably defined matrices b) Give a formula for the eigenvalues of B using the general formula for the eigenvalues of a tridiagonal matrix (For simplicity of notation you can use the abbreviation r = β 2 t/ x 2 ) d) The Crank-Nicholson scheme is defined by θ = 1/2 State the eigenvalue criterion for the difference scheme to be stable and use b) to derive the condition for r and θ for the Crank-Nicholson scheme

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21 Question 5 18 points Consider the linear advection equation u t + au x = 0 a) Discretise it using the Crank-Nicholson scheme (CN ) defined by θ = 1/2 as a special case of the θ-method Use the forward time approximation and the central difference for the space discretisation u n j+1 un j 1 2 x b) Outline the difference stencil for the above discretisation scheme Explain the meaning of the parameter θ as a weighting factor within the drawing How does the difference stencil change for the extreme values of θ c) Perform a stability analysis for your discretisation via the von-neumann method for the CN scheme to derive stability conditions

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25 Question 6 20 points The equation contains a source term Take f = a u u t β2 2 u x 2 = f a) Discretise this equation in space and time and take a finite difference scheme of your choice b) Show that your discretisation in a) is consistent and give the order of error in space and time c) For t the stationary(=steady) state is reached What is the equation the continuous stationary state must satisfy? Write down the discretised version in matrix form Au = 0 by showing what the matrix A looks like

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29 Question 7 14 points a) State the two general requirements for a difference scheme to be convergent b) Show that the Crank-Nicholson scheme with u j+1 2u j + u j+1 x 2 as the approximation of the second space derivative is convergent for the heat diffusion equation

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