MATH 450/550 Test 1 Spring 2015

Transcription

1 MATH 450/550 Test 1 Spring 015 Note the problems are separated into two sections a set for during class and an additional set to take home and complete that involves some programming. Please read and follow all of the directions. You may use any calculator you have at your disposal (however, not your phone). Show all of your work to get full credit. Answer all questions neatly. Work alone! Please return typed solutions to the take home portion of the test at the start of class on Wednesday March 5th. NAME: 1

2 In Class Questions 1. (10 pts) Explain the difference between a Forward Euler and Backward Euler time stepping technique. What are the advantages and disadvantages of each method. Forward Euler is an explicit time-stepping method where u t + F (u, x, t) = 0 is discretized: u n+1 u n + F n = 0 t A backward Euler time-stepping method is implicit and has a discretization that resembles u n+1 u n + F n+1 = 0 t Forward Euler has a restrictive time step typically having t Ch. The advantage here as the solution can be advanced in time with out a solve and non-linear problems may be approximated easily. The Backward Euler method requires a linear solve and is stable with t Ch, and can advance a solution more quickly by using larger time steps.. (5 pts) If a numerical method is said to be 4th order accurate with respect to the mesh spacing h what does this mean? As the spatial mesh is refined the error in the solution is going at a rate 4 times that of the mesh refinement.

3 3. (15 pts) Given the following governing PDE, and ignoring boundary conditions set up a Forward Euler discretization of the system using second order finite differences for the spatial derivatives: u t + u u x u x = 0 Explain how your system would be advanced in time. Why would using a Backward Euler technique be difficult for this problem. Note here we have Burger s Equation. This has a non-linear transport term. A forward Euler discretization would look like: ( ) ( ) u n+1 u n u + u n n i+1 u n i 1 u n i+1 u n i + u n i 1 = 0 t x x Thus we could advance the solution in time using an algebraic expression for each discrete point on the mesh as follows: ( ) ( ) u u n+1 = u n tu n n i+1 u n i 1 u n + t i+1 u n i + u n i 1 x x The nonlinear term would make the problem difficult. An iterative solver such as Newton s Method would need to be used to advance the solution in time. 3

4 4. (5 pts) Explain the difference between using GMRES to solve a linear system and using Gaussian Elimination. GMRES is an iterative KSP solver that essentially reduces the matrix solve to Grahm- Schmidt orthogonalization of a growing set of basis vectors and applying givens rotations to create a upper-triangular matrix that may be easily back solved. The GMRES algorithm when implemented correctly will often save time and memory especially when used to solve sparse matrix systems like those that arise from discretization of partial differential equations. 5. (5 pts) A numerical method is used to compute an approximation to a partial differential equation with a known analytic solution. The error on a mesh with h = 0.15 is e 06. After a mesh refinement to h = the error between the analytic solution and the approximation is found to be e 07. What is the order of convergence for the solution method used? Note that the mesh is being refined by a factor of. Thus the convergence rate can be found using the familiar: r = ln( e 06/ e 07)/ ln()

5 6. (5 pts) Given the Navier-Stokes equations in dimensional form: ( ) u ρ t + u u + p µ u = f in Ω u = 0 in Ω. Match the proper term from the Navier-Stokes Equations with its roll in the mathematical model (more than one letter may apply to each term). 1. Advective Transport: c a. µ u. Diffusion : a b. f 3. incompressibility: e c. u + u u t 4. Material Derivative: c d. p 5. Conservation of Mass: e e. u = 0 6. Conservation of Momentum: a, b, c, d 7. Sources and Sinks: b 7. (10 pts) Use Taylor s Theorem to find an approximation that can be used for the first derivative. Explain the order of accuracy of the approximation you have found. f(x + h) = f(x) + f (x)h + O(h ) = f f(x + h) f(x) (x) + O(h) h This is a first order approximation for a derivative. 5

6 550 Graduate Problem 1. (10 pts) Use Taylor s Theorem expansions to show that the centered difference formula: is O(h ). f (x) f(x + h) f(x h) h The centered difference formula can be obtained by subtracting the first sequence (f(x + h) form the second sequence (f(x h)). This gives: f (x h) f (x + h) = f (x) f (x) h+ f (x) f (x) f (x) h f (x) h f (x) 3! h f (x) 3! h 3 + f (4) (x) 4! h 3 f (4) (x) 4! h 4 f (5) (x) h 5 + 5! h 4 f (5) (x) h 5 5! f (x h) f (x + h) = f (x) h f (x) h 3 f (5) (x) h 5 3! 5! Swapping terms to different sides we obtain, f (x) h = f (x + h) f (x h) f (x) 3! h 3 f (5) (x) h 5 5! f (x) = f (x + h) f (x h) f (x) h f (5) (x) h 4 h 3! 5! f f (x + h) f (x h) (x) + O(h ) h 6

7 Take Home Questions 450/550 This is the take home portion of the test. Please submit your neatly typed solutions at the start of Class on Wednesday March 5th, 015. (30 pts) Use a Forward Euler discritization technique to solve Burger s equation defined as follows: u t + u u x u c x = f u(0, x) = g(0, x) for x [0, 4] the initial condition. u(t, 0) = g(t, 0) for x = 0 L.H.S. boundary condition u(t, 4) = g(t, 4) for x = 4 R.H.S. boundary condition with g(t, x) = ae (x (b+t)) k were a > 0, b, and k > 0 are fixed real constants. Note that a, b and k can adjust the values of your functions initial profile. When coding it will help to start by setting a = 1.0, b = 0.5, and k = 0.1 This gives a Gaussian pulse of height of 1, centered at 0.5 with a spread set by k = 0.1 or something similar. Set the forcing function f in the governing PDE to be such that the true solution is given by g(t, x) (I would use a computational algebra system when finding the proper forcing function). Set the final time T to be something appropriate for the task at hand (T = ). Use second order finite-differences for both the first and second derivative approximations. Include your code as a listing or multiple listings, and plots of your approximate and true solutions at different times throughout the simulation. What is the order of convergence for the approximation method you have implemented (in time and in space). Do a convergence study that verifies the order of your method. Seethecodepostedtotheclasswebsitef orthis. Take Home Questions 550 (10 pts) Derive the approximation formula f (x) 1 (4f(x + h) 3f(x) f(x + h)) h 7

8 What is the order of accuracy of this formula? In a finite difference code where would an approximation of this form be useful? Here the goal is to use Taylor Series Expansions. f(x + h) = f(x) + f (x)h + f (x)h + O(h 3 ) (0.0.1) f(x + h) = f(x) + f (x)h + f (x)4h + O(h 3 ) (0.0.) Consider the combination of two times (0.0.1) and subtract one half times (0.0.) yielding: Rearrange the terms to obtain: f(x + h) 1 f(x + h) = 3 f(x) + f (x)h + O(h 3 ) f (x) = 3 f(x + h) h h f(x) 1 h f(x + h) + O(h ) Showing that the expression in O(h ). An approximation of this form would be helpful at the boundary of a code to make sure that the code keeps second order with out having information about the solution outside of the computational domain. 8