Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Jan07 667

Problem 4 Let , suppose and assume is nonsingular . Consider the following iteration Problem 4a Derive the following error equation for Solution 4a Note the following identity The error is given by Problem 4b Let be a fixed matrix. Find conditions on B that guarentee local convergence. What rate of convergence do you expect and why? Solution 4b Assume is invertible, is bounded, and is Lipschitz. This implies local superlinear convergence. Problem 4c Find sufficient conditions on for the convergence to be superlinear. What choice of corresponds to the Newton method and what rate of convergence do you expect? Solution 4c Super linear convergence as Find condition for super linear convergence From part(b) Since , if as , we have super linear convergence i.e. Problem 5 Let be uniformly Lipschitz with respect to . Let be the solution to the initial value problem : . Consider the trapezoid method . Problem 5a Find a condition on the stepsize that ensures (1) can be solved uniquely for . Solution 5a The implicit method can be viewed as a fix point iteration: We want which implies Problem 5b Define a local truncation error and estimate it. Examine the additional regularity of needed for this estimate. Solution 5b Re-writing (1) and replacing we have a formula for consistency of order p: For uniform stepsize h Expanding in Taylor Series about gives Problem 5c Prove a global error estimate for (1) Solution 5c Problem 6 Consider the 2-point boundary value problem , with constants and . Let be a uniform partition of [0,1] with meshsize . Problem 6a Use centered finite differences to discretize (2). Write the system as and identify . Prove that A is nonsingular. Solution 6a Using Taylor Expansions, we can approximate the second derivative as follows We can eliminate two equations from the n+2 equations by substituting the initial conditions into the equations for and respectively. We then have the system is nonsingular since it is diagonally dominant. Problem 6b Define truncation error and derive a bound for this method in terms of . State without proof an error estimate of the form and specify the order s. Solution 6b Local Truncation Error Bound for Local Truncation Error Derive Bound for Max Error Let , , and is the local truncation error. Then Subtracting the two last equations gives Hence, , that is the error has order 2. Problem 6c Prove the following discrete monotonicity property: If is the solution corresponding to a forcing , for , and then componentwise. Solution 6c Note that is a matrix and hence the discrete maximum principle applies. (See January 05 667 test for definition of matrix) Discrete Maximum Principle If , then . Specifically let , then which implies