Exam
6620-2022-05
Summer 2022 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
-
Question 1
Suppose \(f(x) \in C^{n+1}([a,b])\), and consider the interpolation problem: For \(a \le x_0 < x_1 <\) \(x_2 < \cdots < x_{n-1} < x_n \le b\), find a polynomial \(p(x)\) of degree at most \(n\) for which \(p(x_j) = f(x_j)\), \(j = 0,1,\ldots ,n\).
- Show that this problem has at most one solution.
- Show that this problem has a solution.
- Show, for any \(x \in [a,b]\), that \[ f(x)-p(x)=\frac {f^{n+1}(\xi )}{(n+1)!}(x-x_0)(x-x_1)\cdots (x-x_n), \] for some \(\xi \in [a,b]\).
-
Question 2
Suppose you have a program for calculating approximations to a quantity \(A\) which takes a value of \(h\) as input and produces a value \(A_h\) as output. Suppose that \begin{equation} \tag {1} A = A_h + a_1 h + a_2 h^2 + a_3 h^3 + a_4 h^4 + O(h^5). \end{equation}
Here, \(h\) should be thought of as a small positive number, and \(a_1\), \(a_2\), \(a_3\), and \(a_4\) are fixed (but unknown) numbers.
- What is the order of the approximation \(A_h\) for \(A\)?
- Use Eq. (1) to derive an approximation \(B_h\) to \(A\) for which the error \(A - B_h = O(h^3)\) and which uses only your program and some simple algebra.
-
Question 3
Consider the 9-point discrete Laplacian \[ (\Delta ^h_9 U)_{j,l} = \frac {1}{6h^2}\left ( U_{j-1,l-1}+U_{j+1,l-1}+U_{j-1,l+1}+U_{j+1,l+1} +4U_{j,l-1}+4U_{j-1,l}+4U_{j+1,l}+4U_{j,l+1}-20U_{j,l} \right ) \]
- State and prove a maximum principle for \(\Delta ^h_9 U\). (Hint: Think about what the key observation is in stating and proving a maximum principle for the standard 5-point discrete Laplacian.)
- Show that the following problem has a solution and that it is unique: \begin{equation} \tag {2} \begin {aligned} \frac {1}{6h^2}\big (&U_{j-1,l-1}+U_{j+1,l-1}+U_{j-1,l+1}+U_{j+1,l+1}\\ &+4U_{j,l-1}+4U_{j-1,l}+4U_{j+1,l}+4U_{j,l+1}-20U_{j,l}\big ) = f_{j,l}, \end {aligned} \end{equation} for \(j = 1,\ldots ,m\) and \(l = 1,\ldots ,m\) (where \((m+1)h = 1\)) along with the conditions \(U_{j,l} = 0\) at grid points on the boundary of the domain.
- The local trucation error in using the scheme in Eq. (2) to find an approximate solution of \(\Delta u = f\) for \(0 < x < 1\) and \(0 < y < 1\) with homogeneous Dirichlet boundary conditions is given by \[ \tau _{j,l} = \frac {1}{12}(u_{xxxx}+u_{xxyy}+u_{yyyy})h^2 + O(h^4), \] where the derivatives of \(u\) are evaluated at \(x_j = jh\), \(y_l = lh\). Prove that the values \(U_{j,l}\) obtained from Equations (2), along with zero discrete boundary values, converge to the values \(u(x_j,y_l)\) of the true solution to the PDE BVP as \(h \to 0\). (Hint: Use the relation between the set of global error values \(E_{j,l} = U_{j,l} - u(x_j,y_l)\) and the values of the local truncation error.)
-
Question 4
Suppose you are trying find an approximation \(u(t)\) to a function \(v(t)\) and you know that the following two equations hold:
\[ u(t_{n+1}) = C(k)u(t_n) \]\[ v(t_{n+1}) = C(k)v(t_n) + k\tau (t_n) \]
for some smooth function \(\tau (t,k)\) that is not identically 0. Here, \(k > 0\), \(t_n = nk\), and \(C(k)\) is a linear operator which depends on \(k\) but not on \(u(t)\) or \(v(t)\). State and prove conditions on \(C(k)\) and \(\tau (t,k)\) which are sufficient to imply that \[ \lim _{k \to 0,\ nk=t_n=t} u(t_n) = v(t). \]
-
Question 5
For the initial value problem \(u'(t) = f(u(t),t)\), \(u(0) = \eta \), where \(f(u,t)\) is continuous with respect to \(t\) and Lipschitz continuous with respect to \(u\), consider the scheme \[ \frac {U^{n+2}-\frac {4}{3}U^{n+1}+\frac {1}{3}U^n}{k} = \frac {2}{3}f(U^{n+2},t_{n+2}), \] where \(t_n = nk\), and \(U^n\) is supposed to approximate \(u(t_n)\).
- Analyze the consistency, zero-stability, and convergence of this scheme.
-
If you apply this scheme to the initial value problem
\[ u'(t) = -10^{12}(u(t)-\cos (t))-\sin (t),\qquad u(0) = 2, \]what issues should you consider in choosing the timestep \(k\)? Are there time intervals in which \(k\) must be very small to get a reasonable solution and others in which it does not? Explain your answers.