Exam
6620-2022-08
Fall 2022 · 5 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Suppose \(a < b\) and \(f \in C^3([a,b])\). Consider the interpolation problem: Find a polynomial \(p(x)\) of degree two or less which satisfies the conditions \(p(a) = f(a)\), \(p'(a) = f'(a)\), and \(p(b) = f(b)\).
- Prove that this problem has a solution and that the solution is unique. (You do not need to find the solution.)
- Prove that the error in using \(p(x)\) to approximate \(f(x)\) for \(x \in (a,b)\) is given by the formula \begin{equation} \tag {1} f(x)-p(x)=\frac {f^{(3)}(\xi )}{3!}(x-a)^2(x-b), \end{equation} for some \(\xi \in (a,b)\).
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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 {2} A = A_h + a_2 h^2 + a_4 h^4 + a_6 h^6 + O(h^8). \end{equation}
Here, \(h\) should be thought of as a small positive number, and \(a_1\), \(a_2\), \(a_4\), and \(a_6\) are fixed nonzero, but otherwise unknown, numbers.
- What is the order of the approximation \(A_h\) for \(A\)?
- Use Eq. (2) to derive an approximation \(B_h\) to \(A\) for which the error \(A - B_h = O(h^4)\) and which uses only your program and some simple algebra.
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Question 3
Consider the boundary value problem \[ u''(x) = f(x),\quad \text {for }0 < x < 1, \] \[ u(0) = 0,\quad u(1) = 0, \]
where \(f\) is a smooth function on \([0, 1]\), and the finite difference scheme \[ \frac {U_{j-1}-2U_j+U_{j+1}}{h^2} = f(x_j), \qquad j = 1,2,\ldots ,m, \] for finding approximations \(U_j\) to \(u(x_j)\). Here, \(x_j = jh\) and \((m+1)h = 1\).
Analyze the accuracy, stability, and convergence properties of the scheme.
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Question 4
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.
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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.
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Question 5
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Consider the initial value problem for the constant-coefficient diffusion equation (with \(\beta > 0\)) \[ v_t = \beta v_{xx},\quad t > 0 \]
with initial data \(v(x,0) = f(x)\). A scheme for this problem is: \[ \frac {u_j^{n+1}-u_j^n}{k} = \frac {\beta }{h^2}\left \{u_{j-1}^{n+1}-2u_j^{n+1}+u_{j+1}^{n+1}\right \}. \]
Analyze the 2-norm stability of this scheme. For which values of \(k > 0\) and \(h > 0\) is the scheme stable? (Note that there are no boundary conditions here.)
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Consider the variable coefficient diffusion equation \[ v_t = (\beta (x)v_x)_x,\quad 0 < x < 1,\quad t > 0 \]
with Dirichlet boundary conditions \[ v(0,t) = 0,\qquad v(1,t) = 0 \]
and initial data \(v(x,0) = f(x)\). Assume that \(\beta (x) \ge \beta _0 > 0\), and that \(\beta (x)\) is smooth. Let \(\beta _{j+1/2} = \beta (x_{j+1/2})\). A scheme for this problem is: \[ \frac {u_j^{n+1}-u_j^n}{k} = \frac {1}{h^2}\left \{\beta _{j-1/2}u_{j-1}^{n+1} -(\beta _{j-1/2}+\beta _{j+1/2})u_j^{n+1} +\beta _{j+1/2}u_{j+1}^{n+1}\right \}. \]
Analyze the 2-norm stability of this scheme for solving this initial boundary value problem. DO NOT NEGLECT THE FACT THAT THERE ARE BOUNDARY CONDITIONS!
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