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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6620 • Analysis of Numerical Methods II

Fall 2024 • Question 1

Question

Math 6620 • Analysis of Numerical Methods II Fall 2024 • Question 1

(20 pts) On an equidistant mesh \(x_j = jh\), define the operator \(\widetilde {D}_0\) as, \[ \widetilde {D}_0u(x_j) = \frac {u(x_j+h/2)-u(x_j-h/2)}{h}, \] For a given smoooth \(\kappa \) and \(f\), consider the scheme, \[ -\widetilde {D}_0\left (\kappa (x_j)\widetilde {D}_0u_j\right ) = f(x_j), \] for the ODE \[ -\frac {d}{dx}\left (\kappa (x)\frac {d}{dx}u(x)\right ) = f(x). \] What order is the local truncation error?

Raw LaTeX

(20 pts) On an equidistant mesh \(x_j = jh\), define the operator \(\widetilde{D}_0\) as,
\[
\widetilde{D}_0u(x_j) = \frac{u(x_j+h/2)-u(x_j-h/2)}{h},
\]
For a given smoooth \(\kappa\) and \(f\), consider the scheme,
\[
-\widetilde{D}_0\left(\kappa(x_j)\widetilde{D}_0u_j\right) = f(x_j),
\]
for the ODE
\[
-\frac{d}{dx}\left(\kappa(x)\frac{d}{dx}u(x)\right) = f(x).
\]
What order is the local truncation error?

Topic Tags

Local Truncation Error and Global Error