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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 1999 • Question 6

Question

Math 6610 • Analysis of Numerical Methods I Fall 1999 • Question 6

Numerical PDEs Consider the PDE \[ u_t+u_x=0 \tag {5} \] and the usual discretization \[ U_m^n\approx u(x_m,t_n) \qquad \text {where}\qquad x_m=mh\quad \text {and}\quad t_n=nk. \tag {6} \] The Leap Frog Method is given by \[ U_m^{n+1}=U_m^{n-1}-\frac {k}{h}\left (U_{m+1}^n-U_{m-1}^n\right ). \tag {7} \] Compute the local truncation error and discuss the stability of this method.

Raw LaTeX

{Numerical PDEs}
Consider the PDE
\[
  u_t+u_x=0 \tag{5}
\]
and the usual discretization
\[
  U_m^n\approx u(x_m,t_n)
  \qquad \text{where}\qquad x_m=mh\quad \text{and}\quad t_n=nk. \tag{6}
\]
The \textit{Leap Frog Method} is given by
\[
  U_m^{n+1}=U_m^{n-1}-\frac{k}{h}\left(U_{m+1}^n-U_{m-1}^n\right). \tag{7}
\]
Compute the local truncation error and discuss the stability of this method.

Topic Tags

Absolute Stability and A-Stability Consistency, Stability, and Convergence Linear Programming and Optimization Local Truncation Error and Global Error