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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2017 • Question 5

Question

Math 6610 • Analysis of Numerical Methods I Fall 2017 • Question 5

Unstable Multistep Method

Consider the numerical method \[ y_{p+1}=3y_p-2y_{p-1} +\frac {h}{2}\left (f(x_p,y_p)-3f(x_{p-1},y_{p-1})\right ), \qquad p\ge 1, \]

for the solution of the initial value problem \[ y'=f(x,y),\qquad y(x_0)=y_0. \] Illustrate with an example of a simple initial value problem that the above numerical scheme is unstable.

Raw LaTeX

{Unstable Multistep Method}

Consider the numerical method
\[
  y_{p+1}=3y_p-2y_{p-1}
  +\frac{h}{2}\left(f(x_p,y_p)-3f(x_{p-1},y_{p-1})\right),
  \qquad p\ge1,
\]

\newpage

for the solution of the initial value problem
\[
  y'=f(x,y),\qquad y(x_0)=y_0.
\]
Illustrate with an example of a simple initial value problem that the above
numerical scheme is unstable.

Topic Tags

Linear Multistep Methods Linear Programming and Optimization Numerical Differentiation and Finite Difference Operators Numerical Quadrature