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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Spring 2006 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Spring 2006 • Question 4

ODE Consider the initial value problem for a system of ODEs \[ y'=Ay \] where \(A\) has eigenvalues ranging from \(-10^4\) to \(-1\). Which of the following schemes would be the best choice for solving this problem? Justify your answer in terms of stability, accuracy, and efficiency. \[ y^{n+1}=y^n+hAy^n, \tag {1} \] \[ y^{n+1}=y^n+hAy^{n+1}, \tag {2} \] \[ y^{n+1}=y^n+\frac {h}{2}(Ay^n+Ay^{n+1}). \tag {3} \]

Raw LaTeX

{ODE} Consider the initial value problem for a system of ODEs
\[
  y'=Ay
\]
where \(A\) has eigenvalues ranging from \(-10^4\) to \(-1\). Which of the following
schemes would be the best choice for solving this problem? Justify your answer in
terms of stability, accuracy, and efficiency.
\[
  y^{n+1}=y^n+hAy^n, \tag{1}
\]
\[
  y^{n+1}=y^n+hAy^{n+1}, \tag{2}
\]
\[
  y^{n+1}=y^n+\frac{h}{2}(Ay^n+Ay^{n+1}). \tag{3}
\]

Topic Tags

Absolute Stability and A-Stability Consistency, Stability, and Convergence Eigenvalues, Spectral Radius, and Gershgorin Theorem Linear Programming and Optimization