Test Questions on A-stability of Numerical Integration Schemes

📚 Quick Study Guide

• 📈 A-Stability Definition: A numerical method is A-stable if its region of absolute stability contains the entire left half-plane, i.e., all complex numbers $z$ with $Re(z) < 0$.
• 🧭 Implications: A-stability implies that the numerical solution remains bounded for any linear test problem of the form $y' = \lambda y$, where $Re(\lambda) < 0$.
• 💡 Linear Test Problem: The linear test problem is given by $y' = \lambda y$, where $\lambda$ is a complex number. The stability of a method is often analyzed using this test problem.
• 🎯 Region of Absolute Stability: This is the set of complex numbers $z = h\lambda$ for which the numerical solution remains bounded as $n \rightarrow \infty$, where $h$ is the step size.
• ⭐ Examples: Backward Euler method is A-stable, while Forward Euler is not. Some Runge-Kutta methods can be A-stable depending on their coefficients.
• 📝 Importance: A-stability is particularly important when solving stiff ordinary differential equations (ODEs), where explicit methods may require extremely small step sizes to maintain stability.
• 🧮 Stiff ODEs: Stiff ODEs are problems where some components of the solution decay much faster than others. A-stable methods are preferred for solving these problems.

Practice Quiz

1. Which of the following best describes A-stability?

   1. A) The numerical solution converges to the exact solution as the step size approaches zero.
   2. B) The region of absolute stability contains the entire left half-plane.
   3. C) The method is stable for all step sizes.
   4. D) The method is only stable for certain types of differential equations.

2. For the linear test problem $y' = \lambda y$, what condition on $\lambda$ is necessary for A-stability to guarantee bounded solutions?

   1. A) $Re(\lambda) > 0$
   2. B) $Re(\lambda) < 0$
   3. C) $Im(\lambda) > 0$
   4. D) $Im(\lambda) < 0$

3. Which of the following methods is A-stable?

   1. A) Forward Euler
   2. B) Backward Euler
   3. C) Explicit Runge-Kutta method
   4. D) Leapfrog method

4. What is the primary reason A-stability is important?

   1. A) It simplifies the implementation of numerical methods.
   2. B) It guarantees high accuracy for all ODEs.
   3. C) It ensures stability when solving stiff ODEs.
   4. D) It reduces the computational cost of solving ODEs.

5. The region of absolute stability is a set of complex numbers $z = h\lambda$. What do $h$ and $\lambda$ represent?

   1. A) $h$ is the step size, and $\lambda$ is the eigenvalue of the Jacobian matrix.
   2. B) $h$ is the eigenvalue of the Jacobian matrix, and $\lambda$ is the step size.
   3. C) $h$ is the initial condition, and $\lambda$ is the step size.
   4. D) $h$ is the final time, and $\lambda$ is the initial condition.

6. Which type of ODE benefits most from using an A-stable numerical method?

   1. A) Linear ODEs
   2. B) Nonlinear ODEs
   3. C) Stiff ODEs
   4. D) Autonomous ODEs

7. What is a key characteristic of stiff ODEs?

   1. A) All components of the solution decay at the same rate.
   2. B) Some components of the solution decay much faster than others.
   3. C) The solution is always oscillatory.
   4. D) The solution is always constant.