1.8 Multi-Step Methods | 1.8 Multi-Step Methods | Unit 1: Numerical Integration of Ordinary Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

<Backwards Differentiation Methods
1.8.1Adams-Bashforth Methods
1.8.2Adams-Moulton Methods
1.8.3Backwards Differentiation Methods
1.8.4Backwards Differentiation Excercise
>Runge-Kutta Methods

1.8.4 Backwards Differentiation Exercise

Suppose we wish to simulate the ODE \(u_ t = Au\) defined by the 2-by-2 matrix \(A\) for \({\Delta t}=0.1\). For which of the following \(A\) matrices will the fourth-order accurate backwards differentiation method be eigenvalue stable?

\(A = \left[\begin{array}{cc} 0 & 1 \\ -180 & 0 \end{array}\right]\)

\(A = \left[\begin{array}{cc} 0 & 1 \\ -290 & -2 \end{array}\right]\)

\(A = \left[\begin{array}{cc} 0 & 1 \\ -8 & -4 \end{array}\right]\)

\(A = \left[\begin{array}{cc} -5 & 0 \\ 0 & 5 \end{array}\right]\)

Answer: The third matrix is the only one with both eigenvalues inside of the stability region for \({\Delta t}=0.1\).