# 1.8 Multi-Step 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?

Exercise 1

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