# 1.7 Stiffness and Implicit Methods

## 1.7.5 Apply Newton-Raphson

The value of $$u_1$$ will be (3 significant digits):

Consider the system of ODEs $$\dot{u}_1 = \sin (u_2), \dot{u_2}=\cos (u_1)$$ starting from an initial condition of $$u_1=0, u_2=\pi /2$$. Implement trapezoidal integration with a timestep of $${\Delta t}=0.1.$$ Ensure that the 2-norm of the residual is less than $$10^{-5}$$ to ensure that the Newton-Rhapson iteration has converged. After one timestep

The value of $$u_1$$ will be:

Exercise 1

The value of $$u_2$$ will be:

Exercise 2