Problem 1

When updating $u_n$ from $u_{n-1}$ we use Newton iterations,

$$u^1_n = u^0_n - (\nabla g(u^0_n))^{-1}g(u^0_n)$$ (7)

where

$$g(u^0_n) = u^0_n - u_{n-1}-k_nf(u^0_n)$$ (8)