Study guide: Solving nonlinear ODE and PDE problems

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Linearization by explicit time discretization

Forward Euler method: $$ \frac{u^{n+1} - u^n}{\Delta t} = u^n(1 - u^n)$$

gives a linear algebraic equation for the unknown value $ u^{n+1} $!

Explicit time integration methods will (normally) linearize a nonlinear problem.

Another example: 2nd-order Runge-Kutta method $$ \begin{align*} u^* &= u^n + \Delta t u^n(1 - u^n),\\ u^{n+1} &= u^n + \Delta t \half \left( u^n(1 - u^n) + u^*(1 - u^*)) \right)\tp \end{align*} $$