$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
Explicit time integration
Explicit time integration methods remove the nonlinearity
Forward Euler method:
$$ [D_t^+ u = \nabla\cdot (\dfc(u)\nabla u) + f(u)]^n$$
$$ \frac{u^{n+1} - u^n}{\Delta t} = \nabla\cdot (\dfc(u^n)\nabla u^n)
+ f(u^n)$$
This is a linear equation in the unknown \( u^{n+1}(\x) \), with solution
$$ u^{n+1} = u^n + \Delta t\nabla\cdot (\dfc(u^n)\nabla u^n) +
\Delta t f(u^n) $$
Disadvantage: \( \Delta t \leq (\max\alpha)^{-1}(\Delta x^2 + \Delta y^2 + \Delta z^2) \)
