$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\x}{\boldsymbol{x}}
\newcommand{\X}{\boldsymbol{X}}
\renewcommand{\u}{\boldsymbol{u}}
\renewcommand{\v}{\boldsymbol{v}}
\newcommand{\e}{\boldsymbol{e}}
\newcommand{\f}{\boldsymbol{f}}
\newcommand{\dfc}{\alpha} % diffusion coefficient
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\Iz}{\mathcal{I}_z}
\newcommand{\If}{\mathcal{I}_s} % for FEM
\newcommand{\Ifd}{{I_d}} % for FEM
\newcommand{\Ifb}{{I_b}} % for FEM
\newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}}
\newcommand{\basphi}{\varphi}
\newcommand{\baspsi}{\psi}
\newcommand{\refphi}{\tilde\basphi}
\newcommand{\psib}{\boldsymbol{\psi}}
\newcommand{\sinL}[1]{\sin\left((#1+1)\pi\frac{x}{L}\right)}
\newcommand{\xno}[1]{x_{#1}}
\newcommand{\Xno}[1]{X_{(#1)}}
\newcommand{\xdno}[1]{\boldsymbol{x}_{#1}}
\newcommand{\dX}{\, \mathrm{d}X}
\newcommand{\dx}{\, \mathrm{d}x}
\newcommand{\ds}{\, \mathrm{d}s}
$$
A Forward Euler scheme; stages in the discretization
- \( \uex(\x,t) \): exact solution of the PDE problem
- \( \uex^n(\x) \): exact solution of time-discrete problem (after applying
a finite difference scheme in time)
- \( \uex^n(\x)\approx u^n = \sum_{j\in\If}c_j^n\baspsi_j = \)
solution of the time- and space-discrete problem
(after applying a Galerkin method in space)
$$
\begin{equation*}
\frac{\partial \uex}{\partial t} = \dfc\nabla^2 \uex + f(\x, t)
\end{equation*}
$$
$$
\begin{equation*}
\uex^{n+1} = \uex^n + \Delta t \left( \dfc\nabla^2 \uex^n + f(\x, t_n)\right)
\end{equation*}
$$
$$
\uex^n \approx u^n = \sum_{j=0}^{N} c_j^{n}\baspsi_j(\x),\quad
\uex^{n+1} \approx u^{n+1} = \sum_{j=0}^{N} c_j^{n+1}\baspsi_j(\x)
$$
$$ R = u^{n+1} - u^n - \Delta t \left( \dfc\nabla^2 u^n + f(\x, t_n)\right)$$
