Solving PDEs
Partial differential equations describe quantities that vary in both time and space. A diffusion equation is
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}.\]Finite differences
On a spatial grid \(x_i=i\Delta x\),
\[\frac{\partial^2u}{\partial x^2}\approx\frac{u_{i+1}-2u_i+u_{i-1}}{(\Delta x)^2}.\]An explicit time update is
\[u_i^{n+1}=u_i^n+\frac{D\Delta t}{(\Delta x)^2}(u_{i+1}^n-2u_i^n+u_{i-1}^n).\]Boundary conditions
The PDE is incomplete without appropriate initial and boundary conditions. Numerical stability also constrains permissible grid and time-step choices.
Key idea. Numerical PDE methods replace continuous space and time by a grid while preserving the governing equation and boundary conditions as accurately as possible.