2.2.5 Parabolic PDE: discretization

For discrete time-steps on a spatial grid

$$t^n = \Delta t n + t_0$$ (81)

$$x_i = \Delta x i + x_0$$ (82)

the discretization in time and space

$$\frac{\partial y}{\partial t} \rightarrow \frac{y^{n+1}_i-y^n_i}{\Delta t}$$ (83)

$$\frac{\partial^2 y}{{\partial x}^2} \rightarrow \frac{y^n_{i+1}-2y^n_i+y^n_{i-1}}{\Delta x^2}$$ (84)

gives the explicit Euler scheme

$$y^{n+1}_i = y^n_i + \frac{\Delta t}{\Delta x^2} K \left( y^n_{i+1}-2y^n_i+y^n_{i-1} \right) \enspace .$$ (85)