11 Numerical solutions of PDEs
Like the subject of numerical solutions of ODEs, you will only be given a very brief introduction to how PDEs are solved based on the simplest (Euler) scheme.
Let us consider the solution of the heat equation on a finite interval: \[ \begin{gathered} \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}, \\ u(0, t) = 1, \quad u(1, t) = 2, \quad u(x, 0) = 2. \end{gathered} \]
11.1 Finite difference problem
We consider a discretisation of the spatial domain \(x \in [0, 1]\) via the \((N+1)\) points \[ x_0 = 0, \quad x_1 = \Delta x, \quad x_2 = 2\Delta x, \quad \ldots , \quad x_{N+1} = 1. \] Typicall \(\Delta x\) is small. We also have time steps of size \(\Delta t\), and thus going from \(t_0 = 0\), \(t_1 = \Delta t\), and so forth. We represent the solution by the indexing scheme: \[ u(x_j, t_k) = u_{j}^j. \]
Like for Euler’s method, we consider the approximation of the time derivative as follows: \[ \left(\frac{\partial u}{\partial t}\right)_j^k \approx \frac{u_j^{k+1} - u_j^k}{\Delta t}, \qquad \text{for $0 \leq j \leq N+1$, $k \geq 0$}. \]
In order to approximate the second-order derivative, Hwe will use \[ \left(\frac{\partial^2 u}{\partial x^2}\right)_j^k \approx \frac{u_{j+1}^k - 2 u_j^k + u_{j-1}^k}{(\Delta x)^2}, \qquad \text{$1 \leq j \leq N$, $k \geq 0$}. \] Note that the finite difference cannot be applied directly to the first and last points (\(j = 0\) and \(j = N+1\)). Substituting the above finite differences into the heat equation, we see that we now have the prescription: \[ u_j^{k+1} = u_j^k + \frac{\Delta t}{(\Delta x)^2} \left[u_{j+1}^k - 2 u_j^k + u_{j-1}^k \right], \] which applies for \(j = 1, 2, \ldots, N\).
11.2 Boundary and initial conditions
The initial conditions are implemented by setting \[ u_j^0 = 2, \qquad \text{for $j = 0, 1, 2, \ldots, N+1$}. \] The boundary conditions require \[ u_0^k = 1 \quad \text{and} \quad u_{N+1}^k = 2, \quad \text{for all $k \geq 0$}. \]
During the lectures, we will examine script lecture14-SolvingPDEs.ipynb on the course scripts to see an example implementation. A similar application is applied in lecture15-winecellar.ipynb to solve a heat equation.