5. Partial Differential Equations#
This chapter focuses on numerical methods for solving partial differential equations (PDEs), which involve derivatives in multiple dimensions.
We can write a general, linear 2nd-order PDE for a variable \(u(x,y)\) as
(5.1)#\[\begin{equation}
A \frac{\partial^2 u}{\partial x^2} + 2 B \frac{\partial^2 u}{\partial x \, \partial y} + C \frac{\partial^2 u}{\partial y^2} = F \left( x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y} \right)
\end{equation}\]
where \(A\), \(B\), and \(C\) are constants. Depending on their value, we can categorize a PDE into one of three categories:
\(B^2 - AC < 0\): elliptic
\(B^2 - AC = 0\): parabolic
\(B^2 - AC > 0\): hyperbolic
The different PDE types will exhibit different characteristics and will also require slightly different solution approaches.