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)$$A\frac{{\partial }^{2}u}{\partial x^2}+2B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}=F(x,y,u,\frac{\partial u}{\partial x},\frac{\partial u}{\partial y})$$

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.