Basic Methods for Numerical Approximation of PDE

1. Finite Difference Techniques (we'll concentrate on these) $\equiv FD$
2. Spectral Methods $\equiv SM$
3. FEM/Galerkin $\equiv FEM$

CLASSIFICATION OF PDE'S

PDE's come in all shapes and forms. Each type, you will find, will require a different numerical approximating method. We will concentrate on three types of problems, which are ubiquitous in physics and engineering

Type Classification	Canonical Example
Hyperbolic	Wave equation $U_t+AU_x=0$
Parabolic	heat equation $U_t=kU_{xx}$
Elliptic	Poisson equation $\nabla^2U=f$

We will also consider an equation of mixed type, the ``Advection-Diffusion equation.''

Type Classification: Within our purview the type classification is not a tremendously important concern. Perhaps more useful is to make the association between the names and the canonical examples. Nevertheless the type name comes from the classification of linear second-order pde's. Take $u=u(x,y)$ and $a, b, c, d, e, f$ real constants. The equation

$$au_{xx}+bu_{xy}+cu_{yy}+du_{x}+eu_y+fu=g$$

is

$$b^2-4ac<0$$

$$\mbox {parabolic if } b^2-4ac=0$$

$$\mbox {hyperbolic if } b^2-4ac>0$$

The names come from an analogy with conic sections. Consult a PDE book for more details.

BASIC PROBLEM

Every PDE type equation requires a special or particular strategy of numerical approximation. Traditionally courses on numerical methods for PDE's have been organized by method (i.e. FD, SM, FEM, etc.) rather than by equation type. The reasons have to do with the fact that this relatively young subject has been taught by researchers who specialize in the methods rather than in the equation types.

We will follow tradition here: we will cover mostly the FD method of mostly linear hyperbolic, parabolic, and elliptic equations.

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In many physical situations one might also find equations of mixed type, or problems that couple equations of various types.