Initial and boundary conditions

It should be clear that, since these equations are first-order in time, it is necessary to specify initial conditions at some initial time (say) $t=0$. Another important condition which must be stated at this point, is the boundary condition, which must typically be defined for the spatial boundaries in many physically interesting situations. The specification of these depends on the transverse partial differential operator involved. Thus, in the advection equation, we can specify a boundary condition at $x = -X/2$, but not simultaneously one at $x = X/2$. In the diffusion equations, which are second order in space, we typically need to specify two boundary conditions.

The equation is not fully specified unless the boundary conditions are known.

For example, on a one-dimensional transverse space, with boundaries at $x = \pm X/2$, there are three common types of boundary condition:

Dirichlet boundary conditions

The field is fixed at the boundaries, so,

$$\begin{aligned}\mbf{A}(-X/2) & = \mbf{A}^{(1)} \\ \mbf{A}(X/2) & = \mbf{A}^{(2)}. \end{aligned}$$

Neumann boundary conditions

The field gradient is fixed at the boundaries, so,

$$\begin{aligned}\frac{\partial }{\partial x}\mbf{A}(-X/2) & = ... ...tial }{\partial x}\mbf{A}(X/2) & = \mbf{B}^{(2)}. \end{aligned}$$

Periodic boundary conditions

The field is periodic at the boundaries, so,

$$\begin{aligned}\mbf{A}(-X/2) & = \mbf{A}(X/2) \\ \frac{\part... ...2) & = \frac{\partial }{\partial x} \mbf{A}(X/2). \end{aligned}$$

Infinite boundaries

For Maxwell's equations, as with many other examples in physics, it is often the case that we wish to define the boundaries to be at infinity. This option, however, leads to numerical difficulties in defining the transverse lattice. In some cases, we choose to use a finite spatial region or `window' to model an infinite domain, with the `window-size' being increased until the results change less than a prescribed accuracy criterion. Another technique is to transform the transverse (i.e., typically spatial) coordinates in some way to map them onto a finite domain. One example is the mapping $x = w \tan(y)$, which maps an infinite range of $x$ to a finite range of $y$, from $y=-\pi /2$ to $y=\pi /2$. Note that $w$ can be adjusted to increase or decrease the central region where the mapping is approximately linear; for best accuracy, this should be adjusted so it contains most of the ``interesting'' region!

To use infinite boundaries in $x$ one has to rewrite the differential equation in terms of the new variable $y$, which then satisfies an equation defined on a finite domain. This involves using the chain-rule of calculus, so that

$$\frac{\partial }{\partial x} = \frac{\partial y}{\partial x}\frac{\partial }{\partial y},$$ (2.22)

$$\frac{\partial ^2}{\partial x^2} = \frac{\partial ^2 y}{\partial x^2} \frac{\partial }{\partial y}... ...left( \frac{\partial y}{\partial x} \right)^2 \frac{\partial ^2}{\partial y^2}.$$ (2.23)

Thus, for example, in the $y$ variables, the scalar diffusion equation becomes:

$$\frac{\partial }{\partial t} A(y,t) = \kappa \left( \frac{\partia... ...artial y}{\partial x} \right)^2 \frac{\partial ^2}{\partial y^2}\right) A(y,t).$$ (2.24)