Boundary Conditions

Numerical schemes may require points outside of computational domain. This happens at boundary conditions. Suppose we are solving a problem over a space grid indexed by $m= 0, 1, \cdots M$. Let $u^n_m$ be an approximation to $U(x_m, t_n)$. Hence the edge variables are $u^n_0$ and $u^n_M$. For example, suppose we have a scheme that requires $u^{n}_{M+1}$ in order to determine $u^n_m$. This can happen when $m=M$, i.e. at the edge of the domain and we will have that $u^n_m$ is determined by $u^n_{M+1}$ as well as by interior grid quantities.

Numerical boundary conditions should be some form of extrapolation that determines the solution on the boundary in terms of the solution in the interior. For example:

some numerical b.c.'s $\left\{\begin{array}{ll} u^{n+1}_M=u_0^{n+1} & \mbox{Periodic}\\ u^{n+1}_M=u_... ...-a\lambda(u^n_M-u^n_{M-1}) & (\mbox{quasi-characteristics)} \end{array}\right.$

We will consider boundary conditions further when we discuss Parabolic equations (see 0.5.1).

There are three aspects to numerical boundary conditions

• The real boundary conditions, along with initial data and the pde, should lead to a well-posed problem.
• Need to come up with a numerical approximation of the physical boundary conditions, and this may lead to approximate boundary conditions which have their own inherent truncation error as well as round-off errors.
• A stable numerical scheme may become unstable by a bad choice of boundary data or a bad choice of approximating scheme for the boundary data!!

Recall that von Neumann stability only gives stability of IVP, so we need to consider stability of the scheme in the neighborhood of the boundaries: always work out the stability issues on each boundary. First do these separately, and then check that they all work together. When we do parabolic problems we will use elementary matrix methods to infer whether or not a particular choice of b.c. will lead to instabilities when coupled to a particular scheme.

But now we revert to two important aspects of the quality of an approximating scheme, which are most important in hyperbolic problems but that also are considerations in other types of evolutionary problems. These are: Dissipation and Dispersion.