Dirichlet and Neumann conditions: reflecting and mirroring boundaries

The first two animations demonstrates the differences between a
Dirichlet condition \( u=0 \) at the boundary and a Neumann condition
\( \partial u/\partial x=0 \).

Reflecting boundaries (\( u_x=0 \)) for a Gaussian wave.

Mirroring boundaries (\( u=0 \)) for a Gaussian wave.

Instead of a Gaussian wave profile, we can test the geometrically simple
plug profile.

Numerical noise

All of the above computations were run with unit Courant number, which
means that the solutions are exact without any numerical errors. (This
is a remarkable property of the numerical solution method.)
For smaller Courant numbers, numerical noise may be visible, depending
on the smoothness of the initial profile.
Below is a smooth Gaussian profile and the almost discontinuous plug profile.

Reflecting boundaries (\( u_x=0 \)) for a Gaussian wave, computed with \( C=0.5 \), which implies numerical noise on a coarse grid.

A plug wave generates very significant numerical noise (\( C=0.5 \)).

Effect of impulsive start of waves

The previous demonstrations had an initial condition with a prescribed
\( u=0 \) profile at rest (\( u_t=0 \)). Alternatively, one may start with
a flat profile \( u=0 \) and use an initial condition \( u_t=V\neq 0 \) to
impulsively start the wave motion. For example, if we think of \( u \)
as the displacement of a string on a string instrument, this set of
initial conditions corresponds to an undeformed string that is given
an impulsive start from an impact.

Impulsive start of a wave motion: \( I=0 \), \( V\neq 0 \).

Feeding of waves from the boundary

We can also start with a flat profile, \( u=0 \), at rest, \( u_t=0 \), and
create propagating signals by moving \( u \) at the boundary. That is,
we have a time-varying Dirichlet condition \( u(0,t)=U_0(t) \) at the
left boundary. The movements lead to signals that propagate to the
right. In the movie, the movements are paused to make separate
signals.

Feeding of waves from the boundary by a time-dependent Dirichlet condition \( U_0(t) \).

Open and periodic boundary conditions

Demonstrations of periodic boundary condition on the left combined with an open boundary condition on the right: waves passing out of the domain enter the left end again.

Demonstrations of periodic boundary condition on the left combined with an open boundary condition on the right: waves passing out of the domain enter the left end again.

Error in the open boundary condition

In 2D and 3D...

Periodic boundary condition combined with a slight wrong open boundary condition at the right end (20% wrong wave velocity leads to small reflections back into the domain).

Summary of boundary conditions.

Condition

Formula

Effect

Dirichlet

\( u(0,t)=0 \)

mirror wave

Dirichlet

\( u(0,t)=U_0(t) \)

feed incoming wave

Neumann

\( u_x(0,t)=0 \)

reflect wave

Open

\( u_t \pm cu_x=0 \)

let wave out of the domain

Periodic

\( u(0,t)=u(L,t) \)

turn outgoing wave to incoming

Science is a differential equation. Religion is a boundary condition.
Alan Turing, computer scientist, 1912-1954.