Hyperbolic 1D

Solves the 1D advection equation with periodic boundary conditions.

\[\frac{\partial }{\partial t} U + a \frac{\partial }{\partial x} U = 0\]

where $U = u(x,t)$ and $a = 1$ is the advection velocity. The domain $x \in [0,1]$ and $t \in [0,1]$ with initial condition

\[u(x, 0) = \sin(2\pi x)\]

Periodic boundary conditions are used

\[u(0, t) = u(1, t)\]

Using finite differences for the time derivative

\[\frac{\partial U}{\partial t} = \frac{U^{n+1}_{i}-U^{n}_{i}}{\delta t}\]

where $U_{i}^{n}$ is $u(x_{i},t_{n})$ and the mimetic operator $\mathbf{D}$ for the space derivative.

\[\frac{U^{n+1}_{i}-U^{n}_{i}}{\delta t} + a \mathbf{D}U_{i}^{n} = 0\\ \frac{U^{n+1}_{i}-U^{n}_{i}}{\delta t} = - a \mathbf{D}U_{i}^{n}\\ U^{n+1}_{i} = U^{n}_{i} - a \delta t \mathbf{D} U_{i}^{n}\\\]

This example is implemented in hyperbolic1D.jl