Hyperbolic1D Lax Friedrichs

Solves the 1D advection equation with periodic boundary conditions using a sided nodal mimetic operator and the Lax-Friedrichs scheme for time integration. The main feature of this code is the lack of an interpolator. This is first order in both time and space, even though a second order mimetic operator is used.

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

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.

\[ \frac{U^{n+1}_i - U^{n}_i}{\Delta t} + a \mathbf{D} U^n_i = 0 \]

\[ \frac{U^{n+1}_i - U^{n}_i}{\Delta t} = -a \mathbf{D} U^n_i \]

\[ U^{n+1}_i = \mathbf{U}^n_i - a \Delta t \mathbf{D} U^n_i \]

The last line \(\mathbf{U}\) is the derived (average) value of the U from the \(U^n\) timestep.