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Peano
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Peano
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This is one of the simplest equations we can define and it appears as follows:
\( \frac{\partial}{\partial t}\left( \begin{array}{lr} q_x \\ q_y \\ q_z \end{array} \right) + \nabla \begin{pmatrix} q_x & 0 & 0 \\ 0 & q_y & 0 \\ 0 & 0 & q_z \end{pmatrix} = \vec{0} \)
This equation describes the transport of a quantity q by the motion of a fluid, such as the movement of a particle in a river.
To implement this equation, the following things are needed: the initial condition, the boundary conditions, the maximal eigenvalue, the flux, and (optionally) the refinement criterion.
We start with the initial conditions: here we define a starting configuration in which the quantity in direction i is equal to the position in that direction x_i.
Then we can define our boundary conditions. You can choose any boundary conditions you prefer but we have chosen to use the homogeneous Neumann boundary conditions.
Next, we can define the largest eigenvalue of our system. This is important because it defines a lot of the behaviour of the PDE, and among others the maximum allowable timestep for which a lot of solvers remain stable. The eigenvalues of the advection equations are equal to the velocity of the fluid, which here we set uniformly to 1.
Finally, let's define our flux in x- and y-direction according to our advection equation. The directions of the flux are defined by the normal parameter. This parameter defines through which surface, and therefore in which direction, the flux is being computed. Here if the normal is 0, we are computing the flux in the x-direction and if the normal is 1, we are computing the flux in the y-direction. For higher dimensions, there would logically be additional directions to implement the flux in.
Our fluxes in x- and y-directions are:
\( F_x = \begin{pmatrix} u_x \\ 0 \\ 0 \end{pmatrix}, F_y =\begin{pmatrix} 0 \\ u_y \\ 0 \end{pmatrix} \)
| t=0.0 s | t=0.5 s |
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The measured quantity in x-direction respectively at time t=0.0 s and t=0.5 s.
1.9.1 
