# The div B = 0 constraint

One thing that makes integrating the equations of magnetohydrodynamics more complicated is the additional constraint equation that we have to deal with. The no-monopole constraint that comes from the Maxwell equations 
    
$$\nabla \cdot \vec{B} = 0$$

has to be satisfied when we integrate our evolution equations for the magnetic field. This is trivially true in the limit of infinite resolution (and via the CFL condition infinitely-small timesteps) but turns out to be a problem when we integrate the equations without any additional measures employed. 

There are multiple strategies used by application codes solving the MHD equations:

 - hyperbolic divergence cleaning
 - constrained transport
 - evolving the magnetic vector potential
 - projecting the divergence out

They each have pros/cons. We'll focus here on the most commonly used ones, hyperbolic divergence cleaning and constrained transport.

# Constrained transport

In constrained transport we make use of our flux-conservative formulation to guarantee that even in the discrete system we are not generating any divergence of the magnetic field during the evolution. To do so we need to use a staggered mesh to store the magnetic field $$\vec{B}$$. This adds complications to our finite-volume grid as it involves keeping track of different variable centering in the state vector and requires additional interpolation operations. You can see a sketch of where magnetic and electric field (not part of the state vector in ideal MHD but used for calculating the fluxes in constraint transport) are defined/stored in a fully-staggered mesh used for constrained transport here.
    
![Staggered mesh](mhd-ct.png)
    
$\vec{B}$ and $\vec{E}$ are the staggered magnetic and electric field vectors. Similarly to how we interpreted the cell-centered variables of the state vector as volume-averages/volume-integrals over the cell, we can view the magnetic field as the surface integral/average over the cell face and the electric field as the line integral over the cell edge. 

The key strategy of constrained transport schemes is using the electric field components stored at the cell edges to implement the discrete induction equation for the magnetic field. For example, in

![Discrete induction equation](mhd-induction.png)

we can use the fact that in ideal MHD the electric field at the cell edges can be expressed as $B^k v^i - v^k B^i$.

In code this looks like this

![Constrained transport in action](mhd-code.png)
  
# Hyperbolic divergence cleaning

One of the main altervatives to constrained transport is hyperbolic divergence cleaning. In this approach we add a scalar field $\Phi$ to the induction equation. If we define $\Phi$ to be the discrete $\nabla \vec{B}$ violation on our grid, we can add an additional evolution equation to our system and use it to transport and damp any arising constraint violations. This can be written in the following form (via a hyperbolic equation):

$\frac{\partial\vec{B}}{\partial t} + \frac{\partial}{\partial x} (\vec{B}\cdot\vec{u} -\vec{u}\cdot\vec{B}) + \nabla \Phi = 0$

and

$\frac{1}{c_h^2} \frac{\partial\Phi}{\partial t} + \nabla \cdot \vec{B} = 0$ with $c_h \epsilon [0,\infty].$

In the limit of $\Phi -> 0$ and $\nabla \cdot \vec{B}$ we recover the ideal-MHD induction equation. The characteristic velocities of the evolution system for MHD without and with divergence cleaning differ. In the former, the largest-magnitude wave speed is the fast magnetosonic wave speed. The inclusion of divergence cleaning introduces an additional mode, corresponding to eigenvalue for the evolution of the divergence cleaning field $\Phi$. We can use this additional eigenvalue to set the hyperbolic damping speed $c_h$ of the scheme, for example we can set it to the speed of light.

We can also add an additional term 

$\frac{1}{c_h^2} \frac{\partial\Phi}{\partial t} + \frac{1}{c_p^2} \Phi + \nabla \cdot \vec{B} = 0$ with $c_p \epsilon [0,\infty].$

This additional term will now also damp away any existing constraint violations. In this way we can both damp and transport away constraint violations on our grid.

# Vector potential

Another alternative is evolving the vector potential instead of the magnetic field in the MHD equations. The vector potential is defined via $\vec{B} = \nabla \times \vec{A}$ and we can formulate an evolution equation for the vector potential instead of the induction equation:

$\frac{\partial\vec{A}}{\partial t} = (\vec{B}\cdot\vec{u} -\vec{u}\cdot\vec{B}).$

When we take the divergence of a curl we automatically have $\nabla \cdot \nabla \times \vec{B} = 0$ so we get the no-monopole constraint satisfied by construction. One complexity with the vector potential in applications in relativity is that we need to choose a gauge due to the extra degrees of freedom in the vector potential definition. We can for example add the electromagnetic scalar potential 

$\frac{\partial\vec{A}}{\partial t} = (\vec{B}\cdot\vec{u} -\vec{u}\cdot\vec{B}) - \frac{\partial (\Phi - \vec{A})}{\partial x}$ to our evolution equation without changing the principal part. Care has to be taken when choosing a suitable gauge as some choices are better behaved than others.

# Conservative to Primitive variable conversion in MHD

Another part of our solver for the Hydrodynamics equations that we have developed as part of the class is the Conservative to Primitive variable conversion. While we could do this analytically in the example in class, this is generally not possible. Especially in MHD the conversion problem becomes significantly more complex with standard methods employing multi-dimensional root solvers to iteratively find a solution for these equations. You can find a detailed comparison of different Conservative to Primitive methods (in General Relativity) here