## Random initial conditions for LimitingADERDGSolver pose a problem

Currently, random initial conditions likely lead to undefined behaviour for the LimitingADERDGSolver.

## Background

The LimitingADERDGSolver replaces the ADER-DG solution locally by a FV solution if the former does not satisfy certain conditions (PAD, DMP). During the imposition of initial conditions, we then allocate a new FV patch and impose the initial conditions on its FV solution degrees of freedom.

## Random initial conditions

We have to ensure that FV and ADER-DG solution are consistent. This is currently only ensured if we have deterministic initial conditions.

In case of random initial conditions, the imposition of initial conditions on the ADER-DG solution and on the FV solution might lead to two different outcomes. A cell might encounter rough initial conditions while imposing the initial conditions on the ADER-DG solution but it then might encounter smooth initial conditions for the FV solution. But more severely, both differ from each other.

## A solution?

In case a cell allocates an FV patch and has a limiter status such that it will compute with Finite Volumes in the first iteration, we will then treat this FV solution as the correct one and will project it back on the ADER-DG solution. This way ADER-DG and FV solution are consistent.

On the other hand, if the limiter status of the cell is such that the cell will compute with ADER-DG in the first iteration, we will treat the ADER-DG solution as the correct one, we will project the ADER-DG solution onto the FV solution. We will not impose initial conditions on the FV solution. FV and ADER-DG solution are consistent in this case as well.