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Implement a balanced Burgers equation

Florian Atteneder requested to merge fa/scalareq-balance into main

BalancedBurgers equation is a simple test problem in which one can test HRSC methods for a scalar equation with stationary solutions. This 'is just one example of a balanced law.

About the choice of source term: Given an initial data profile for $u_0(x)$ which should remain stationary, we compute the source term via

s(x) = \partial_x f(u_0(x))

Based on this test problem I could make a few improvments to the ScalarEq project and AV computations:

  • Added a av_drag option to ScalarEq project, which allows to slow down the rate of change of AV. This helps in reducing oscillations in the final stationary state.
  • Change time step computation for AV method to only use the smallest suggested step and not a combination thereof. This should allow for larger speeds in general.
  • In mda (and should probably be extended to other AVs) use local instead of global speed to compute baseline AV. This avoid AV being generated and then propagate in regions where the solution is constant. This also fixed excessive use of AV (even in smooth regions) in the burgers_sine_avmda ref test.

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