weak_form_kernel! implemented only for nonconservative_terms::False

[DanielDoehring]

Across solvers, weak_form_kernel! is only implemented for nonconservative_terms::False, i.e.,

Trixi.jl/src/solvers/dgsem_tree/dg_1d.jl

Lines 147 to 150 in 7fd4503

	@inline function weak_form_kernel!(du, u,
	element, mesh::Union{TreeMesh{1}, StructuredMesh{1}},
	nonconservative_terms::False, equations,
	dg::DGSEM, cache, alpha = true)

while the calling function

Trixi.jl/src/solvers/dgsem_tree/dg_1d.jl

Lines 133 to 137 in 7fd4503

	function calc_volume_integral!(du, u,
	mesh::Union{TreeMesh{1}, StructuredMesh{1}},
	nonconservative_terms, equations,
	volume_integral::VolumeIntegralWeakForm,
	dg::DGSEM, cache)

suggests that there is an implementation for nonconservative_terms::True at least possible.
Is this true, or can this particular volume integral not be computed in a meaningful way for non-conservative terms?

[ranocha]

It could in principle but there is probably not much to be gained since we would like to use some split form anyway - but you may better ask our nonconservative wizard @andrewwinters5000

[andrewwinters5000]

In principle one can probably derive and implement the non-conservative terms in a weak form style. But, as @ranocha pointed out, we do not really have a use case for this, as we typically need a split form anyway for entropy stability or for well-balancedness in the case of the shallow water equations. There are also some subtleties that arise if you want to do the non-conservative term of the shallow water equations weakly. What I mean is given a test function $\phi$ then the inner product form after one application of integration-by-parts would be $$(ghb_x,\phi) = ghb\phi|_{\partial E} - (gb, (h\phi)_x)$$
because the weak derivative now falls on the product $h\phi$ instead of $b$, deriving a well-balanced approximation becomes tricky. Not to say it is impossible, I just have never investigated it further.

[DanielDoehring]

Okay, interesting! Maybe it makes sense to add a comment explaining why there is only one implementation available.

[ranocha]

Please feel free to make a PR with such a comment, linking to this discussion

[DanielDoehring]

Opinions on this? @ranocha @andrewwinters5000

"""
`weak_form_kernel!` is only implemented for conserved terms as 
non-conserved terms are supposed to be always applied in conjunction with a flux-splitting scheme, 
see `flux_differencing_kernel!`.
This treatment is required to achieve e.g. entropy-stability and well-balancedness.
"""

[ranocha]

Something like that could work as a comment for the weak form kernel itself - wuth a link to this issue. You can also mention something similar in the docstring of the volume integral type - we should have a docstring for it, don't we?