Finite difference SBP methods

[ranocha]

This is to track the progress of porting finite difference methods to Trixi. I propose to use my package SummationByPartsOperators.jl to get the basic methods. Then, we could use the following approach (implemented in #617).

• Pack an SBP derivative operator from SummationByPartsOperators.jl as basis in Trixi's DG struct
• Organize methods written for general DG operators such that they can use these new basis types
• Implement the weak-form volume integral in 2D for these SBP operators and the TreeMesh with conforming interfaces (no mortar, no AMR)

We shouldn't look at performance too much at this stage. It will probably be okay to just compute the fluxes and use mul!. However, that will require multiple mul!s for each component, reinterpreting arrays to have SVectors as contents, or to use something like StructArrays.jl, see also #557.

Some limitations of this first approach and open questions are

• We will use the same number of nodes per coordinate direction
• Efficient parallelization will probably require new/more code. Will we have enough elements such that threading across elements is still optimal? Or will we have a few huge elements such that we also need threading inside of the elements?
• Truly multi-dimensional implementations can be more (cache-) efficient than dimension-by-dimension versions implemented at first
• How should we handle a single truly periodic block? The basic option above will give us one SBP element coupled via numerical fluxes. Do we also want to enable a single truly periodic FD block?
• Shall we also allow upwind SBP operators? If so, how can we do that efficiently?

This will be a basic proof-of-concept and allows us to see whether/how our current abstractions need to be changed. Afterwards, some possible next steps are

• Rename volume/surface integrals to terms (Naming convention: volume/surface *integral* → *terms* #616)
• Implement purely periodic operators
• Move code to appropriate places (cf. Consolidate mesh infrastructure after adding structured and unstructured ones #542). We need more levels in our solver hierarchy.
  • DG: Current sub-methods are FDSBP and DGSEM
  • For each mesh type, we could have a dg_common.jl, a dgsem_common.jl, and a fdsbp_common.jl
  • More specific methods can be implemented in tree_2d_dgsem, tree_2d_fdsbp etc.
• Add FDSBP methods to the benchmarks
• Add references
• Add docs (including surface terms/integrals, e.g., First step towards FD methods and surface integrals/terms #617 (comment))
• Port stuff to 1D and 3D
• Extend convergence_test to different refinement types (number of elements, number of nodes per element)
• Implement split-form volume integrals. This could either be done in Trixi or moved (partially) to SummationByPartsOperators.jl.
• Implement curvilinear meshes Central SBP finite difference solver for UnstructuredMesh2D #1773
• Adapt visualization to FDSBP methods
• Implement non-conforming meshes and AMR
• Implement shock-capturing stuff
• Create keyword constructors with default mortar=nothing for DG (https://github.com/trixi-framework/Trixi.jl/pull/617/files#r643664755) and add more explanation of mortar and the other terms (https://github.com/trixi-framework/Trixi.jl/pull/617/files#r644461598)
• Optimize the heck out of this stuff