DG and Trixi tutorial in the docs

[ranocha]

It would be nice to have a brief explanation of DG methods and their implementation in Trixi.jl in the docs. That part is different from https://github.com/trixi-framework/Trixi.jl/issues/169, since it should contain an introduction to the mathematical parts of the algorithms instead of "just" the combination of Trixi's API components to run something. Here is a preliminary list of topics to be covered:

• [x] Simple and clear scalar linear advection example in 1D (#943)
  • Describe basics of weak form DG
  • Implement a simple weak form DG specialized to scalar linear advection in 1D on a uniform mesh (maybe using the TreeMesh? or probably better creating a simple uniform 1D mesh type on the fly)
  • Use OrdinaryDiffEq.jl with adaptive time stepping (simple and easy to use)
  • Show how to write the same example using Trixi's capabilities (TreeMesh etc.)
• [x] Flux differencing aka split form DG: Compressible Euler equations based on existing elixirs, EC flux of Ranocha and split form of Shima et al. or Kennedy & Gruber (#1002)
• [x] Shock capturing with flux differencing based on an existing elixir (#1024)
• [x] Stage limiter (positivity preserving limiter of Zhang & Shu) (#1024)
• [x] Adaptive mesh refinement based on an existing elixir (#1025)
• [x] Non-periodic boundary conditions based on existing elixirs (Dirichlet BC, wall BC) (#1059)
• [x] Structured mesh with mapping, e.g., 2D flow with compressible Euler equations (flow around a cylinder?) (#1084)
• [x] Mention KPP elixir introduced in #1035 in "Adding a new equation" (#1025)
• [x] Explicit time stepping, CFL-based step size control and error-based adaptive step sizes (#1116)
• [x] DGMulti solver and mesh
  • [x] Triangular meshes with DG (see examples/dgmulti_2d/elixir_euler_triangulate_pkg_mesh.jl for an example using a "ready-made" geometry from StartUpDG.jl)
  • [x] Polynomial and nodal SBP approximation spaces
  • [x] "Esoteric" methods such as FD SBP, CGSEM via SummationByPartsOperators.jl, and GaussSBP

[jlchan]

It looks like all of these are completed in the Tutorials section now? https://trixi-framework.github.io/Trixi.jl/dev/tutorials/introduction/

[ranocha]

👍