Add 3D Cartesian Shallow Water Equations #36
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… the curl invariant form (we need the surface Jacobian, not the volume Jacobian...)
…ve for Cartesian solvers (not used)
amrueda
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amrueda
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| struct MetricTermsCrossProduct end | ||
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| """ | ||
| MetricTermsCurlInvariant() |
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Just a small thing regarding terminology - In his 2006 paper, David Kopriva originally called this the "invariant curl form" rather than the "curl-invariant form". Some more recent papers (not by Kopriva) have indeed called it the "curl-invariant form", but I don't see why the terminology should be changed. It also may be worth adding references to the following papers:
- Kopriva, D. A. (2006). Metric identities and the discontinuous spectral element method on curvilinear meshes. Journal of Scientific Computing 26, 301-327. https://doi.org/10.1007/s10915-005-9070-8
- Vinokur, M. and Yee, H. C. (2001). Extension of efficient low dissipation high order schemes for 3-D curvilinear moving grids. In Caughey, D. A., and Hafez, M. M. (eds.), Frontiers of Computational Fluid Dynamics 2002, World Scientific, Singapore, pp. 129–164. https://doi.org/10.1142/9789812810793_0008
| Shallow water equations (SWE) in three space dimensions in conservation form (with constant bottom topography). | ||
| The equations are given by | ||
| ```math | ||
| \begin{aligned} |
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My understanding is that the 3D Cartesian form of the shallow water equations first appeared in the following paper, which we should reference here:
J. Coté (1988). A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere. Quarterly Journal of the Royal Meteorological Society 114, 1347-1352. https://doi.org/10.1002/qj.49711448310
I also think it would also make sense to mention the Lagrange multiplier source term and perhaps link to where that is implemented.
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| # Custom RHS that applies a source term that depends on du (used to convert apply Lagrange multiplier) | ||
| function rhs_semi_custom!(du_ode, u_ode, semi, t) |
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Is it possible to have the custom RHS and Lagrange multiplier source terms defined in dg_2d_manifold_in_3d_cartesian.jl and called by dispatch on the ShallowWaterEquations3D equation type rather than specified in the elixir? This is how I do it for the covariant form, for example:
| # for performance reasons. | ||
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| # First summand 0.5 * (Xₘ Xₗ_ζ - Xₗ Xₘ_ζ)_η | ||
| @turbo for j in eachnode(basis), i in eachnode(basis) |
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The LoopVectorization.jl @turbo macro is used here to compute the metrics, but it is not used anywhere else (e.g. in the volume integral). With the future support for LoopVectorization.jl uncertain, I am hesitant to use it in new code unless it's really critical for performance.
This PR adds the 3D Shallow Water Equations (SWE) in Cartesian form, which can be solved on a 2D manifold embedded in 3D. We add two strategies to obtain entropy (total energy) conservation based on two-point fluxes: (i) using an entropy correction term, and (ii) using an entropy-consistent projection of the momentum. Both strategies require divergence-free metric terms on the 2D manifold. In this PR, we implement the curl-invariant form of Kopriva (2006). We also add the possibility to discretize the equations with the weak-form kernel.