3d eulerian

A brief note on indexing and notation

In order to keep track of tensors e.g. in point space and spectral space, I like to do the following:

• The index for cubes will be $m$
• the index for cubical indices will be $l_1,l_2,l_3$
• The index for spectral coefficients will use variable $l$

Design simplifications:

• 3D cubical domain.
• rigid among z and y boundaries, periodic among x boundary.
• $\nu = \nu(T, S)$ with potentially wildly different stiffnesses depending on both tracer (glass color) and temperature values.
• design for multiple GPUs (i.e. one MPI per GPU. This generalizes well to the cluster that I have)

Construction and derivatives

Assume that $\Omega = [-1, 1]^3$. Then $~_{lL}\mathbf{D}$ is the derivative of 1D interpolant polynomials on the GLL points. Then if we have a scalar quantity $u(x_1, x_2, x_3)$ located at the GLL points $~_{l_1,l_2,l_3}\mathbf{U}$ then $\partial_{x_i} u$ is $~_{l_1l_2l_3}\mathbf{U}_{L=l_i}~_{lL}\mathbf{D}_{L}.$ This comes from the fact that $u(x_1, x_2, x_3) = \sum_{l_1,l_2,l_3=1}^{n_{GLL}} u_{l_1,l_2,l_3} p_{l_1}(\xi_{l_1})p_{l_2}(\xi_{l_2})p_{l_3}(\xi_{l_3}).$

Righthand side terms are computed within an element, then undergo DSS (direct stiffness summation) which involves intra-element communication. Since DSS is a jacobian-weighted average over all redundant points, and our Eulerian grid (which is a subset of Euclidean space) has a Jacobian which is just the identity matrix, this means that our DSS procedure is just a sum which must take into account the number of redundant points.

Given that our viscosity has the form $\nu(T, S)$ (in particular, it isn't flow dependent), we get some choice in of prognostic equations. For the sake of education, I'm going to try to follow along with this paper

But I think I can basically just use the Navier Stokes equations as is.

Code:

Create abstract base class for an equation set