Tests on the 2D polar advection operator

The tests implemented in this folder test the 2D polar advection operator implemented in the src/advection/ folder ( advection ).

The tests of the convergence order are made for constant CFL which means it checks the slope of the errors (infinity norm of the difference at the final time) for \((N_r\times N_\theta, dt) = (N_{r,0}\times N_{\theta,0}, dt_0)\) and then \((n*N_{r,0}\times n*N_{\theta,0}, dt_0/n)\) for \(n = 1, 2, 4, 8, ...\).

The tests are made for different parameters which are:

- The mapping and the domain used for the advection

Circular mapping in the physical domain (CircularToCartesian);
Czarny mapping in the physical domain (CzarnyToCartesian);
Czarny mapping in the pseudo-Cartesian domain (CzarnyToCartesian);
Discrete mapping of the Czarny mapping in the pseudo-Cartesian domain (DiscreteToCartesian).

- The time integration method used to solve the characteristic equation

Explicit Euler (Euler);
Crank-Nicolson (CrankNicolson);
Runge-Kutta 3 (RK3);
Runge-Kutta 4 (RK4).

- The test simulation

simulation 1: translation of Gaussian function (AdvectionFieldSimulation)
\(f_0(x,y) = \exp\left( - \frac{(x- x_0)^2}{2 \sigma_x^2} - \frac{(y- y_0)^2}{2 \sigma_y^2} \right)\),
\(A(t, x, y) = (v_x, v_y)\) .
simulation 2: rotation of Gaussian function (AdvectionFieldSimulation)
\(f_0(x,y) = \exp\left( - \frac{(x- x_0)^2}{2 \sigma_x^2} - \frac{(y- y_0)^2}{2 \sigma_y^2} \right)\),
\(A(t, x, y) = J_{\mathcal{F}_{\text{circular}}}(v_r, v_\theta)\).
simulation 3: decentred rotation (test given in Edoardo Zoni's article [1]) (AdvectionFieldSimulation)
\(f_0(x,y) = \frac{1}{2} \left( G(r_1(x,y)) + G(r_2(x,y))\right)\),
with
\(G(r) = \cos\left(\frac{\pi r}{2 a}\right)^4 * 1_{r<a}(r)\),
\(r_1(x, y) = \sqrt{(x-x_0)^2 + 8(y-y_0)^2}\)
\(r_2(x, y) = \sqrt{8(x-x_0)^2 + (y-y_0)^2}\)
\(A(t, x, y) = \omega(y_c - y, x - x_c)\).

Python tests

animated_curves.py: create .mp4 video(s) of the advected function for the selected configurations among the 48 test ones.
Command to launch the test in this folder: python3 animated_curves.py ../../../build/tests/geometryRTheta/advection_rtheta/advection_ALL or python3 animated_curves.py ../../../build/tests/geometryRTheta/advection_rtheta/<selected advection test case>
display_all_errors_for_gtest.py: Google test which tests the convergence order for the 48 configurations.
Command to launch the test in this folder: python3 display_all_errors_for_gtest.py ../../../build/tests/geometryRTheta/advection_rtheta/advection_ALL
display_curves.py: display the curve of the function for 9 time steps between the initial and the final state.
Command to launch the test in this folder: python3 display_curves.py ../../../build/tests/geometryRTheta/advection_rtheta/<selected advection test case>
display_feet_errors.py: compute the convergence order of the characteristic feet and display the computed and the exact feet.
Command to launch the test in this folder: python3 display_feet_errors.py ../../../build/tests/geometryRTheta/advection_rtheta/<selected advection test case>
advection_functions.py: define all the useful functions used in the other python files.

References

[1] Edoardo Zoni, Yaman Güçlü. "Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements". (https://doi.org/10.1016/j.jcp.2019.108889.) Journal of Computational Physics (2019).

Tests on the 2D polar advection operator