Tests on the 2D polar advection operator

Tests on the 2D polar advection operator

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

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).

Contents

  • advection_all_tests.cpp : launch the 48 configurations on an uniform mesh.
  • advection_selected_test.cpp : launch 1 configuration. The selection is made in the CMakeLists.txt file.
  • advection_maths_tools.cpp : define functions used in the test files.
  • test_cases.hpp : define the three test simulations.