Add space and time tests

Author: riccardotosi

convection_diffusion/validation/space_time_convergence_test/README.md

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+# Space and time convergence tests
+
+**Author:** [Riccardo Tosi](https://github.com/riccardotosi)
+
+**Kratos version:** 9.0
+
+**Source files:** [section 2.7 of [1]](https://github.com/KratosMultiphysics/Documentation/blob/master/Resources_files/convection_diffusion_explicit_elements/Eulerian_convection_diffusion_explicit_element.pdf)
+
+## Space convergence
+
+We solve the transient convection diffusion equation
+<img src="https://render.githubusercontent.com/render/math?math=\frac{\partial \phi}{\partial t} %2B v \cdot  \nabla \phi %2B \phi \nabla \cdot v - \nabla \cdot k \nabla \phi = f"> and validate its reference implementation. We refer to section 2.7.1 of [1] for details.
+
+We validate the implementation by computing the <img src="https://render.githubusercontent.com/render/math?math=l^2"> norm of the error as <img src="https://render.githubusercontent.com/render/math?math=\varepsilon = \sqrt{\int_{\Omega} (\phi - \phi_h)^2 }">, where <img src="https://render.githubusercontent.com/render/math?math=h"> is the mesh size, <img src="https://render.githubusercontent.com/render/math?math=\phi"> and <img src="https://render.githubusercontent.com/render/math?math=\phi_h"> are the analytic and FEM solutions, respectively.
+
+<p align="center">
+  <img src="convergence_error_convection_diffusion_explicit_solution.jpg"alt="velocity" style="width: 500px;"/>
+</p>
+
+The figure shows that the error <img src="https://render.githubusercontent.com/render/math?math=\varepsilon"> converges as expected for both quasi-static ASGS and quasi-static OSS.
+
+## Time convergence
+
+We solve the transient convection diffusion equation
+<img src="https://render.githubusercontent.com/render/math?math=\frac{\partial \phi}{\partial t} %2B v \cdot  \nabla \phi %2B \phi \nabla \cdot v - \nabla \cdot k \nabla \phi = f"> and validate its reference implementation. We refer to section 2.7.2 of [1] for details.
+
+The analytic solution at time <img src="https://render.githubusercontent.com/render/math?math=t"> is <img src="https://render.githubusercontent.com/render/math?math=\phi(t) = x - \sin(t)">. Therefore, it is possible to compute the <img src="https://render.githubusercontent.com/render/math?math=l^2"> norm of the error as
+<img src="https://render.githubusercontent.com/render/math?math=\varepsilon = \sqrt{\int_{\Omega} (\phi - \phi_h)^2 }">, where <img src="https://render.githubusercontent.com/render/math?math=h"> is the mesh size, <img src="https://render.githubusercontent.com/render/math?math=\phi"> and <img src="https://render.githubusercontent.com/render/math?math=\phi_h"> are the analytic and FEM solutions, respectively. It is expected to obtain an order four accuracy for the Runge-Kutta 4 time integration scheme.
+
+<p align="center">
+  <img src="convergence_error_time_convection_diffusion_bar.jpg"alt="velocity" style="width: 500px;"/>
+</p>
+
+The figure shows that the time accuracy is of order four, as expected.
+
+
+## References
+
+[1] Tosi, R. (2020). Eulerian convection diffusion explicit elements (p. 27). p. 27. Retrieved from https://github.com/KratosMultiphysics/Documentation/blob/master/Resources_files/convection_diffusion_explicit_elements/Eulerian_convection_diffusion_explicit_element.pdf