In order to verify a two-phase subsonic compressible isothermal Euler code, I am trying to implement a manufactured solution following what is discussed here and references therein. Also, as an extension of the question already answered here, I am looking for a more detailed explanation on the choice of a manufactured solution for a specific case and especially on the choice of the related parameters.
For instance, in this paper, the authors suggest for the density a manufactured solution of the form (in 2D): $$ \rho(x,y)=\rho_0 + \rho_x \sin \left( \frac{a_{\rho x}\pi x}{L} \right) + \rho_y \cos \left( \frac{a_{\rho y}\pi y}{L} \right) + \rho_{xy} \cos \left( \frac{a_{\rho xy}\pi xy}{L^2} \right) $$ where $\rho_0,\rho_x,\rho_y,\rho_{xy},a_{\rho x},a_{\rho y},a_{\rho x y} $ are given just like that without any explanation.
I find this expression really complicated and my question is why the authors have chosen such a complicated function ? I know that trigonometric functions are good candidate because $C^\infty$ but still why the proposed manufactured solution is such a complex combination of such functions ? Why a simple cosinus cannot be enough ? And above all, how do they choose the value for the seven parameters written above ? What make you choose between setting $\rho_{xy}$ to 2 and 500 for instance ?