How to choose a simple manufactured solution for Euler equation?

Question

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 ?

Answer 1

With manufactured solutions, you pick the form of the solutions you want, crank them through the governing equations, and generate right-hand side functions that you can go implement to drive the solution to the solution you manufactured. Some solutions are chosen to satisfy other properties that you want. Spencer's comment suggests that boundary conditions are one set of properties you might desire to also include in your manufactured solution. Periodicity is another one. Also, you need to pick solutions that are smooth enough to have enough derivatives to pass neatly through your operator. There are numerous other conditions you might want to impose on your solution, so the reasons for choices may not ever be guessable from the outside. This paper does not give the authors' reasons for selecting this particular form.

Transcendental functions are often used since they may excite multiple modes of the operator simultaneously. A sum or product of them is an easy way to combine such functions into something that explores a multi-dimensional space effectively.

Answer 2

Augmenting what Bill Barth already said:

1/ You don't want to use polynomials because many methods (such as finite element methods) are exact for at least lower order polynomials.

2/ You want to choose the solution to be analytic or at least $C^\infty$ so that the solution can be well approximated.

These two requirements naturally lead to trig functions.

For the coefficients: 3/ You want to choose $a_{\rho x}$ large enough so that you get more than, say, half of an oscillation into your domain width.

4/ But you want to choose $a_{\rho x}$ small enough so that you have a chance of resolving the solution with your mesh. For example, even on the coarsest mesh, you probably want at least 2 mesh sizes per period of the (co)sine, and then go to more and more elements per period as you refine the mesh.

These last two points lead to pretty obvious choices for the frequency of the (co)sines.

As for the fore-factors: they typically do not matter. At least for linear problems, they just scale right through (if you choose the solution twice as large, the error will also be twice as large). For nonlinear problems, however, you will want to choose these factors so that the nonlinearity is there and important, but not overwhelming your algorithm.