A General Methodology for Symbolically Generating Manufactured Solutions Satisfying Prescribed Conditions: Application to Two-phase Flows Equations

Abstract

The Method of Manufactured Solution (MMS) is a powerful technique for code verification. It provides a systematic procedure for generating analytical solutions to be discretized by a numerical solver. Usually, an arbitrary continuous solution is selected before being inserted into the governing equations. Although sufficient in many situations, this procedure may be inappropriate if the problem at hand imposes some constraints on the shape of the manufactured solutions. In such cases, some unknown parameters should be included in the continuous solution and computed to satisfy the imposed constraints. This limitation of the standard MMS has already been recognized in previous work. However, the way to handle it is most of the time case-dependent and based on ad-hoc strategies. In this work, we develop a generic framework based on the Sympy library to produce manufactured solutions complying with arbitrary complex Dirichlet and Neumann-type conditions. It is made available through an open-source Python software. As a challenging illustrative application, analytical solutions obeying the interface jumps conditions of a two-phase compressible Navier–Stokes system are built.

Appendices

Example of Manufactured Solutions Incompatible with the Proposed Methodology

To illustrate the limitations of the current framework concerning the imposition of Neumann or Robin boundary/interface conditions, discussed in Sect. 2.2.2, we consider the example of the test case in Sect. 4.2. Instead of an horizontal interface, let’s consider the most general case of an interface whose normal vector depends on both spatial coordinates,

$$\begin{aligned} \textbf{n}_\Gamma = (n_{x,\Gamma }, \, n_{y,\Gamma }) (x,y) . \end{aligned}$$

Let’s focus on the shape of the manufactured temperature field in phase 2 that we keep dependent only on the y coordinate,

$$\begin{aligned} \widetilde{T}_{2}(y) = \lambda _{T,2} \, f(y) \, \end{aligned}$$

with f(y) a general function. The energy jump condition, i.e. the last equation in Eq. (21), can be rewritten by isolating the jump of the normal heat flux on one side,

$$\begin{aligned} \lambda _{T,2} \, \kappa _2 \, \left[ \frac{\text {d} f}{\text {d} y} n_y \right] \big |_{x_{\Gamma }, y_{\Gamma }} - \kappa _1 \, \left[ \frac{\text {d} \widetilde{T}_{1}}{\text {d}y} n_y \right] \big |_{x_{\Gamma }, y_{\Gamma }} = J(x,y, \lambda _{T,2}) \big |_{x_{\Gamma }, y_{\Gamma }} . \end{aligned}$$

(29)

Because of the dependence to (x, y) of all the terms in Eq. (29), the assumption that $\lambda _{T,2}$ is a constant was wrong, i.e.

$$\begin{aligned} \lambda _{T,2} = \text {constant} \Rightarrow \lambda _{T,2} = \lambda _{T,2}(x,y) . \end{aligned}$$

(30)

This forces us to correct the energy jump, Eq. (29), such that

$$\begin{aligned} \kappa _2 \, \left[ \frac{\partial \lambda _{T,2}}{\partial y} \, f \, n_y \right] \big |_{x_{\Gamma }, y_{\Gamma }} + \kappa _2 \, \left[ \lambda _{T,2} \, \frac{\text {d} f}{\text {d} y} n_y \right] \big |_{x_{\Gamma }, y_{\Gamma }} - \kappa _1 \, \left[ \frac{\text {d} \widetilde{T}_{1}}{\text {d}y} n_y \right] \big |_{x_{\Gamma }, y_{\Gamma }} = J(x,y, \lambda _{T,2}) \big |_{x_{\Gamma }, y_{\Gamma }} . \end{aligned}$$

(31)

It is now clear from Eq. (31), which involves the unknown $\lambda _{T,2}$ and one of its derivatives, that the system that would need to be solved is a non-linear differential algebraic system. As the solution of such system in a symbolic manner appears to be particularly intricated, we decided to select a priori the shape of the manufactured solution and the geometry of the interface or domain boundaries such that a standard algebraic system needs to be solved instead.

Example of PyManufSol MS Generation

An example of Python script importing the various modules of the PyManufSol software is given below. It is the one used to create the test case in Sect. 4.2.