A projection scheme for phase change problems with convection

A fast and accurate method for transport and dispersion of phosphogypsum in coastal zones: Application to Jorf Lasfar

“…To solve the stokes problem (6), we formulate the coupled projection scheme based on a rotational pressure correction with a BDF2 time integration method as those studied in Reference 22. Thus, we solve the Stokes problem (6) for updating the velocity field and pressure at every step using the coupled projection scheme in order to complete the implementation of semi‐Lagrangian finite element approach for solving incompressible Navier–Stoke equations (1).…”

“…Comparing to the standard projection methods, the presence of previous gradient step in the velocity prediction problem improves the order of the scheme. Moreover, the rotational pressure correction projection enhanced the method to avoids artificial boundary conditions on the pressure and improves its rate of convergence 22 . Hence, assuming that $$$ {\hat{\boldsymbol{U}}}^n $$$, $$$ {\hat{\boldsymbol{U}}}^{n-1} $$$, $$$ {\hat{C}}^n $$$, $$$ {\hat{C}}^{n-1} $$$ and $$$ {p}^n $$$ are known, the Stokes equations (6) are solved using the following steps: Solve for $$$ {C}^{n+1}\in {\mathbf{V}}_h $$$ $$$$ \frac{3}{2\Delta t}{C}^{n+1}-\nabla \cdotp \left(D\nabla {C}^{n+1}\right)=-\frac{4}{2\Delta t}{\hat{C}}^n+\frac{1}{2\Delta t}{\hat{C}}^{n-1}+{S}^n.$$…”

“…Moreover, the rotational pressure correction projection enhanced the method to avoids artificial boundary conditions on the pressure and improves its rate of convergence. 22 Hence, assuming that Ûn , Ûn−1 , Ĉn , Ĉn−1 and p n are known, the Stokes equations (6) are solved using the following steps:…”

“…However, the main drawback of this method is the computational cost especially when solving the large systems of algebraic equations associated with the discretized problem. To overcome this drawback, we proposed in the current work a coupled projection scheme based on a rotational pressure correction with a second‐order backward difference formula for the time integration presented in Reference 22. Compared to the standard projection methods, the presence of the previous gradient step in the velocity prediction problem improves the accuracy order of the method.…”

On the convergence of a low order Lagrange finite element approach for natural convection problems

“…A common approach is to use a single domain method [1,2] for the momentum equation. Projection schemes [3] or Newton based algorithms [1,4] have been shown to give accurate results. Enthalpy models have been proven to be suitable for phase change, even with different thermophysical properties between the two phases, especially when combined with adaptive mesh refinement [5,6].…”