Enriched Galerkin-Characteristics Finite Element Method for Incompressible Navier--Stokes Equations

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

“…This solution can be formulated as $$$$ {\boldsymbol{\chi}}_j^n={\boldsymbol{x}}_j-{\int}_{t_n}^{t_{n+1}}{\boldsymbol{U}}_h\left({\boldsymbol{\chi}}_j(t),t\right)\kern2.56804pt dt. $$$$ In the Lagrangian framework, the Equation (7) is usually evaluated using numerical integration techniques such as the Runge‐Kutta and Euler approaches 15‐19 . However, while these solvers are generic, they have the inconvenient feature of quick system energy development (Hamiltonian) in long‐time computations and they are inappropriate for particle tracking since they are not structure‐preserving (SP) methods, see for example Reference 20.…”

“…where is the velocity solution calculated using (26) at the departure point $$$ {\boldsymbol{\chi}}_{q,k}^n $$$. Note that the search‐locate algorithm used in References 17‐19 is accurate and suitable for the conventional semi‐Lagrangian method. However, in our current enriched method for each quadrature point in each element, we compute the corresponding departure point.…”

“…Illustration of the search‐locate algorithm used in References 17‐19 (left plot) and the modified search‐locate algorithm adopted in the present study (right plot). [Colour figure can be viewed at wileyonlinelibrary.com]…”

“…However, when dealing with real applications, solving convection‐dominated flow problems using this class of methods becomes computationally expensive due to the huge density of quadratures needed by the $$$ {\mathrm{L}}^2 $$$‐projection for the computation of solutions at the departure points. To overcome this drawback, a novel class of adaptive enriched semi‐Lagrangian finite element method has been proposed for convection‐diffusion problems in Reference 17, for incompressible Navier–Stokes equations in Reference 18 and for coupled Navier–Stokes/Boussinesq equations in Reference 19. This approach enriches locally the approximation of integrals in the $$$ {L}^2 $$$‐projection using a family of quadrature rules which allows for spatial discretization on coarse fixed meshes.…”

“…To evaluate the departure points, a second-order extrapolation based on the midpoint rule is used in previous works. [17][18][19] However, these typical solvers are inappropriate to compute departure points because of the feature of the quick expansion of the invariants in large-time computations. 20 Consequently, the overall accuracy of the semi-Lagrangian schemes is highly sensitive to the evaluation of the departure points.…”