Light aircraft dynamics, car and Formula One designs, low speed wind tunnels or petroleum reservoir simulations are some of the numerous engineering examples involving incompressible viscous flows. Obtaining a good level of precision in numerical results is then highly interesting, especially when dealing with zones such as boundary layers. In the context of Finite Element Methods, an alternative to using a finer mesh to increase precision and obtain high-fidelity is to define high-order approximations in the mesh elements, using for instance a Discontinuous Galerkin (DG) formulation.

High-order DG spatial discretization

In the context of high-fidelity DG for incompressible flow, the number of degrees of freedom of the problem is significantly reduced using solenoidal approximations, whose definition and implementation is straightforward.

Several competitive DG methods with solenoidal approximations where velocity and pressure are partially or completely decoupled have been developed within the LaCàn research group.

See the following published articles for more information:

Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations, Montlaur, A., Fernández-Méndez, S., Huerta, A., International Journal for Numerical Methods in Fluids, Vol. 57, Issue 9, pp. 1071-1092, 2008.

Discontinuous Galerkin methods for the Navier-Stokes equations using solenoidal approximations, Montlaur, A., Fernandez-Mendez, S., Peraire, J. and Huerta, A., International Journal for Numerical Methods in Fluids , Vol. 64, Issue 5, pp. 549-565, 2010.

High-order temporal methods

High-order time approximations, such as Runge-Kutta and Rosenbrock methods for Differential Algebraic Equations, are also considered for unsteady problems so that the resulting scheme presents the attractive features of high-accuracy both in space and in time.

High-order unconditionally stable time integration methods have been obtained in 2D, allowing maximum flexibility even in meshes where the variation of size of elements is critical. Current researchs aims to adapt the code to 3D engineering problems.

See the following published articles for more information:

High-order implicit time integration for unsteady incompressible flows, Montlaur, A., Fernández-Méndez, S., Huerta, A., to appear in International Journal for Numerical Methods in Fluids, 2011.

Métodos Runge-Kutta Implícitos de alto orden para flujo incompresible, Montlaur, A., Fernández-Méndez, S., Huerta, A., Revista Internacional Métodos Numéricos para Cálculo y Diseño en Ingeniería, Vol. 27, Issue 1, pp-77-01, 2011.