Structure preserving discretization for CFD

Structure-preserving numerical methods aim to preserve physical conservation laws exactly in a discrete setting. Examples are: exact conservation of mass, conservation of kinetic energy and helicity in time.  In order to establish this one needs a formulation (Lagrangians, Hamiltonians) which includes the conservation properties and appropriate finite dimensional sub-spaces in which the conserved quantities can be represented.

Spatial discretization

  • Mimetic spectral element methods: Higher order methods which satisfy a discrete De Rham sequence. This allows for the conservation of many physical properties at the discrete level, see or
  • The use of algebraic dual polynomial spaces: In finite element methods one looks for the (optimal) solution in a linear vector space. Associated to linear vector spaces are their algebraic dual spaces. The combination of primal and dual spaces leads to:
    1. very sparse spectral element methods;
    2. parts of the system matrix which are independent of the size and shape of the mesh;
    3. low condition number of the resulting system

For more information see:

Figure 1: General shape of a hybridized stiffness matrix. Once the interface variables λ are found, the system reduces to fully independent problems in the sub-domains.
  • Hybrid finite element methods: Hybrid finite element methods are based on domain decomposition techniques. These methods were proposed midway the 1960’s. By solving variables in a suitably chosen trace space, the method allows for full parallelization of the solution in the sub-domains. Furthermore, hybrid methods admit a larger class of admissible boundary conditions. In solid mechanics, for instance, both the displacements and the forces at the boundary are available. Such methods will play a role in multi-physics applications such as fluid-structure interaction or conjugate heat transfer problems.
Figure 2: Solution of the Poisson equation where we impose continuity of the fluxes between elements. Left: exact solution, middle: degrees of freedom, right: the (discontinuous) numerical solution)

In order to conserve certain physical quantities in time special time stepping procedures need to be developed. In the Aerodynamics group research in time-stepping techniques entails:

  • Investigation of the application of Spectral Deferred Corrections (SDC), Integral Deferred Corrections (IDC) and the Picard Integral Exponential Solver (PIES) as a means to perform time integration for unsteady flow and fluid-structure interaction problems. By dividing a time interval into a number of sub-steps, an arbitrary order of accuracy can be achieved by performing a number of sweeps with a low order method over the time interval. Each sweep results in a correction to the solution. This allows the use of parallel-in-time discretization that can potentially speed up unsteady simulations on cluster computers. A second benefit is the possibility to reconstruct the solution at any point in the interval, which can be beneficial when coupling to another solver that uses a more refined time discretization, as one could encounter in fluid-structure interaction or fluid-acoustic coupling.
  • Dual time integration methods to ensure conservation in time. Staggered time stepping schemes,, energy-preserving time-stepping and symplectic time-stepping methods,

For more information: M.I. (Marc) Gerritsma A.H. (Sander) van Zuijlen