Hybridised Mimetic Discretisation and Variational Multiscale Theory for Advection-Dominated Problems

Abstract

Structure-preserving or mimetic discretisations are a class of advanced discretisation techniques derived by employing concepts from differential geometry. Such techniques can attain specific conservation properties at the discrete level such as conservation of mass, kinetic energy, etc when applied to conservation laws. However, like traditional techniques, they are not entirely robust in specific multiscale cases such as under-resolved simulations which are often encountered in industrial applications. This ties into the concept of Large Eddy Simulations (LES) of which an enticing approach is the Variational Multiscale (VMS) method. Both Mimetic methods and VMS approaches have been extensively studied independently, however, their combination offers the potential of achieving a favorable robust solver capable of handling complicated industrial problems. Moreover, the extension of the Hybridised variant of the Mimetic methods towards advection-dominated problems is an interesting avenue yet to be explored. Therefore, this thesis focuses on the extension of the Hybridised Mimetic method and its combination with the VMS theory. The said combination is tested on simple linear equations, namely the advection-diffusion equation in both steady and unsteady cases. Additionally, the Hybridised Mimetic method is studied in more complex test cases such as the Burgers' equation and the incompressible Euler/Navier-Stokes equations.