Reference : Numerical schemes for tracer studies in shallow shelf seas
Dissertations and theses : Doctoral thesis
Physical, chemical, mathematical & earth Sciences : Earth sciences & physical geography
Numerical schemes for tracer studies in shallow shelf seas
Mercier, Christophe mailto [Université de Liège - ULiège > Département d'aérospatiale et mécanique > Mathématiques générales >]
Université de Liège, ​Liège, ​​Belgium
Doctorat en sciences de l'ingénieur
Delhez, Eric mailto
Ponthot, Jean-Philippe mailto
Pietrzak, Julie
Pirotton, Michel mailto
Beckers, Jean-Marie mailto
Deleersnijder, Eric
[en] TVD schemes ; monotonicity preservation ; CART ; Age ; Sediment model ; advection
[en] Nowadays, tracers have revealed to be an invaluable integrated tool to investigate the different thematics related to coastal areas, shallow shelf seas or estuaries. In this work, they are used in combination with a parallel scalable Fortran90 implementation of a 3D hydrodynamic model to study the transport of mud within the Belgian Coastal Zone. The results obtained illustrate the need for tools to understand the interactions between such highly energetic coastal areas and the sediment dynamics. The Constituent Age and Residence time Theory (CART), originally applied to passive tracers, is therefore extended to describe the sediments and used to quantify their rates of resuspension and horizontal transport.

This application reveals some numerical issues that may degrade the reliability of the results. The different numerical parameterizations are thus revisited starting with the use of advection schemes, and especially, the Total Variation Diminishing (TVD) schemes built along the line defined by Harten. It is shown that the blind application of the usual TVD schemes and associated flux limiters can lead to non-TVD solutions when applied to complex geometries. Spatial and/or temporal variations of the bathymetry can indeed break the TVD property. Hopefully, really TVD schemes can be recovered by taking these variations into account in the formulation of the flux limiters.

Even if its Eulerian formulation eases the implementation of CART into existing models, it is shown that CART's computation procedure lacks a numerical framework for a robust application. To address this, we enforce the consistency, defined as the requirement that the integration scheme does not introduce errors in the age field when aging is the only active process. Unfortunately, this does not guarantee the absence of overshootings. While it does not seem feasible to ensure a TVD behavior of the age field, appropriate modifications of the flux/slope limiters are derived here to prevent the occurrence of age values outside the physically acceptable range.

On the bases of the solutions obtained after the advection of any numbers of tracers using high-order non-linear schemes, several control or global variables can be built. Even if the independent advection schemes used are specifically built to produce profiles free from the numerical artifacts that can spoil the physical meaning of the results, such global/control variables can exhibit these unwanted behaviors. This problematic is firstly treated with regard to the TVD schemes built from Harten's theorem. But guaranteeing the TVD property is not always feasible. Alternatively, appropriate modifications of the flux/slope limiters are implemented here to enforce a maximum principle for any global/control variable built as first-order homographic function of its components.
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