• Speaker: Jerome Droniou, Monash University
  • Title: Numerical schemes for diffusion equations: how to construct them, and how to analyse their convergence under real-world constraints
  • Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
  • Time and Date: 4:00 pm, Wed, 23rd Jul 2014
  • Abstract:

    A vast amount of natural processes can be modelled by partial differential equations involving diffusion operators. The Navier-Stokes equations of fluid dynamics is one of the most popular of such models, but many other equations describing flows involve diffusion processes. These equations are often non-linear and coupled, and theoretical analysis can only provided limited information on the qualitative behaviours of their solutions. Numerical analysis is then used to obtain a prediction of the fluid's behaviour.

    In many circumstances, the numerical methods used to approximate the models must satisfy engineering or computational constraints. For examples, in underground flows in porous media (involved in oil recovery, carbon storage or hydrogeology), the diffusions properties of the medium vary a lot between geological layers, and can be strongly skewed in one direction. Moreover, the available meshes used to discretise the equations may be quite irregular. The sheer size of the domain of study (a few kilometres wide) also calls for methods that can be easily parallelised and give good and stable results on relatively large grids. These constraints make the construction and study of numerical methods for diffusion models very challenging.

    In the first part of this talk, I will present some numerical schemes, developed in the last 10 years and designed to discretise diffusion equations as encountered in reservoir engineering, with all the associated constraints. In the second part, I will focus on mathematical tools and techniques constructed to analyse the convergence of numerical schemes under realistic hypotheses (i.e. without assuming non-physical smoothness on the data or the solutions). These techniques are based on the adaptation to the discrete setting of functional analysis results used to study the continuous equations.

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