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Philippe Helluy


6 Project: the Discontinous Galerkin Lattice Boltzmann Method (DGLBM)


As explained in section 2.3 we have developed in Tonus very efficient numerical methods for solving kinetic equations. Recently, I have launched a research project with my team for harnessing kinetic solvers for general systems of conservation laws.

6.1 Kinetic modeling

Indeed, I have recently realized that all conservative systems of conservation laws admit a minimalist kinetic interpretation. The kinetic formalism mimics the Boltzmann theory of gas. The ingredients of a kinetic model are the following:

When the relaxation time τ is small, the kinetic equation provides an approximation of the hyperbolic conservative system

∂tw + ∇ ⋅F(w) = 0,

with

        ∫
Fi(w) =   viMw  (v)K (v)dv.
         v

The main idea is that numerical solvers for the linear scalar transport equation lead to natural solvers for the non-linear hyperbolic system. This approach is very general and very fruitful for theoretical reasons. For instance, it permits to construct numerical fluxes with good mathematical properties for general finite volume methods.

The kinetic model can also be solved directly when the velocity space V is small, typically a lattice with a few points. With small velocity lattices, the method presents many advantages for parallelism, generic implicit solvers, stability, asymptotic properties, etc. The standard Lattice Boltzmann Method (LBM) consists in solving the transport equation (1 ) exactly with the characteristic method. Its main drawback is that this imposes Cartesian space grids and that the time step Δt is fixed by the gris step Δx. In the Discontinuous Galerkin LBM (DGLBM) the transport equation is solved with a Discontinuous Galerkin method. This is very interesting because then the time step is free, the mesh can be unstructured and the method can easily be made implicit without the actual resolution of a large linear system.

Two preprints on the DGLBM are available here:

https://hal.archives-ouvertes.fr/hal-01422922

https://hal.archives-ouvertes.fr/hal-01451393

6.2 DGLBM: future works

In the next year, my main objective is to explore many aspects of the DGLBM methods and applications. Many questions arise and many software developments are needed: