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.
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:
- a distribution function: f(x,v,t), x ∈ ℝd, v ∈ V ⊂ ℝd, t ∈,
- a microscopic “collision vector” K(v) ∈
ℝm and macroscopic conserved data w
- a microscopic “Maxwellian” Mw(v)
- a kinetic equation with relaxation source term:
When the relaxation time τ is small, the kinetic equation provides an approximation of the hyperbolic conservative system
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:
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:
- physical models: I have applications in view in MHD for computing tokamak instabilities (PhD of Conrad Hillairet) or low Mach flows, for instance;
- time integration: using composition methods coming from the geometric integration community we can achieve arbitrary order in the time integration. The resulting scheme has excellent low storage properties;
- boundary conditions: in its simplest form the implicit solver does not accept complicated boundary conditions. Using optimal control methods it should be possible to generalize to arbitrary boundary conditions;
- parallelism: we have started to implement in SCHNAPS the DGLBM solver. We use StarPU for distributing automatically, in parallel, the tasks on a MPI cluster of multicore CPU nodes. Much work remains to be done for addressing also GPU or larger supercomputers;
- theoretical aspects: for small finite relaxation time τ the kinetic model is an approximation of a conservative system with second order diffusive terms. With adaptations of the kinetic model, it is often possible to obtain adapted kinetic models for representing Navier-Stokes equations or MHD with the correct resistive terms.
- Finally, the DGLBM is a direct method for solving time-dependent PDE. It is very well adapted to new computer architectures and fast computations. It is thus also a good candidate for being the building block of higher level problematic, such as uncertainty quantification or data assimilation modeling that require to solving a large number of direct problems. These topics also have to be investigated.