*High order and adaptive methods for plasma physics
* Boundary Integral Equations in the Time Domain
* Discretization of Boundary Integral Equations with reduced quadrature.
* Time-domain scattering of linear elastic waves.
* Finite volume methods for non-linear conservation laws.
Plasma physicists and engineers working on nuclear fusion have to deal with complicated problems involving hydrodynamical simulations, kinetic equations, electromagnetic equilibria, etc. Due to the complexity of these problems the availability of accurate and robust numerical methods for their solution is vital for the further development of the field.
Working jointly with Antoine Cerfon and Manuel Solano we are applying high order hybridizable discontinuous Galerkin methods to determine magnetic equilibrium configurations on reactors with axial symmetry. The methods feature spatial adaptivity and can handle complex geometries using embedded polygonal meshes. We have also worked on spectral collocation methods using non-standard grids tailored for the solution of some Fokker-Plank drift kinetic equations appearing in plasma physics.
Boundary integral formulations exploit the possibility of transferring a linear "volume" problem into the boundary of the region via the fundamental solution, effectively decreasing the dimensionality of the problem by one and are particularly well suited for the treatment of exterior problems in unbounded domains.
In the time domain we make heavy use of Christian Lubich's Convolution Quadrature (CQ) , which allows us to use only the Laplace-domain fundamental solution in combination with time-domain boundary data to produce approximations of the time-domain solution.
deltaBEM is an inexpensive and simple way to discretize BIE's in 2D domains with smooth parametrizable boundaries. The method can be understood as a Petrov-Galerkin discretization with piecewise constant and delta trial and test functions whose shape has been optimized for third order convergence. An extension of this method for 3D scattering is the subject of ongoing research.
In collaboration with Víctor Domínguez, Matthew Hassell, Sijiang Lu, Tianyu Qiu and Francisco-Javier Sayas, this method was implemented for all the operators of the Calderón projector for the Laplace, Helmholtz and Navier-Lamé equations. The matlab code is available and can be downloaded here. Below is a sample of frequency domain acoustic scattering by open arcs and elastic scattering by multiple rigid obstacles. Both simulations were done with deltaBEM.
Numerical simulation of a plane pressure wave interacting with a rigid scatterer with homogeneous Dirichlet boundary conditions. deltaBEM was used for space discretization and Trapezoidal Rule CQ was used for time evolution. Thanks to the boundary integral formulation, the computations are done solely on the boundary of the scatterers and post processed for visualization without the need for absorbing boundary conditions or perfectly matched layers. The magnitude of the displacement is shown on the left, while the right image shows its direction.
We've proposed pure boundary integral formulations and coupled boundary and variational formulations for the problem and showed the system is well posed for Lipschitz scatterers and sufficiently smooth incident waves. BEM/FEM coupling can then be used to deal with inhomogeneous anisotropic solids embedded in homogeneous fluids or BEM/BEM for the case of homogeneous properties. The simulation below, for the case of a homogeneous scatterer, was done using deltaBEM for both acoustic and elastic wavefields and Trapezoidal Rule CQ. The triangulation inside the scatterer is given only as a reference for the direction of the elastic displacement.
The simulation below shows the case of an inhomogeneous solid scatterer, it is handled with the coupled BEM/FEM formulation using Trapezoidal Rule CQ and time stepping. The computation used P2 Lagrangian finite elements for the elastic displacement and P2/P1 Continuous/Discontinuous Galerkin boundary elements for the acoustic field. The right half shows a close-up of the elastic body and the computational mesh.
The case of a piezoelectric scatterer introduces a third unkown in the problem, the electric potential inside the solid, and results in a loss of isotropy. In applications, the electric potential is not assumed to satisfy a wave equation, but only the usual Poisson equation in a dielectric. We have formulated a coupled boundary integral and variational formulation aiming to discretize the acoustic wave with boundary elements and the elastic and electric fields with finite elements.
In the past I was involved on the development of a FORTRAN code for the solution of Euler's equations of gas dynamics in vaccuum. The 3D implementation was based on dimensional splitting and explored different choices of flux limiters to increase the order of the approximation in the regions far from the shock profiles. The simulation below shows a density countour plot of two gas spheres colliding at supersonic speed.