Uniform convergent scheme for discrete-ordinate radiative transport equation with discontinuous coefficients on unstructured quadrilateral meshes

Higher order time-implicit bounds preserving discontinuous Galerkin discretizations for partial differential equations

“…Nonlinear, possibly degenerate, parabolic equations, describe many problems in science and engineering, such as radiative transport in the diffusive limit, flow of electrons and holes in semi-conductor devices, heat and mass transfer, combustion, flow in porous media, displacement of oil by water in oil reservoirs and the evolution of a gas of fermionic and Bose-Einstein particles. These phenomena are modelled for instance by the radiative transport equation in the diffusive limit [65,119], the drift-diffusion equation for semiconductors [13,64], the heat equation [19], the porous media equation [2,112,131], the Buckley-Leverett equation [13,73], and the nonlinear Fokker-Plank equation modelling fermion and boson gases [21,109]. Preserving bounds on the numerical solution of these parabolic and degenerate parabolic equations is non-trivial, but is further complicated by the fact that many bounds preserving numerical discretizations for parabolic equations also have a severe time step constraint.…”