On a convergent DSA preconditioned source iteration for a DGFEM method for radiative transfer. (English) Zbl 1446.65177
Summary: We consider the numerical approximation of the radiative transfer equation using discontinuous angular and continuous spatial approximations for the even parts of the solution. The even-parity equations are solved using a diffusion synthetic accelerated source iteration. We provide a convergence analysis for the infinite-dimensional iteration as well as for its discretized counterpart. The diffusion correction is computed by a subspace correction, which leads to a convergence behavior that is robust with respect to the discretization. The proven theoretical contraction rate deteriorates for scattering dominated problems. We show numerically that the preconditioned iteration is in practice robust in the diffusion limit. Moreover, computations for the lattice problem indicate that the presented discretization does not suffer from the ray effect. The theoretical methodology is presented for plane-parallel geometries with isotropic scattering, but the approach and proofs generalize to multi-dimensional problems and more general scattering operators verbatim.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 65F08 | Preconditioners for iterative methods |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 78A45 | Diffraction, scattering |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
| 82D75 | Nuclear reactor theory; neutron transport |
Keywords:
radiative transfer; discontinuous angular approximation; discrete ordinates method; diffusion synthetic acceleration; convergence ratesReferences:
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