2018/10/16 by Wolfgang Dahmen, Dahmen, Wolfgang, Felix Gruber +3 · 1 citation
Mathematics · Engineering · #Numerical methods in inverse problems #Gas Dynamics and Kinetic Theory #Radiative Heat Transfer Studies
paper · pdf · doi:10.48550/arxiv.1810.07035
We propose a new approach to the numerical solution of radiative transfer\nequations with certified a posteriori error bounds. A key role is played by\nstable Petrov--Galerkin type variational formulations of parametric transport\nequations and corresponding radiative transfer equations. This allows us to\nformulate an iteration in a suitable, infinite dimensional function space that\nis guaranteed to converge with a fixed error reduction per step. The numerical\nscheme is then based on approximately realizing this iteration within\ndynamically updated accuracy tolerances that still ensure convergence to the\nexact solution. To advance this iteration two operations need to be performed\nwithin suitably tightened accuracy tolerances. First, the global scattering\noperator needs to be approximately applied to the current iterate within a\ntolerance comparable to the current accuracy level. Second, parameter dependent\nlinear transport equations need to be solved, again at the required accuracy of\nthe iteration. To ensure that the stage dependent error tolerances are met, one\nhas to employ rigorous a posteriori error bounds which, in our case, rest on a\nDiscontinuous Petrov--Galerkin (DPG) scheme. These a posteriori bounds are not\nonly crucial for guaranteeing the convergence of the perturbed iteration but\nare also used to generate adapted parameter dependent spatial meshes. This\nturns out to significantly reduce overall computational complexity. Since the\nglobal operator is only applied, we avoid the need to solve linear systems with\ndensely populated matrices. Moreover, the approximate application of the global\nscatterer accelerated through low-rank approximation and matrix compression\ntechniques. The theoretical findings are illustrated and complemented by\nnumerical experiments with non-trivial scattering kernels.\n