A new Implicit Monte-Carlo scheme for photonics (without teleportation error and without tilts)

Abstract

In this paper, we present a new implicit Monte-Carlo scheme for photonics. The new solver combines the benefits of both the IMC solver of Fleck & Cummings and the SMC solver of Ahrens & Larsen. It is implicit hence allows taking affordable time steps (as IMC) and has no teleportation error (as SMC). The paper also provides some original analysis of existing schemes (IMC, tilted IMC, SMC), especially with respect to the teleportation error in the equilibrium diffusion regime. In particular, we demonstrate that any small spatial inaccuracies during the sampling of source particles for IMC lead to a competing behaviour between the spatial and time discretisation parameters. The new scheme we suggest is implicit, conservative, has no teleportation error (and as a consequence does not need tilting), does not rely on source sampling for the emission of source particles, captures the equilibrium diffusion limit (provided a small enough time step), can be used with arbitrary equations of state and does not suffer the above competing behaviour. All those properties are either demonstrated or numerically highlighted in the paper.

Efficient uncertainty propagation for photonics: Combining Implicit Semi-analog Monte Carlo (ISMC) and Monte Carlo generalised Polynomial Chaos (MC-gPC)

“…The choice of the linearisation considerably affects the structure of the code and the choice of the MC scheme. Several linearisations can be chosen (explicit MC [56], IMC [57,58], titled-IMC [59,29,27], SMC [60], ISMC [9]). We would like, the more possible, to preserve the relevant properties of the chosen linearisation once combined with MC-gPC.…”

“…fast) convergence of the built hierachical models has been numerically [10] and theoretically [30] demonstrated. The approach in [10,31,33,30], denoted by MC-gPC in the following, enters the class of intrusive 9 gPC methods. It consequently demands some code modifications.…”

“…In this paper, we design an ISMC-gPC solver. It relies on the combined material of [10] (for MC-gPC) and [9] (for ISMC 12 ) plus some additional key ingredients proper to the photonic problem we aim at addressing in this paper (mainly positiveness, boundedness and moment preserving strategies).…”

“…Those are mainly inspired from the hyperbolicity-preserving strategies of the literature [1,2,3,4,5,6,7,36,8] as similar difficulties are encountered for the reduced models we build in this paper. In section 3, we recall the ISMC solver from [9] and explain in which way we think it is a good candidate to be combined to MC-gPC. The general sketch of an ISMC solver is also recalled.…”

“…In order to enforce the aforementioned wellposedness conditions, several strategies, inspired from the hyperbolicity-preserving ones [1,2,3,4,5,6,7,8] are reviewed, adapted and analysed. The resolution of the reduced models is performed thanks to an astute combination of the Implicit Semi-analog MC (ISMC, see [9]) scheme for photonics and of MC-gPC (see [10]) for uncertainty propagation. This work demonstrates that MC-gPC can be efficiently applied to a stiff nonlinear set of partial derivative equations if the MC resolution allows a fast convergence with respect to both the time and spatial discretisation (the latter properties being allowed by ISMC).…”