The Discontinuous Asymptotic Telegrapher’s Equation (P₁) Approximation

Abstract

Modeling the propagation of radiative heat-waves in optically thick material using a diffusive approximation is a well-known problem. In optically thin material, classic methods, such as classic diffusion or classic P1, yield the wrong heat wave propagation behavior, and higher order approximation might be required, making the solution harder to obtain. The asymptotic P1 approximation [Heizler, NSE 166, 17 (2010)] yields the correct particle velocity but fails to model the correct behavior in highly anisotropic media, such as problems that involve sharp boundary between media or strong sources. However, the solution for the two-region Milne problem of two adjacent halfspaces divided by a sharp boundary, yields a discontinuity in the asymptotic solutions, that makes it possible to solve steady-state problems, especially in neutronics. In this work we expand the timedependent asymptotic P1 approximation to a highly anisotropic media, using the discontinuity jump conditions of the energy density, yielding a modified discontinuous P1 equations in general geometry. We introduce numerical solutions for two fundamental benchmarks in plane symmetry. The results thus obtained are more accurate than those attained by other methods, such as Flux-Limiters or Variable Eddington Factor. * avnerco@gmail.com † highzlers@walla.co.il deterministic methods, and they are both exact when N → ∞ [13]. Alternatively, a statistical implicit Monte Carlo (IMC) approach can also be used [14], which is exact when the number of particles (histories) goes to infinity. Although these three methods approach the exact solution, their application requires extensive numerical calculations that might be difficult to carry out, especially in multi-dimensions. Hence, there is an extensive body of literature dealing with the search for approximate models which will be relatively easy to simulate, and yet produce solutions that are close to the exact problem (for example, see [15,16]).The classical (Eddington) diffusion theory, as a specific case of the P 1 is relatively easy to solve and is commonly used [1,4,13]. The diffusion equation is parabolic, and thus yields infinite particle velocities. The full P 1 equations, that give rise to the Telegrapher's equation, has a hyperbolic form, but with an incorrect finite velocity, c/ √ 3 [17]. Possible solutions, such as flux-limiters (FL) solution (in the form of a non-linear diffusion notation), or Variable Eddington Factor (VEF) approximations (in the form of full P 1 equations), yielding a gradient-dependent nonlinear diffusion coefficients (or a gradient-dependent Eddington factor), are harder to solve, especially in multi-dimensions [15,16,[18][19][20][21][22][23][24].In previous work, Heizler [17] offered a modified P 1 approximation, based on the asymptotic derivation (both in space and time), the asymptotic P 1 approximation (or the asymptotic Telegrapher's equation approximation) [17]. In steady state, it tends to the well-known asymptotic diffusion approximation [13,25,26]. This approximation shar...

Multi-Group Discontinuous Asymptotic $P_1$ Approximation in Radiative Marshak Waves Experiments

“…when A g = B g = 3. In the asymptotic P 1 approximation instead, the coefficients A g ( r, t) and B g ( r, t) are media-dependent, which have closed sets of functions (not free parameters), of ω eff,g ( r, t), the group-dependent mean number of particles that are emitted per collision or source terms and (for the non-scattering case) is defined by [3,17]:…”

“…The coefficients are derived from an asymptotic derivation, both in space and time, as detailed in [13,18]. For the exact functional dependence of A g (ω eff,g ) and B g (ω eff,g ), please see [13,17].…”

“…This approximation assumes continuous radiation flux F A = F B (between semi-infinite medium A and semiinfinite medium B) and discontinues energy density, µ A E A = µ B E B , when µ(ω eff,g ) is a medium dependent function, and derived by applying the Marshak boundary condition on the asymptotic distribution near a boundary between two different media [3]. We [17] expanded these these boundary jump conditions to the whole medium for the time-dependent asymptotic P 1 equation (Eq. 8):…”

“…The discontinuous asymptotic P 1 approximation is being derived to approximate the Boltzmann solution in heterogeneous media [17]. It rests on two foundations: (1) the asymptotic P 1 approximation [13,18,19,20];…”

“…These modifications allow the use of a P 1 form that is accurate even in highly-anisotropic scenarios. This approximation was tested via wellknown benchmarks, showing better results than other common approximations [17]. In addition, it was also compared to the results in one of the main Marshak wave experiments (using SiO 2 foam [6]) and IMC gray (mono-energetic) simulations of the same problem, showing good agreements [16].…”