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Register a new userAn Implicit Finite Volume Scheme to Solve the Time Dependent Radiation Transport Equation Based on Discrete Ordinates
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Yan-Fei Jiang
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We describe a new algorithm to solve the time dependent, frequency integrated radiation transport (RT) equation implicitly, which is coupled to an explicit solver for equations of magnetohydrodynamics (MHD) using {sf Athena++}. The radiation filed is represented by specific intensities along discrete rays, which are evolved using a conservative finite volume approach for both cartesian and curvilinear coordinate systems. All the terms for spatial transport of photons and interactions between gas and radiation are calculated implicitly together. An efficient Jacobi-like iteration scheme is used to solve the implicit equations. This removes any time step constrain due to the speed of light in RT. We evolve the specific intensities in the lab frame to simplify the transport step. The lab-frame specific intensities are transformed to the co-moving frame via Lorentz transformation when the source term is calculated. Therefore, the scheme does not need any expansion in terms of $v/c$. The radiation energy and momentum source terms for the gas are calculated via direct quadrature in the angular space. The time step for the whole scheme is determined by the normal Courant -- Friedrichs -- Lewy condition in the MHD module. We provide a variety of test problems for this algorithm including both optically thick and thin regimes, and for both gas and radiation pressure dominated flows to demonstrate its accuracy and efficiency.
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The time-dependent radiation transport equation is discretized using the meshless-local Petrov-Galerkin method with reproducing kernels. The integration is performed using a Voronoi tessellation, which creates a partition of unity that only depends on the position and extent of the kernels. The resolution of the integration automatically follows the particles and requires no manual adjustment. The discretization includes streamline-upwind Petrov-Galerkin stabilization to prevent oscillations and improve numerical conditioning. The angular quadrature is selectively refineable to increase angular resolution in chosen directions. The time discretization is done using backward Euler. The transport solve for each direction and the solve for the scattering source are both done using Krylov iterative methods. Results indicate first-order convergence in time and second-order convergence in space for linear reproducing kernels.
In a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a scheme for obtaining very close seeds for solving the elliptic Kepler Equation with the classical and the modified Newton-Rapshon methods. This implied an important reduction in the number of iterations needed to reach a given accuracy. However, RPP also made strong claims about the errors of their method that are incorrect. In particular, they claim that their accuracy can always reach the level $sim5varepsilon$, where $varepsilon$ is the machine epsilon (e.g. $varepsilon=2.2times10^{-16} $ in double precision), and that this result is attained for all values of the eccentricity $e<1$ and the mean anomaly $Min[0,pi]$, including for $e$ and $M$ that are arbitrarily close to $1$ and $0$, respectively. However, we demonstrate both numerically and analytically that any implementation of the classical or modified Newton-Raphson methods for Keplers equation, including those described by RPP, have a limiting accuracy of the order $simvarepsilon/sqrt{2(1-e)}$. Therefore the errors of these implementations diverge in the limit $eto1$, and differ dramatically from the incorrect results given by RPP. Despite these shortcomings, the RPP method can provide a very efficient option for reaching such limiting accuracy. We also provide a limit that is valid for the accuracy of any algorithm for solving Kepler equation, including schemes like bisection that do not use derivatives. Moreover, similar results are also demonstrated for the hyperbolic Kepler Equation. The methods described in this work can provide guidelines for designing more accurate solutions of the elliptic and hyperbolic Kepler equations.
We present a new algorithm for radiative transfer, based on a statistical Monte-Carlo approach, that does not suffer from teleportation effects on the one hand, and yields smooth results on the other hand. Implicit-Monte-Carlo (IMC) techniques for modeling radiative transfer exist from the 70s. However, in optically thick problems, the basic algorithm suffers from `teleportation errors, where the photons propagate faster than the exact physical behavior, due to the absorption-black body emission processes. One possible solution is to use semi-analog Monte-Carlo, in its new implicit form (ISMC), that uses two kinds of particles, photons and discrete material particles. This algorithm yields excellent teleportation-free results, however, it also results with nosier solutions (relative to classic IMC) due to its discrete nature. Here, we derive a new Monte-Carlo algorithm, Discrete implicit Monte-Carlo (DIMC) that uses the idea of the two-kind discrete particles and thus, does not suffer from teleportation errors. DIMC implements the IMC discretization and creates new radiation photons each time step, unlike ISMC. This yields smooth results as classic IMC, due to the continuous absorption technique. One of the main parts of the algorithm is the avoidance of population explosion of particles, using particle merging. We test the new algorithm in both one and two-dimensional cylindrical problems, and show that it yields smooth, teleportation-free results. We finish in demonstrating the power of the new algorithm in a classic radiative hydrodynamic problem, an opaque radiative shock wave. This demonstrates the power of the new algorithm in astrophysical scenarios.
We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations based on finite volume with piecewise constant projections and domain decomposition techniques. Next we present an iterative method for solving the composite-grid system that reduces to solution of standard problems with standard time stepping on the coarse and fine grids. At every step of the algorithm, conservativity is ensured. Finally, numerical results illustrate the accuracy of the proposed methods.
In this paper, a fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and $L^2$ error estimate for the linear case with the convergence rate $O(h^{k+1}+(Delta t)^2+(Delta t)^frac{alpha}{2}h^{k+1/2})$. Numerical examples are presented to show the efficiency and accuracy of our scheme.
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