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Register a new userMeshless discretization of the discrete-ordinates transport equation with integration based on Voronoi cells
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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.
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We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrodinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the systems Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to $Napprox;$70 sites. An advantage of the latter methods is the better conservation of the systems second integral, i.e. the wave packets norm.
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.
The self-adjoint angular flux and streamline-upwind Petrov-Galerkin transport equations are discretized using reproducing kernels with the collocation method to produce a discretization that is compatible with conservative reproducing kernel smoothed particle hydrodynamics. A novel second derivative is derived for the diffusion-like term in the self-adjoint angular flux equation. The resulting equations involve only evaluations of kernels and physical data at the nodal centers.
Torsional modes within a complex molecule containing various functional groups are often strongly coupled so that the harmonic approximation and one-dimensional torsional treatment are inaccurate to evaluate their partition functions. A family of multi-structural approximation methods have been proposed and applied in recent years to deal with the torsional anharmonicity.However, these methods approximate the exact almost periodic potential energy as a summation of local periodic functions with symmetric barrier positions and heights. In the present theoretical study, we illustrated that the approximation is inaccurate when torsional modes present non-uniformly distributed local minima. Thereby, we proposed an improved method to reconstruct approximate potential to replace the periodic potential by using information of the local minima and their Voronoi tessellation. First, we established asymmetric barrier heights by introducing two periodicity parameters and assuming that the exact barrier positions are at the boundaries of Voronoi cells. Second, we used multiplicatively weighted Voronoi tessellation to refine the barrier heights and positions by defining a structure-related distance metric. The proposed method has been tested for a few higher-dimensional cases, all of which show promising improved accuracy.
This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered.
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