A “Consistent” Quasidiffusion Method for Solving Particle Transport Problems

A Consistent Hybrid Finite-Volume/Particle Method for the PDF Equations of Turbulent Reactive Flows

The scattering source iterative (SI) scheme is traditionally applied to converge fine-mesh numerical solutions to fixed-source discrete ordinates (SN) neutron transport problems. The SI scheme is very simple to implement under a computational viewpoint. However, the SI scheme may show very slow convergence rate, mainly for diffusive media (low absorption) with several mean free paths in extent (low leakage). In this work we describe an acceleration technique based on an improved initial guess for the scattering source distribution within the slab. In other words, we use as initial guess for the fine-mesh scattering source, the coarse-mesh solution of the neutron diffusion equation with special boundary conditions to account for the classical SN prescribed boundary conditions, including vacuum boundary conditions. Therefore, we first implement a spectral nodal method that generates coarse-mesh diffusion solution that is completely free from spatial truncation errors, then we reconstruct this coarse-mesh solution within each spatial cell of the discretization grid, to further yield the initial guess for the fine-mesh scattering source in the first SN transport sweep (forward and backward) across the spatial grid. We consider a number of numerical experiments to illustrate the efficiency of the offered diffusion synthetic acceleration (DSA) technique.

The second-moment (SM) method is a linear variant of the quasi-diffusion (QD) method for accelerating the iterative convergence of Sn source calculations. It has several significant advantages relative to the QD method, diffusion synthetic acceleration, and nonlinear diffusion acceleration. Here, we define a variant of this method for k-eigenvalue calculations that retains the advantages of the original method, and we computationally demonstrate the efficacy of the method for simple example calculations. In particular, this method has two important properties. First, it is a linear acceleration scheme requiring only the solution of a pure k-eigenvalue diffusion equation with a corrective source term as opposed to a k-eigenvalue drift-diffusion equation. Second, unconditional stability is achieved even when the diffusion equation is not discretized in a manner consistent with the Sn spatial discretization. We are unaware of any other scheme that has these properties. We also show a connection between our method and the k-eigenvalue acceleration technique of Barbu and Adams, which motivated us to develop our SM method.

An investigation is conducted of the acceleration of the iterative implementation of the two-dimensional infinite impulse response (IIR) filter proposed by D.E. Dudgeon (IEEE Trans. Acoust. Speech Signal Proc., vol.ASSP-28, no.6, p.666-71, 1980). The conventional procedure results in a sequence of estimations that converge linearly to the desired filter output. Applying an accelerated convergence algorithm for linearly distorted signal reconstruction problems, the authors use a product expansion of the denominator of the transfer function of an IIR filter to derive a class of iterative algorithms that have a pth-order rate of convergence. Specifically, k iterations of the proposed pth-order algorithm are equal to the p/sup k/ iterations of the linear algorithm. Therefore, the number of iterations required to obtain a given approximation is reduced from p/sup k/ to k. The authors show that the spatial truncation error does not affect the computation of the desired result in the region of interest, provided that the boundary values outside the area of interest are known. Finally, the authors present two examples in order to compare the convergence rates for the linear and the accelerated algorithms. >

The grid partial analytical solution method is a newly developed unconditionally stable explicit numerical solution method for solving parabolic partial differential equations. This method discretizes only the spatial domain and predicts a continuous time dependent function at each spatial nodal point. As such, instead of conventionally predicting the solution from a number set, this method predicts from a functional domain. The typical properties of the grid partial analytical solution method can be summarized as the following: (1) It predicts a continuous nodal time dependent function rather than a discrete nodal value. (2) The prediction is unconditionally stable. And unlike any other unconditionally stable finite difference schemes which will lose accuracy when Fourier number becomes large, the proposed method allows single step time marching and unlimited reduction in the spatial step size Δx. (3) For a fixed time step, the higher value of the grid Fourier number resulting from decreasing Δx, the higher the accuracy is achieved in the predicted solution. (4) The grid partial analytical solution converges uniformly to the full analytical solution as the spatial truncation error is infinitely decreased by reducing the spatial step size Δx. This unique characteristic of the analytical treatment of time also makes it possible to treat other time dependent nonhomogeneities involved in heat conduction problem analytically. In this paper, a moving source heat conduction problem is posed and its grid partial analytical solution method developed.

The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization mechanism is added. The additional computational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle computationally expensive smooth forcing terms. Using the mathematical analysis developed in a previous manuscript, it is shown that splitting schemes are not appropriate for badly skewed hyperbolic systems. In a number of atmospheric models, the semi-implicit method is used to treat the badly skewed vertical sound wave terms. This leads to the excitation of the high-frequency waves in a nonphysical manner. It is also shown that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale cast will be ill posed at the lateral boundaries. The multiscale system for meteorology was introduced by Browning and Kreiss to overcome exactly these problems.

Progress in spectral nodal methods applied to discrete ordinates transport problems

A hybrid spectral nodal method for multigroup X,Y-geometry discrete ordinates eigenvalue problems

Exact analytical solution of time-independent neutron transport equation, and its applications to systems with a point source

ABSTRACTIn this work, the adjoint to the transport operator is used to estimate the spatial distribution of an isotropic neutral particles source in a homogeneous one-dimensional medium, from readings of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. Simulations are performed for reconstructing Gaussian and piecewise localized sources. Numerical results indicate that the source estimates may be considered satisfactory although scalar fluxes are reconstructed more precisely.

Second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage.Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal.In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. We show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

A Discontinuous Galerkin Finite Element Method based axial SN for the 2D/1D transport method

In this paper we describe, analyse and implement a parallel iterative method for the solution of the steady-state drift diffusion equations governing the behaviour of a semiconductor device in two space dimensions. The unknowns in our model are the electrostatic potential and the electron and hole quasi-Fermi potentials. Our discretisation consists of a finite element method with mass lumping for the electrostatic potential equation and a hybrid finite element with local current conservation properties for the continuity equations. A version of Gummel's decoupling algorithm which only requires the solution of positive definite symmetric linear systems is used to solve the resulting nonlinear equations. We show that this method has an overall rate of convergence which only degrades logarithmically as the mesh is refined. Indeed the (inner) nonlinear solves of the electrostatic potential equation converge quadratically, with a mesh independent asymptotic constant. We also describe an implementation on a MasPar MP-1 data parallel machine, where the required linear systems are solved by the preconditioned conjugate gradient method. Domain decomposition methods are used to parallelise the required matrix-vector multiplications and to build preconditioners for these very poorly-conditioned systems. Our preconditioned linear solves also have a rate of convergence which degrades logarithmically as the grid is refined relative to subdomain size, and their performance is resilient to the severe layers which arise in the coefficients of the underlying elliptic operators. Parallel experiments are given.

Wavelets and the Numerical Solution of Partial Differential Equations

In this paper, we give a rigorous justification for the derivation of the primitive equations with only horizontal viscosity and magnetic diffusivity (PEHM) as the small aspect ratio limit of the incompressible three-dimensional scaled horizontal viscous magnetohydrodynamics (SHMHD) equations. Choosing an aspect ratio parameter ε∈(0,∞) , we consider the case that if the orders of the horizontal and vertical viscous coefficients µ and ν are μ=O(1) and ν=O(εα) , and the orders of magnetic diffusion coefficients κ and σ are κ=O(1) and σ=O(εα) , with α > 2, then the limiting system is the PEHM as ɛ goes to zero. For H1 -initial data, we prove that the global weak solutions of the SHMHD equations converge strongly to the local-in-time strong solutions of the PEHM, as ɛ tends to zero. For H1 -initial data with additional regularity (∂zA~0,∂zB~0)∈Lp(Ω)(2<p<∞) , we slightly improve the well-posed result in Cao et al (2017 J. Funct. Anal. 272 4606–41) to extend the local-in-time strong convergences to the global-in-time one. For H2 -initial data, we show that the global-in-time strong solutions of the SHMHD equations converge strongly to the global-in-time strong solutions of the PEHM, as ɛ goes to zero. Moreover, the rate of convergence is of the order O(εγ/2) , where γ=min{2,α−2} with α∈(2,∞) . It should be noted that in contrast to the case α > 2, the case α = 2 has been investigated by Du and Li in (2022 arXiv:2208.01985), in which they consider the primitive equations with magnetic field (PEM) and the rate of global-in-time convergences is of the order O(ε) .