Low-order Equations Research Articles

Accurate simulation of thin dielectric sheet by higher order surface integral equation

ABSTRACTA higher order approach for the accurate simulation of a thin dielectric sheet is proposed in this paper. Due to the small thickness of thin dielectric sheet, an equivalent surface integral equation containing tangential and normal fields can be derived from the conventional volume integral equation. Higher order bases are then applied to approximate both components. Although the programming complexity increases slightly compared with the traditional low-order thin dielectric sheet surface integral equation method, the number of unknowns required in the same problem can be drastically reduced in the proposed higher order approach. At the same time, a higher accuracy can be achieved. Numerical results are presented to show the high efficiency and stability of the proposed method.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Abstract In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

Multigroup Radiation Hydrodynamics with a High-Order–Low-Order Method

Recent efforts at Los Alamos National Laboratory to develop a moment-based, scale-bridging [or high-order (HO)–low-order (LO)] algorithm for solving large varieties of the transport (kinetic) systems have shown promising results. A part of our ongoing effort is incorporating this methodology into the framework of the Eulerian Applications Project to achieve algorithmic acceleration of radiation-hydrodynamics simulations in production software. By starting from the thermal radiative transfer equations with a simple material-motion correction, we derive a discretely consistent energy balance equation (LO equation). We demonstrate that the corresponding LO system for the Monte Carlo HO solver is closely related to the original LO system without material-motion corrections. We test the implementation on a radiative shock problem and show consistency between the energy densities and temperatures in the HO and LO solutions as well as agreement with the semianalytic solution. We also test the approach on a more challenging two-dimensional problem and demonstrate accuracy enhancements and algorithmic speedups. This paper extends a recent conference paper by including multigroup effects.

Nonlinear Diffusion Acceleration Method with Multigrid in Energy for k-Eigenvalue Neutron Transport Problems

We present a multilevel method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional Cartesian geometry. It is based on the nonlinear diffusion acceleration (NDA) method. The multigroup low-order NDA (LONDA) equations are formulated on a sequence of energy grids. Various multigrid cycles are applied to solve the hierarchy of multigrid LONDA equations. The algorithms developed accelerate transport iterations and are effective in solving the multigroup NDA low-order equations. We present numerical results for model reactor-physics problems with a large number of groups to demonstrate the performance of different iterative schemes.

Low-order penalty equations for semidefinite linear complementarity problems

We extend the power penalty method for linear complementarity problem (LCP) (Wang and Yang, 2008) to the semidefinite linear complementarity problem (SDLCP). We establish a family of low-order penalty equations for SDLCPs. Under the assumption that the involved linear transformation possesses the Cartesian P-property, we show that when the penalty parameter tends to infinity, the solution to any equation of this family converges to the solution of the SDLCP exponentially. Numerical experiments verify this convergence result.

Improved Bell-polynomial procedure for the higher-order Korteweg–de Vries equations in fluid dynamics

Korteweg–de Vries (KdV)-typed equations are seen in fluid dynamics, plasma physics and other fields. By means of the Bell-polynomial procedure, we take two higher-order members of the KdV hierarchy, the seventh- and ninth-order Lax’s KdV equations in fluid dynamics, as the examples for studying the integrable properties of the higher-order equations. Different from lower-order equations, two new partial differential operators in the Bell-polynomial procedure are introduced to construct the “multi-dimensional” bilinear forms of such equations with several auxiliary independent variables. Through the procedure simplified via the algebraic operation of the polynomials, the Bäcklund transformations, Lax pairs and infinite conservation laws of such equations are deduced.

A unified compatibility method for exact solutions of non-linear flow models of Newtonian and non-Newtonian fluids

Criteria are established for higher order ordinary differential equations to be compatible with lower order ordinary differential equations. Necessary and sufficient compatibility conditions are derived which can be used to construct exact solutions of higher order ordinary differential equations subject to lower order equations. We provide the connection to generalized groups through conditional symmetries. Using this approach of compatibility and generalized groups, new exact solutions of non-linear flow problems arising in the study of Newtonian and non-Newtonian fluids are derived. The ansatz approach for obtaining exact solutions for non-linear flow models of Newtonian and non-Newtonian fluids is unified with the application of the compatibility and generalized group criteria.

New fundamental dynamical equation for higher derivative quantum field theories

In space-time with the Minkowski metric, the group of the metric is the inhomogeneous Lorentz group, which is also known as the Poincar e group. A dynamical equation is called fundamental if it is invariant with respect to the group of the metric, which consists of all transformations that leave the metric invariant. A method based on this group is used to derive two innite sets of Poincar e invariant linear dynamical equations for scalar and analytical functions that represent free, spin-zero, massive elementary particles. The sets contain two dierent types of fundamental higher derivative dynamical equations, which are used to construct higher derivative quantum eld theories. One of these sets contains the original Klein-Gordon equation and it is shown that all physically acceptable solutions to the higher derivative equations in this set are the same as the solutions to the Klein-Gordon equation. This means that none of the higher order equation in this set can be considered as new and that the Klein-Gordon is the only fundamental dynamical equation available to construct local quantum eld theories. However, for the other set, it is demonstrated that all physically acceptable solutions to the higher derivative equations are the same as the solutions to the lowest order fundamental equation in this set. As a result, this lowest order equation is the only new fundamental equation in the set, and it is used to construct higher derivative (non-local) quantum eld

Goldstones with extended shift symmetries

We consider scalar field theories invariant under extended shift symmetries consisting of higher order polynomials in the spacetime coordinates. These generalize ordinary shift symmetries and the linear shift symmetries of the galileons. We find Wess–Zumino Lagrangians which transform up to total derivatives under these symmetries, and which possess fewer derivatives per field and lower order equations of motion than the strictly invariant terms. In the nonrelativistic context, where the extended shifts are purely spatial, these theories may describe multi-critical Goldstone bosons. In the relativistic case, where the shifts involve the full spacetime coordinate, these theories generally propagate extra ghostly degrees of freedom.

AN ORDER SEVEN CONTINUOUS EXPLICIT METHOD FOR DIRECT SOLUTION OF GENERAL FIFTH ORDER ORDINARY DIFFERENTIAL EQUATIONS

This paper provides an explicit continuous method of order seven for direct integration of fifth-order general differential equations without reduction to systems of lower order equations. The explicit method requires no main predictor as a starting value for implementation. In the derivation of the method, collocation and interpolation procedures are adopted using power series as the basis function. The purpose of this work is to produce a high order explicit method that could be used as a predictor for the implementation of implicit methods in predictor-corrector mode in order to enhance the accuracy and efficiency of such methods. The method is symmetric, normalized, consistent and zero-stable. It is suitable for both non-stiff and mildly-stiff problems.

A Multilevel Projective Method for Solving the Space-Time Multigroup Neutron Kinetics Equations Coupled with the Heat Transfer Equation

We present a computational method for adequate and efficient coupling of the multigroup neutron transport equation with the precursor and heat transfer equations. It is based on the multilevel nonlinear quasi-diffusion (QD) method for solving the multigroup transport equation. The system of equations includes the time-dependent high-order transport equation and time-dependent multigroup and effective one-group low-order QD equations. We also apply the α-approximation for the time-dependent high-order transport equation. This approach enables one to avoid storing the angular flux from the previous time step. Numerical results for model transient problems are presented.

A hybrid transport-diffusion method for 2D transport problems with diffusive subdomains

In this paper we present a computational method based on the Simple Corner Balance (SCB) scheme for solving 2D transport problems in diffusive media. It utilizes decomposition of spatial domain into transport and diffusive subregions. This methodology uses the low-order equations of the Second-Moment (SM) method for the first two angular moments of the transport solution. These low-order SM equations are solved globally. The high-order transport solution is computed only in transport subregions. The transport boundary conditions at interfaces with neighbouring diffusion subregions are formulated using asymptotic analysis of SCB. We apply the quasidiffusion (Eddington) tensor to evaluate transport effects in the problem domain and determine spatial ranges of diffusive subregions. Numerical results are presented. They demonstrate the accuracy of the developed methodology for the SCB scheme.

A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation

AbstractWe propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.

Bistatic RCS of Electrically-Large Objects Calculations with High-Order Parabolic Equation Method

A high-order parabolic equation (PE) method is applied to calculate the Bistatic RCS of two-dimensional (2-D) Electrically-large Objects. According to the shapes of objexts, Crank-Nicolson and Pade (1,0) schemes are introduced to solve the high-order PE. The numerical results demonstrate that the method not only extends the accurate angle range but also decreases the rotation times and computational time in comparison with the traditional low-order parabolic equation method.

Hybrid Monte Carlo-CMFD Methods for Accelerating Fission Source Convergence

In this paper, two modifications to improve the efficiency of Lee et al.’s recently proposed “CMFD [coarse-mesh finite difference]-accelerated Monte Carlo” method for neutron criticality problems are presented and tested. This CMFD method employs standard Monte Carlo techniques to estimate nonlinear functionals (ratios of integrals), which are used in low-order CMFD equations to obtain the eigenvalue and discrete representations of the eigenfunction. In a “feedback” procedure, the Monte Carlo fission source is then modified to match the resulting CMFD fission source. The proposed new methods differ from the CMFD-accelerated Monte Carlo method only in the definition of the nonlinear functionals. The new methods are compared with the CMFD-accelerated Monte Carlo method for two high-dominance-ratio test problems. All of the hybrid methods rapidly converge the Monte Carlo fission source, enabling a large reduction in the number of inactive cycles. However, the new methods stabilize the fission source more efficiently than the CMFD-accelerated Monte Carlo method, enabling a reduction in the number of active cycles as well. Also, in all the hybrid methods, the apparent variance of the eigenfunction is nearly equal to the real variance, so the real statistical error is well estimated from a single calculation. This is a major advantage over the standard Monte Carlo method, in which the real variance is typically underestimated due to intercycle correlations.

Linear shoaling in Boussinesq-type wave propagation models

This work focuses on linear shoaling performance of low order Boussinesq-type equations. It is shown that the linear shoaling errors can be important in well known equations in the literature. New sets of coefficients are presented for three well known sets of equations. The sets are found so as to minimize a global linear error that includes celerity and shoaling errors. Finally, a new set of enhanced bilayer low order equations is presented, with much improved linear behavior (errors in wave celerity and wave amplitude below 1% up to kh=20). For completeness, the equations are written in their fully nonlinear version, and the nonlinear coefficients are also given.

Multilevel NDA Methods for Solving Multigroup Eigenvalue Neutron Transport Problems

The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in one-dimensional slab geometry. The proposed method is defined by a multilevel system of equations that includes multigroup and effective one-group low-order NDA equations. The eigenvalue is evaluated in an exact projected solution space of the smallest dimensionality. Numerical results that illustrate the performance of the new algorithm are demonstrated.

Extremal black holes and holographic c-theorem

We found Bogomolʼnyi type of the first order differential equations in three-dimensional Einstein gravity and the effective second order ones in new massive gravity when an interacting scalar field is minimally coupled. Using these equations in Einstein gravity, we obtain analytic solutions corresponding to extremally rotating hairy black holes. We also obtain perturbatively extremal black hole solutions in new massive gravity using these lower order differential equations. All these solutions have the anti-de Sitter spaces as their asymptotic geometries and as the near horizon ones. This feature of solutions interpolating two anti-de Sitter spaces leads to the construction of holographic c-theorem in these cases. Since our lower order equations reduce naturally to the well-known equations for domain walls, our results can be regarded as the natural extension of domain walls to more generic cases.

Hydrogen Effervescence from Aluminum Alloy Melts

With expansion of integrated computational materials engineering, new mathematical models are required for the assessment of processes that traditionally were manually monitored on factory production floors. The Richards logistic function is used to describe the total amount of porosity in an as-cast aluminum alloy sample after treatment under near vacuum and atmospheric pressures. The acquired function constants were correlated to the different process parameters during solidification. It is found that the Richards logistic function describes the best formation of porosity in aluminum alloy melts. The maximum porosity (MP) and lowest porosity level (LP) constants are directly related to the highest and lowest porosity levels in the analyzed samples, while the rate (r) defines the increase of porosity with respect to an increase in dissolved hydrogen between the threshold and mass transfer limits. The Richards logistic function can be used instead of traditional low-order mathematical equations to predict threshold limit and amount of porosity in solidified aluminum alloys by assessing the hydrogen concentration in aluminum alloy melts.

Wide Oscillation Finite Time Blow Up for Solutions to Nonlinear Fourth Order Differential Equations

We give sufficient conditions for local solutions to some fourth order semilinear ordinary differential equations to blow up in finite time with wide oscillations, a phenomenon not visible for lower order equations. The result is then applied to several classes of semilinear partial differential equations in order to characterize the blow up of solutions including, in particular, its applications to a suspension bridge model. We also give numerical results which describe this oscillating blow up and allow us to suggest several open problems and to formulate some related conjectures.