Finite Difference Representation Research Articles

Schur line defect correlators and giant graviton expansion

We consider Schur line defect correlators in four dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) SYM and their giant graviton expansion encoding finite N corrections to the large N limit. We compute in closed form the single giant graviton contribution to correlators with general insertions of 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}-BPS charged Wilson lines. For the 2-point function with fundamental and anti-fundamental Wilson lines, we match the result from fluctuations of two half-infinite strings ending on the giant graviton, recently proposed in arXiv:2403.11543. In particular, we prove exact factorization of the defect contribution with respect to wrapped D3 brane fluctuations representing the single giant graviton correction to the undecorated Schur index. This follows from a finite-difference representation of the Schur line defect index in terms of the index without defects, and similar factorization holds quite generally for more complicated defect configurations. In particular, the single giant graviton contribution to the 4-point function with two fundamental and two anti-fundamental lines is computed and discussed in this perspective.

Finite difference representation of information-theoretic approach in density functional theory

Finite difference representation of information-theoretic approach in density functional theory

Block particle filters for state estimation of stochastic reaction-diffusion systems

In this work, we consider a differential description of the evolution of the state of a reaction-diffusion system under environmental fluctuations. We are interested in estimating the state of the system when only partial observations are available. To describe how observations and states are related, we combine multiplicative noise-driven dynamics with an observation model. More specifically, we ensure that the observations are subjected to error in the form of additive noise. We focus on the state estimation of a Belousov-Zhabotinskii chemical reaction. We simulate a reaction conducted in a quasi-two-dimensional physical domain, such as on the surface of a Petri dish. We aim at reconstructing the emerging chemical patterns based on noisy spectral observations. For this task, we consider a finite difference representation on the spatial domain, where nodes are chosen according to observation sites. We approximate the solution to this state estimation problem with the Block particle filter, a sequential Monte Carlo method capable of addressing the associated high-dimensionality of this state-space representation.

Numerical Solutions of Non-Local Problem for fractional order differential equations

The paper focuses on the study of problem- related to the linear second order boundary value problem (BVP). Three generalization styles are discussed. In the first, the second order BVP with non-local condition of the Dirichlet type is studied. In the second, the fractional order BVP is considered. In the third, the fractional order BVP with non-local condition is treated. The finite difference representation of the BVP is employed to reduce the problem into a system of algebraic equations, in addition, the implicit- explicit treatment to the nonlocal problems is introduced. In the implicit- explicit treatment the computational work is reduced considerably.

Model order reduction of layered waveguides via rational Krylov fitting

Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.

Real-time time-dependent density functional theory implementation of electronic circular dichroism applied to nanoscale metal-organic clusters.

Electronic circular dichroism (ECD) is a powerful spectroscopy method for investigating chiral properties at the molecular level. ECD calculations with the commonly used linear-response time-dependent density functional theory (LR-TDDFT) framework can be prohibitively costly for large systems. To alleviate this problem, we present here an ECD implementation within the projector augmented-wave method in a real-time-propagation TDDFT framework in the open-source GPAW code. Our implementation supports both local atomic basis sets and real-space finite-difference representations of wave functions. We benchmark our implementation against an existing LR-TDDFT implementation in GPAW for small chiral molecules. We then demonstrate the efficiency of our local atomic basis set implementation for a large hybrid nanocluster and discuss the chiroptical properties of the cluster.

Modeling ternary fluids in contact with elastic membranes.

We present a thermodynamically consistent model of a ternary fluid interacting with elastic membranes. Following a free-energy modeling approach for the fluid phases, we derive the governing equations for the dynamics of the ternary fluid flow and membranes. We also provide the numerical framework for simulating such fluid-structure interaction problems. It is based on the lattice Boltzmann method for the ternary fluid (Eulerian description) and a finite difference representation of the membrane (Lagrangian description). The ternary fluid and membrane solvers are coupled through the immersed boundary method. For validation purposes, we consider the relaxation dynamics of a two-dimensional elastic capsule placed at a fluid-fluid interface. The capsule shapes, resulting from the balance of surface tension and elastic forces, are compared with equilibrium numerical solutions obtained by surface evolver. Furthermore, the Galilean invariance of the proposed model is proven. The proposed approach is versatile, allowing for the simulation of a wide range of geometries. To demonstrate this, we address the problem of a capillary bridge formed between two deformable capsules.

Stochastic epidemic modeling with application to the sars-cov-2 pandemic in italy

The Sars-Cov-2 pandemic certainly represents an unprecedented challenge not only for medical researchers fighting against its worldwide diffusion but also for statisticians involved in proposing models to estimate the crucial epidemic parameters and monitor and control its future evolution. The traditional epidemiological SIR model (Brauer, Castillo-Chavez and Feng, 2019) is usually specified in a deterministic way and fitted to empirical data by numerical optimization, neglecting the error component's role so that its performances cannot be statistically tested. This paper introduces uncertainty explicitly in the estimation process (Bailey, 1955;Kendall, 1956). We start providing a finite difference representation of the model. We then introduce stochastic components in the model in the form of a measurement error and a random error. We finally apply the model to the case of the Italian 2020-2021 Sars-Cov-2 pandemic diffusion showing its relative advantages with respect to the deterministic specification. © 2021 Vita e Pensiero. All rights reserved.

A Multilevel Approach for Stability Conditions in Fractional Time Diffusion Problems

The Caputo definition of fractional derivatives introduces solution to the difficulties appears in the numerical treatment of differential equations due its consistency in differentiating constant functions. In the same time the memory and hereditary behaviors of the time fractional order derivatives (TFODE) still common in all definitions of fractional derivatives. The use of properties of companion matrices appears in reformulating multilevel schemes as generalized two level schemes is employed with the Gerschgorin disc theorems to prove stability condition. Caputo fractional derivatives with finite difference representations is considered. Moreover the effect of using the inverse operator which transmit the memory and hereditary effects to other terms is examined. The theoretical results is applied to a numerical example. The calculated solution has a good agreement with the exact solution.

The best parametrization for solving the boundary value problem for the system of differential-algebraic equations with delay

In this paper, we considered the numerical approach for solving a nonlinear boundary value problem for the system of differential-algebraic equations with delay argument. The shooting method is used to solve the boundary value problem. The Newton method is used to find the parameter of shooting. To overcome the difficulties associated with the choice of the initial approximation we apply E. Lahaye’s parameter continuation method. If the curve of the solution contains limit points, the method diverges. Then to find the parameter we used the method of continuation with respect to the best parameter - the length of the curve of the solution set. The solution is constructed by advancing the sequence of values of the parameter. With a discrete continuation, the initial-value problem is transformed by a finite-difference representation of the derivatives and entering the best argument and the corresponding equation of hypersphere. The resulting system is solved using the Newton method. To find the values of the functions at the delay point Lagrange polynomial with three points is used. An example of the behavior of an elastoviscoplastic rod is considered.

A High-order Weighted Finite Difference Scheme with a Multistate Approximate Riemann Solver for Divergence-free Magnetohydrodynamic Simulations

Abstract We design a conservative finite difference scheme for ideal magnetohydrodynamic simulations that attains high-order accuracy, shock-capturing, and a divergence-free condition of the magnetic field. The scheme interpolates pointwise physical variables from computational nodes to midpoints through a high-order nonlinear weighted average. The numerical flux is evaluated at the midpoint by a multistate approximate Riemann solver for correct upwinding, and its spatial derivative is approximated by a high-order linear central difference to update the variables with the designed order of accuracy and conservation. The magnetic and electric fields are defined at staggered grid points employed in the constrained transport (CT) method by Evans & Hawley. We propose a new CT variant, in which the staggered electric field is evaluated so as to be consistent with the base one-dimensional Riemann solver, and the staggered magnetic field is updated to be divergence-free as designed by the high-order finite difference representation. We demonstrate various benchmark tests to measure the performance of the present scheme. We discuss the effect of the choice of interpolation methods, Riemann solvers, and the treatment for the divergence-free condition on the quality of numerical solutions in detail.

MATHEMATICAL MODELING OF METAL FLOW IN CRYSTALLIZER AT ITS SUPPLY FROM SUBMERSIBLE NOZZLE WITH ECCENTRIC HOLES

Flow of liquid melt in the crystallizer is a little-studied process. Analytical solutions of melt flow in general case refer to complex mathematical problems, therefore numerical methods are used to model it. The purpose of this work is to use numerical method proposed by Professor V.I. Odinokov, based on finite-difference representation of the initial system of equations. This method has been successfully used in mechanics of continuous media, in foundry industry in mathematical modeling of strained deformed state of shell molds on investment models,as well as in other technological works, which indicates its universality. In the present study, the object of research is hydrodynamic flows of liquid metal during steel casting into a rectangular section mold when fed from a submerged nozzle with eccentric holes, and the result is a spatial mathematical model describing the flows of liquid metal in the crystallizer. To simulate the processes occurring in the crystallizer, the software complex “Odyssey” was used. The theoretical calculation is based on fundamental equations of hydrodynamics and approved numericalmethod. Solution of differential equations system formulatedin the work was carried out numerically. Investigated area was divided into elements of finite dimensions, for each element the resulting system of equations was written in the difference form. The result of the solution is velocity field of metal flow in crystallizer volume. To solve the system of algebraic equations obtained, a numerical scheme and a calculation algorithm were developed. Based on developed numerical scheme and algorithm, a computation program was compiled in Fortran-4. Mathematical model makes it possible to vary geometric dimensions of the crystallizer and cross-section of metal exit openings from the immersion nozzle, and it can also help to understand the flow pattern of the cast metal that affects heat dissipation of crystallizer walls and to find the optimal parameters for liquid metal outlet from the gravy glass at various casting modes. As an example it is given calculation of steel casting into a rectangular mold with a height of 100 cm and a section of 2000×40 (cm) in plan. Casting was carried out from immersion nozzle eccentrically in both sides in a horizontal plane. The calculation results are presented in graphical form. The movement of liquid metal flows is shown, their magnitudes and intensity are determined.

Self-consistent description of the replacement current driving melt layer motion in fusion devices

The bulk replacement current density triggered by surface charge loss owing to thermionic emission leads to a volumetric Lorentz force which has been observed to drive macroscopic melt layer motion in transient tungsten melting tokamak experiments in which components of different geometries (deliberate leading edges and sloped surfaces) have been exposed to edge localized mode (ELM) pulsed heat loads in high power H-mode discharges. A self-consistent approach is formulated for the replacement current which is based on the magnetostatic limit of the resistive thermoelectric magnetohydrodynamic description of the liquid metal and results in a well-defined boundary value problem for the whole conductor. A new module is incorporated into the incompressible fluid dynamics code MEMOS-3D, which numerically solves the finite difference representation of the problem. The phenomenological approach, employed thus far to describe the replacement current, is demonstrated to be accurate for the sloped geometry but inadequate for the leading edge. MEMOS-3D simulations of very recent ASDEX-Upgrade leading edge experiments with the rigorous as well as the simplified approach are reported. For these simulations, the self-consistent approach predicts a fivefold reduction of the displaced material volume, a sevenfold reduction of the maximum peak height of displaced material and a different eroded surface morphology in comparison with the previously applied simplified approach.

Modeling and Simulation Speed-Up of Plasma Actuators Implementing Reconfigurable Hardware

The objective of the present study is to investigate the capability of field-programmable gate array hardware in numerical simulation of a model of a dielectric barrier discharge plasma actuator to accelerate the calculations. The reconfigurable hardware is designed such that it is possible to reprogram its architecture after manufacturing. This provides the capability to design and implement various architectures for several applications. Two reconfigurable chips are used in the present study, one of which consists of a programmable logic unit and a typical microprocessor. This hybrid architecture makes the high performance of the reconfigurable hardware in custom computing and the efficiency of the microprocessor in data flow control accessible. An automated design procedure is used for the design of the reconfigurable hardware. Further, a finite difference representation of a phenomenological model of a plasma actuator is derived and implemented on the field-programmable gate array hardware. The results are validated against other numerical data, and the computational time is compared to different conventional processors. Using the reconfigurable hardware results in up to 96% computational time reduction compared to a recent Core i7 processor.

A note on exact finite difference schemes for modified Lotka–Volterra differential equations

We construct a class of modified Lotka–Volterra ordinary differential equations (ODE’s) and show that a nonlinear change of the dependent variables transform them into a set of coupled, linear ODE’s. Using the latter equations, we calculate the corresponding exact finite difference schemes using a technique given by Mickens. Next, we show how to reconfigure these relations to obtain the exact finite difference representation of the original modified, nonlinear Lotka–Volterra ODE’s.

NUMERICAL MODELING OF THE PROCESS OF FILLING THE CCM MOLD WITH METAL

Mathematical modeling of the liquid melt flow in the mold of the continuous casting plant is still poorly investigated. Analytic solutions of melt flow in the general case are a complex mathematical problem. Nevertheless, for some cases, exact solutions have been found. Such analytical solutions serve as a means of verifying the results of numerical methods of solution. The purpose of this work is the use of the numerical method proposed by Professor V.I. Odinokov, based on the finite-difference representation of the original system of equations. The method has been successfully used in the mechanics of continuous media, in the lithium manufacturing for the mathematical modeling of the strained deformed state of shell molds of cast models, as well as in other technological works, which indicates its universality. In the present work, hydrodynamic and thermal flows of liquid metal during the steel casting into the rectangular section mold of a continuous cast steel have become objects of research, and the result is a spatial mathematical model describing the flows of liquid metal in the mold. To simulate the processes that take place during filling, the software complex “Odyssey” is used. The basis of theoretical calculations includes the fundamental equations of hydrodynamics, the equations of mathematical physics (the equation of heat conductivity with regard to mass-transfer) and the approved numerical method. The solution of the system of differential equations formulated in this work was carried out numerically. The investigated area was divided into elements of finite dimensions, for each element the resulting system of equations was recorded in the difference form. The solution result is the metal flow velocity fields and the temperature fields in the mold volume. To solve the obtained system of algebraic equations, the numerical schemes and calculation algorithms were developed. Based on the developed numerical schemes and algorithms a computational program was compiled in Fortran-4. The mathematical model allows to vary the geometric dimensions of the crystal-mash and sectional metal exit holes from the submerged nozzle, and may also help to understand the scheme of the cast metal movement that affects the heat sink walls of the mold, and to find the optimal parameters of the output of liquid metal from the submerged nozzle at different modes of casting. An example of calculating the casting of steel into a rectangular mold with a height of 100 cm and cross-sectional dimensions of 2000.40 cm is given. The casting was carried out from a submerged nozzle symmetrically on both sides in a horizontal plane. The result of the solution is presented in the graphical form. The motion of liquid metal flows in different sections of the mold is shown. The areas of the circular flow of metal are revealed, as well as the areas in the mold volume, where the vortex motion of the liquid metal is observed, their magnitudes and intensity are determined. The presented temperature field indicates the presence of a local area with a high temperature at the wall of the mold, which is explained by the directed flow of hot metal emerging from the hole in the submerged nozzle.

SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters

As the first component of SPARC (Simulation Package for Ab-initio Real-space Calculations), we present an accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory (DFT) for isolated clusters. Specifically, utilizing a local reformulation of the electrostatics, the Chebyshev polynomial filtered self-consistent field iteration, and a reformulation of the non-local component of the force, we develop a framework using the finite-difference representation that enables the efficient evaluation of energies and atomic forces to within the desired accuracies in DFT. Through selected examples consisting of a variety of elements, we demonstrate that SPARC obtains exponential convergence in energy and forces with domain size; systematic convergence in the energy and forces with mesh-size to reference plane-wave result at comparably high rates; forces that are consistent with the energy, both free from any noticeable ‘egg-box’ effect; and accurate ground-state properties including equilibrium geometries and vibrational spectra. In addition, for systems consisting up to thousands of electrons, SPARC displays weak and strong parallel scaling behavior that is similar to well-established and optimized plane-wave implementations, but with a significantly reduced prefactor. Overall, SPARC represents an attractive alternative to plane-wave codes for practical DFT simulations of isolated clusters.Program summaryProgram title: SPARCCatalogue identifier: AFBL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFBL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v3No. of lines in distributed program, including test data, etc.: 47525No. of bytes in distributed program, including test data, etc.: 826436Distribution format: tar.gzProgramming language: C/C++.Computer: Any system with C/C++ compiler.Operating system: Linux.RAM: Problem dependent. Ranges from 80 GB to 800 GB for a system with 2500 electrons.Classification: 7.3.External routines: PETSc 3.5.3 (http://www.mcs.anl.gov/petsc), MKL 11.2 (https://software.intel.com/en-us/intel-mkl), and MVAPICH2 2.1 (http://mvapich.cse.ohio-state.edu/).Nature of problem:Calculation of the electronic and structural ground-states for isolated clusters in the framework of Kohn–Sham Density Functional Theory (DFT).Solution method:High-order finite-difference discretization. Local reformulation of the electrostatics in terms of the electrostatic potential and pseudocharge densities. Calculation of the electronic ground-state using the Chebyshev polynomial filtered Self-Consistent Field (SCF) iteration in conjunction with Anderson extrapolation/mixing. Evaluation of boundary conditions for the electrostatic potential through a truncated multipole expansion. Reformulation of the non-local component of the force. Geometry optimization using the Polak–Ribiere variant of non-linear conjugate gradients with secant line search.Restrictions:System size less than ∼4000 electrons. Local Density Approximation (LDA). Troullier–Martins pseudopotentials without relativistic or non-linear core corrections.Running time:Problem dependent. Timing results for selected examples provided in the paper.

Spectral Quadrature method for accurate [formula omitted] electronic structure calculations of metals and insulators

We present the Clenshaw–Curtis Spectral Quadrature (SQ) method for real-space O(N) Density Functional Theory (DFT) calculations. In this approach, all quantities of interest are expressed as bilinear forms or sums over bilinear forms, which are then approximated by spatially localized Clenshaw–Curtis quadrature rules. This technique is identically applicable to both insulating and metallic systems, and in conjunction with local reformulation of the electrostatics, enables the O(N) evaluation of the electronic density, energy, and atomic forces. The SQ approach also permits infinite-cell calculations without recourse to Brillouin zone integration or large supercells. We employ a finite difference representation in order to exploit the locality of electronic interactions in real space, enable systematic convergence, and facilitate large-scale parallel implementation. In particular, we derive expressions for the electronic density, total energy, and atomic forces that can be evaluated in O(N) operations. We demonstrate the systematic convergence of energies and forces with respect to quadrature order as well as truncation radius to the exact diagonalization result. In addition, we show convergence with respect to mesh size to established O(N3) planewave results. Finally, we establish the efficiency of the proposed approach for high temperature calculations and discuss its particular suitability for large-scale parallel computation.

Methods of converting multidimensional measuring information to a number by means of differential equations in partial derivatives

We have proved that finite difference representation of differential equations in partial derivatives is the most effective method of converting multidimensional measuring information to a single number. The devised conversion methods use Laplace’s elliptic equations and Fourier’s parabolic equations. To convert information,we have solved a number of tasks. We have devised a method of elliptic conversion of static images of the measured object by means of finite difference representation of Laplace’s differential equation in the second-order partial derivatives, which means a change of the current value of the pixel brightness to a new one that has two gradations of brightness––black and white. Pixel summation turns the relative number of black pixels into a single number as a result of elliptic conversion. We have improved the method of parabolic conversion of dynamic images of the measured object by means of finite difference representation of Fourier’s differential equation in the second-order partial derivatives,which means a change of the current value of the pixel brightness to a new one that has two gradations of brightness––black and white. Pixel summation turns the relative number of black pixels into a single number as a result of parabolic conversion. We have used the devised methods of industrial testing of metrological support for the technological process of continuous casting of copper ingots. The testing has proved technical and economic effectiveness of the approach. The paper contains examples of technical and economic effectiveness of the devised methods in metallurgical production.

A note on the exact discretization for a Cauchy–Euler equation: application to the Black–Scholes equation

We construct the exact finite difference representation for a second-order, linear, Cauchy–Euler ordinary differential equation. This result is then used to construct new non-standard finite difference schemes for the Black–Scholes partial differential equation.