815 research outputs found

    A Quintic B-Spline Technique for a System of Lane-Emden Equations Arising in Theoretical Physical Applications

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    In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software

    Numerical Solution Of The Heat Equation By Cubic B-Spline Collocation Method

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    This work proposes a numerical scheme for heat parabolic problem by implementing a collocation method with a cubic B-spline for a uniform mesh. The key idea of this method is to apply forward finite difference and Crank–Nicolson methods for time and space integration, respectively. The stability of the presented scheme is proved through the Von-Neumann technique. It is shown that it is unconditionally stable. The accuracy of the suggested scheme is computed through the L_2 and L_∞-norms. Numerical experiments are also given and show that it is compatible with the exact solutions

    Cross-Correlation and Averaging: An Equivalence Based on the Classical Probability Density

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    The averaging method is a widely used technique in the field of nonlinear differential equations for effectively reducing systems with "fast" oscillations overlaying "slow" drift. The method involves calculating an integral, which can be straightforward in some cases, but can also require simplifications such as series expansions. We propose an alternative approach that relies on the classical probability density (CPD) of the "fast" variable. Further, we demonstrate the equivalence between the averaging integral and the cross-correlation product of the CPD and the target function. This equivalence simplifies handling many problems, particularly those involving piecewise-defined target functions. We propose an effective numerical method to calculate the averaged function, exploiting the well-known mathematical properties of cross-correlation products

    Numerical methods for computing the discrete and continuous Laplace transforms

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    We propose a numerical method to spline-interpolate discrete signals and then apply the integral transforms to the corresponding analytical spline functions. This represents a robust and computationally efficient technique for estimating the Laplace transform for noisy data. We revisited a Meijer-G symbolic approach to compute the Laplace transform and alternative approaches to extend canonical observed time-series. A discrete quantization scheme provides the foundation for rapid and reliable estimation of the inverse Laplace transform. We derive theoretic estimates for the inverse Laplace transform of analytic functions and demonstrate empirical results validating the algorithmic performance using observed and simulated data. We also introduce a generalization of the Laplace transform in higher dimensional space-time. We tested the discrete LT algorithm on data sampled from analytic functions with known exact Laplace transforms. The validation of the discrete ILT involves using complex functions with known analytic ILTs

    Swift: A modern highly-parallel gravity and smoothed particle hydrodynamics solver for astrophysical and cosmological applications

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    Numerical simulations have become one of the key tools used by theorists in all the fields of astrophysics and cosmology. The development of modern tools that target the largest existing computing systems and exploit state-of-the-art numerical methods and algorithms is thus crucial. In this paper, we introduce the fully open-source highly-parallel, versatile, and modular coupled hydrodynamics, gravity, cosmology, and galaxy-formation code Swift. The software package exploits hybrid task-based parallelism, asynchronous communications, and domain-decomposition algorithms based on balancing the workload, rather than the data, to efficiently exploit modern high-performance computing cluster architectures. Gravity is solved for using a fast-multipole-method, optionally coupled to a particle mesh solver in Fourier space to handle periodic volumes. For gas evolution, multiple modern flavours of Smoothed Particle Hydrodynamics are implemented. Swift also evolves neutrinos using a state-of-the-art particle-based method. Two complementary networks of sub-grid models for galaxy formation as well as extensions to simulate planetary physics are also released as part of the code. An extensive set of output options, including snapshots, light-cones, power spectra, and a coupling to structure finders are also included. We describe the overall code architecture, summarize the consistency and accuracy tests that were performed, and demonstrate the excellent weak-scaling performance of the code using a representative cosmological hydrodynamical problem with ≈\approx300300 billion particles. The code is released to the community alongside extensive documentation for both users and developers, a large selection of example test problems, and a suite of tools to aid in the analysis of large simulations run with Swift.Comment: 39 pages, 18 figures, submitted to MNRAS. Code, documentation, and examples available at www.swiftsim.co

    A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis

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    Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them.Comment: 28 pages, 24 figures. Accepted manuscrip

    A Powerful Robust Cubic Hermite Collocation Method for the Numerical Calculations and Simulations of the Equal Width Wave Equation

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    In this article, non-linear Equal Width-Wave (EW) equation will be numerically solved . For this aim, the non-linear term in the equation is firstly linearized by Rubin-Graves type approach. After that, to reduce the equation into a solvable discretized linear algebraic equation system which is the essential part of this study, the Crank-Nicolson type approximation and cubic Hermite collocation method are respectively applied to obtain the integration in the temporal and spatial domain directions. To be able to illustrate the validity and accuracy of the proposed method, six test model problems that is single solitary wave, the interaction of two solitary waves, the interaction of three solitary waves, the Maxwellian initial condition, undular bore and finally soliton collision will be taken into consideration and solved. Since only the single solitary wave has an analytical solution among these solitary waves, the error norms Linf and L2 are computed and compared to a few of the previous works available in the literature. Furthermore, the widely used three invariants I1, I2 and I3 of the proposed problems during the simulations are computed and presented. Beside those, the relative changes in those invariants are presented. Also, a comparison of the error norms Linf and L2 and these invariants obviously shows that the proposed scheme produces better and compatible results than most of the previous works using the same parameters. Finally, von Neumann analysis has shown that the present scheme is unconditionally stable.Comment: 25 pages, 9 tables, 6 figure

    The interactions between gold nanoparticles and their self-assembly

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    Gold nanoparticles (AuNPs) are one of the most promising building blocks to fabricate versatile nanostructures. Such nanostructures have the great potential to enable new gold-based nanomaterials or nanocomposites with specific properties by precisely controlling the interactions (potential energies and/or forces) between them. In other words, the interactions between AuNPs are therefore regarded as one of the key factors governing particles’ self-assembly process that can drive multiple AuNPs to form ordered structures as required. Quantifying the interactions between them and understanding of their self-assembly process are of great importance and yet still challenging. In this study, molecular dynamics (MD) simulations are performed to calculate the interactions (e.g., potential energies) between AuNPs. The MD results reveal that a more effective force model between AuNPs can be developed as a function of their surface separation compared with the conventional Hamaker equation. In addition, MD simulations examine several effects (i.e., particle size, shape, rotation, surface patch, surfactant, as well as configuration) on their interactions. The results demonstrate that the different impacts of these factors (e.g., the hindrance of surfactant). Apart from spherical gold nanoparticles, interactions between gold nanorods (AuNRs) are also be quantified by MD simulations. The interparticle forces of AuNRs can be expressed as a function of their surface separation and the rotation angle since the rotational movement is applied on AuNR. Further, the MD-derived interparticle force models of gold nanospheres are integrated into discrete element method (DEM) to explore their self-assembly process. To the best of our knowledge, this might be the first time that the MD-based interparticle force models are integrated into DEM to explore the self-assembly process of gold nanoparticles. The results show that ordered nanostructures are ultimately constructed. Specifically, the mean coordination number (CN) of AuNPs (3 nm in size) is up to 5.99 and two major large clusters is observed under the simulation conditions at the equilibrated state. The completion of this study not only allows us to evaluate the interactions between AuNPs by MD simulation, but profoundly, the MD-DEM coupling approach opens a new window to unfold the self-assembly process of AuNPs

    Super-localised wave function approximation of Bose-Einstein condensates

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    This paper presents a novel spatial discretisation method for the reliable and efficient simulation of Bose-Einstein condensates modelled by the Gross-Pitaevskii equation and the corresponding nonlinear eigenvector problem. The method combines the high-accuracy properties of numerical homogenisation methods with a novel super-localisation approach for the calculation of the basis functions. A rigorous numerical analysis demonstrates superconvergence of the approach compared to classical polynomial and multiscale finite element methods, even in low regularity regimes. Numerical tests reveal the method's competitiveness with spectral methods, particularly in capturing critical physical effects in extreme conditions, such as vortex lattice formation in fast-rotating potential traps. The method's potential is further highlighted through a dynamic simulation of a phase transition from Mott insulator to Bose-Einstein condensate, emphasising its capability for reliable exploration of physical phenomena

    BubbleDet: a Python package to compute functional determinants for bubble nucleation

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    We present a Python package BubbleDet for computing one-loop functional determinants around spherically symmetric background fields. This gives the next-to-leading order correction to both the vacuum decay rate, at zero temperature, and to the bubble nucleation rate in first-order phase transitions at finite temperature. For predictions of gravitational wave signals from cosmological phase transitions, this is expected to remove one of the leading sources of theoretical uncertainty. BubbleDet is applicable to arbitrary scalar potentials and in any dimension up to seven. It has methods for fluctuations of scalar fields, including Goldstone bosons, and for gauge fields, but is limited to cases where the determinant factorises into a product of separate determinants, one for each field degree of freedom. To our knowledge, BubbleDet is the first package dedicated to calculating functional determinants in spherically symmetric backgrounds
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