Accurate Numerical Scheme Research Articles

Gaussian dispersion analysis in the time domain: Efficient conversion with Padé approximants

We present an approach for adapting the Gaussian dispersion analysis (GDA) of optical materials to time-domain simulations. Within a GDA model, the imaginary part of a measured dielectric function is presented as a sum of Gaussian absorption terms. Such a simple model is valid for materials where inhomogeneous broadening is substantially larger than the homogeneous linewidth. The GDA model is the essential broadband approximation for the dielectric function of many glasses, polymers, and other natural and artificial materials with disorder. However, efficient implementation of this model in time-domain full-wave electromagnetic solvers has never been fully achieved. We start with a causal form of an isolated oscillator with Gaussian-type absorption — Causal Dawson-Gauss oscillator. Then, we derive explicit analytical formulas to implement the Gaussian oscillator in a finite-difference time-domain (FDTD) solver with minimal use of memory and floating point operations. The derivation and FDTD implementation employ our generalized dispersive material (GDM) model — a universal, modular approach to describing optical dispersion with Padé approximants. We share the FDTD prototype codes that include automated generation of the approximants and a universal FDTD dispersion implementation that employs various second-order accurate numerical schemes. The codes can be used with non-commercial solvers and commercial software for time-domain simulations of light propagation in dispersive media, which are experimentally characterized with GDA models. Program summaryProgram Title:MADISCPC Library link to program files:https://doi.org/10.17632/x69b4krsy8.1Licensing provisions: GPLv3Programming language: MATLABNature of problem: The problem of efficient time-domain simulation of the Gaussian absorption is essential for wideband modeling of the optical response from materials with inhomogeneous spectral broadening, such as glasses, polymers, and other natural and artificial composite materials with structural or phase disorder.Solution method: A Coupled Oscillator (CO) approximation to the Gaussian absorption in both the frequency and time domains is derived to solve this problem. The time-domain CO approximation is coupled to the finite-difference time-domain (FDTD) solver for the Maxwell equations in the MADIS (MAterial DIspersion Simulator) package. Verification of the Maxwell solvers' accuracy and stability is performed with the FDTD solver coupled to a compact universal implementation of the CO model, employing second-order schemes (either ADE or RC). Code prototypes of these efficient CO schemes can be ported to other methods and platforms for implementing the Gaussian absorption in open-source codes or commercial software.Additional comments including restrictions and unusual features: Patent pending. Restrict any commercial use, including for profit reproduction.

CALLIOPE: Pseudospectral Shearing Magnetohydrodynamics Code with a Pencil Decomposition

Abstract The pseudospectral method is a highly accurate numerical scheme suitable for turbulence simulations. We have developed an open-source pseudospectral code, calliope, which adopts the P3DFFT library to perform a fast Fourier transform with the two-dimensional (pencil) decomposition of numerical grids. calliope can solve incompressible magnetohydrodynamics (MHD), isothermal compressible MHD, and rotational reduced MHD with parallel computation using very large numbers of cores (>105 cores for 20483 grids). The code can also solve for local magnetorotational turbulence in a shearing frame using the remapping method. calliope is currently the only pseudospectral code that can compute magnetorotational turbulence using pencil-domain decomposition. This paper presents the numerical scheme of calliope and the results of linear and nonlinear numerical tests, including compressible local magnetorotational turbulence with the largest grid number reported to date.

The performance of a numerical scheme on the variable-order time-fractional advection-reaction-subdiffusion equations

This paper is concerned with a highly accurate numerical scheme for a class of one- and two-dimensional time-fractional advection-reaction-subdiffusion equations of variable-order α(x,t)∈(0,1). For the spatial and temporal discretization of the equation, a fourth-order compact finite difference operator and a third-order weighted-shifted Grünwald formula are applied, respectively. The stability and convergence of the present scheme are addressed. Some extensive numerical experiments are performed to confirm the theoretical analysis and high-accuracy of this novel scheme. Comparisons are also made with the available schemes in the literature.

Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.

A numerical algorithm for modeling cellular rearrangements in tissue morphogenesis

Among morphological phenomena, cellular patterns in developing sensory epithelia have gained attention in recent years. Although physical models for cellular rearrangements are well-established thanks to a large bulk of experimental work, their computational implementation lacks solid mathematical background and involves experimentally unreachable parameters. Here we introduce a level set-based computational framework as a tool to rigorously investigate evolving cellular patterns, and study its mathematical and computational properties. We illustrate that a compelling feature of the method is its ability to correctly handle complex topology changes, including frequent cell intercalations. Combining this accurate numerical scheme with an established mathematical model, we show that the proposed framework features minimum possible number of parameters and is capable of reproducing a wide range of tissue morphological phenomena, such as cell sorting, engulfment or internalization. In particular, thanks to precise mathematical treatment of cellular intercalations, this method succeeds in simulating experimentally observed development of cellular mosaic patterns in sensory epithelia.

The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations

We propose an accurate data-driven numerical scheme to solve stochastic differential equations (SDEs), by taking large time steps. The SDE discretization is built up by means of the polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Basic error analysis indicates that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression–decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. As a proof of concept, 1D numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.

Stabilization parameter analysis of a second-order linear numerical scheme for the nonlocal Cahn–Hilliard equation

AbstractA second-order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn–Hilliard equation. The backward differentiation formula is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an $O (\varDelta {t}^2)$ artificial regularization term, in the form of $A \varDelta _N (\phi ^{n+1} - 2 \phi ^n + \phi ^{n-1})$, is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience; however, its theoretical analysis turns out to be very challenging, due to the lack of higher-order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the $\ell ^\infty $ bound of the numerical solution is required. To recover such an $\ell ^\infty $ bound, an optimal rate convergence analysis has to be conducted, which combines a high-order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher-order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform $\ell ^\infty $ bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter $A$. Some numerical experiments are carried out to display the behavior of the proposed second-order scheme, including the convergence tests and long-time coarsening dynamics.

A fast Chebyshev method for the Bingham closure with application to active nematic suspensions

Continuum kinetic theories provide an important tool for the analysis and simulation of particle suspensions. When those particles are anisotropic, the addition of a particle orientation vector to the kinetic description yields a 2d−1 dimensional theory which becomes intractable to simulate, especially in three dimensions or near states where the particles are highly aligned. Coarse-grained theories that track only moments of the particle distribution functions provide a more efficient simulation framework, but require closure assumptions. For the particular case where the particles are apolar, the Bingham closure has been found to agree well with the underlying kinetic theory; yet the closure is non-trivial to compute, requiring the solution of an often nearly-singular nonlinear equation at every spatial discretization point at every timestep. In this paper, we present a robust, accurate, and efficient numerical scheme for evaluating the Bingham closure, with a controllable error/efficiency tradeoff. To demonstrate the utility of the method, we carry out high-resolution simulations of a coarse-grained continuum model for a suspension of active particles in parameter regimes inaccessible to kinetic theories. Analysis of these simulations reveals that inaccurately computing the closure can act to effectively limit spatial resolution in the coarse-grained fields. Pushing these simulations to the high spatial resolutions enabled by our method reveals a coupling between vorticity and topological defects in the suspension director field, as well as signatures of energy transfer between scales in this active fluid model.

A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell

We consider the numerical approximation of the binary immiscible mixture of nematic liquid crystals and viscous Newtonian fluids confined in a Hele-Shaw cell, where the free interface motion is simulated by using the phase-field approach via the energy variational method. The governing system is highly complicated nonlinear and coupled, consisting of the Darcy equations for the flow field, the Cahn-Hilliard equations for the free moving interface, and the constitutive equation for the nematic liquid crystal. The numerical scheme developed in this paper is the first “ideal” scheme, namely, it not only has the following characteristics: linearity, second-order time accuracy, unconditional energy stability, and decoupling structure, but also at each time step, only needs to solve a few elliptic equations with constant coefficients. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Various numerical experiments are further carried out to prove the effectiveness of the scheme, in which the influence of the initial orientations and anchoring elastic energy of the liquid crystal on the Saffman-Taylor fingering instability are studied.

A phase field model for compressible immiscible fluids with a new equation of state

In this paper, we propose a new compressible phase field model of two different immiscible fluid components, in which the density of each phase is variable. In order to establish the compressible phase field model, a new P-V-T equation of state is introduced to solve the pressure. The new model is derived by the physics law of conservation, and conforms to the second law of thermodynamics. The model adopts an innovative expression of Helmholtz free energy, taking into account the new state equation of pressure and the varying material properties of each phase. A high-order accurate numerical scheme is introduced to solve the model equations. The convection terms of the governing equations are discretized by the fifth-order WENO scheme, and the residual terms are discretized by the Lax–Friedrichs method. Finally, the reliability and validity of the compressible two-phase model are verified by numerical simulations.

High‐Accurate Numerical Schemes for Black–Scholes Models with Sensitivity Analysis

The significance of both the linear and nonlinear Black‐Scholes partial differential equation model is huge in the field of financial analysis. In most cases, the exact solution to such a nonlinear problem is very hard to obtain, and in some cases, it is impossible to get an exact solution to such models. In this study, both the linear and the nonlinear Black‐Scholes models are investigated. This research mainly focuses on the numerical approximations of the Black‐Scholes (BS) model with sensitivity analysis of the parameters. It is to note that most applied researchers use finite difference and finite element‐based schemes to approximate the BS model. Thus, an urge for a high accurate numerical scheme that needs fewer grids/nodes is huge. In this study, we aim to approximate and analyze the models using two such higher‐order schemes. To be specific, the Chebyshev spectral method and the differential quadrature method are employed to approximate the BS models to see the efficiency of such highly accurate schemes for the option pricing model. First, we approximate the model using the mentioned methods. Then, we move on to use the numerical results to analyze different aspects of stock market through sensitivity analysis. All the numerical schemes have been illustrated through some graphics and relevant discussions. We finish the study with some concluding remarks.

An efficient and accurate numerical scheme for long time statistical properties of the infinite Prandtl number model for convection

In this work we analyze a numerical scheme for the infinite Prandtl number model for convection. There have been many heuristic, experimental, theoretical, and numerical results with the purpose of estimating the scaling of the Nusselt number, which is viewed as a long time statistical quantity. Of the many numerical schemes available, very few are analytically investigated for their ability to adequately capture the long time statistical properties of the model. We present a semi-implicit scheme that treats the advection term explicitly, removing the need for a system solve at each step, with added stability from the implicit treatment of the diffusion term. Given a time step restriction that depends on the Rayleigh number, we show that our scheme falls into the class of “uniformly dissipative” schemes and has the potential to capture long time statistics of the infinite Prandtl number model.

The front runner in roll waves produced by local disturbances

Roll waves produced by a local disturbance comprise a group of shock waves with steep fronts. We used a robust and accurate numerical scheme to capture the steep fronts in a shallow-water hydraulic model of the waves. Our simulations of the waves in clear water revealed the existence of a front runner with an exceedingly large amplitude – much greater than those of all other shock waves in the wave group. The trailing waves at the back remained periodic. Waves were produced continuously within the group due to nonlinear instability. The celerity depended on the wave amplitude. Over time, the instability produced an increasing number of shock waves in an ever-expanding wave group. We conducted simulations for three types of local disturbances of very different duration over a range of amplitudes. We interpreted the simulation results for the front runner and the trailing waves, guided by an analytical solution and the laboratory data available for the smaller waves in the trailing end of the wave group.

Efficient decoupled second-order numerical scheme for the flow-coupled Cahn–Hilliard phase-field model of two-phase flows

We construct in this paper a fully-decoupled and second-order accurate numerical scheme for solving the Cahn–Hilliard–Navier–Stokes phase-field model of two-phase incompressible flows. A full decoupling method is used by introducing several nonlocal variables and their ordinary differential equation to deal with the nonlinear and coupling terms. By combining with some effective methods to handle the Navier–Stokes equation, we obtain an efficient and easy-to-implement numerical scheme in which one only needs to solve several fully-decoupled linear elliptic equations with constant coefficients at each time step. We further prove the unconditional energy stability and solvability rigorously, and present various numerical simulations in 2D and 3D to demonstrate the efficiency and stability of the proposed scheme, numerically.

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with coefficient matrices independent of the magnetization, and fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

Higher-order accurate and conservative hybrid numerical scheme for multi-variables time-fractional Vlasov-Maxwell system: An Atangana-Baleanu Caputo approach

With the historical review, it can determine that plasma scientists devolved many numerical schemes in literature to solve the kinetic plasma models such as Vlasov- Maxwell system, but the fatal problem is that most of the designed algorithms violate the conservation laws. This issue is significant and cannot disregard because the Vlasov Maxwell system is connected with conservations properties and can evaluate after performing some effective mathematical operations. In this prospective, we formulated the Atangana-Baleanu fractional derivative with Caputo concepts based fractional-order plasma model to study the fractional performance of the plasma particles and further suggested a higher-order conservative numerical algorithm that follows the conservation laws, based on operational matrices theory which is evaluated by using the concepts of Shifted Gegenbauer estimations. The numerical convergence of the designed algorithm is also inspected to validate the competency and compatibility. The comprehensive analysis of the fractional performance of the plasma particles is also part of the study. Moreover, this concept can extend to the variable order and multi-dimensional problems for Vlasov and Boltzmann system.

Thermal enhancement of ethylene glycol base material with hybrid nanofluid for oblique stagnation point slip flow

The hybrid nanofluids are more effective and stable to improve the thermal activities of many base liquids. With eminent thermal consequences, the hybrid nanofluids are preferred as compared to simple nanofluids which present the applications in industrial and engineering sectors. This mathematical model described in the thermal applications of hybrid nanofluids due to a non-orthogonal stagnation point addressed over a stretching cylinder. The hybrid nanofluid constrains the multi-walled carbon nanotubes (MWCNT) and single-walled carbon nanotubes (SWCNT). The Ethylene glycol is considered as base fluid. The heat generation features and slip constraints are also introduced to modify the thermal model. The assumed flow is observed due to stretched cylinder with applications of oblique stagnation point flow. The phenomenon of oblique stagnation point flow pattern is quite interesting. The similarity transformations are used to reduce the governing system of partial differential equations into system of ordinary differential equations. The results are exhibited in the present analysis by using efficient and second order accurate numerical scheme Keller box method. The graphical results for different features of flow and heat transfer are presented and discussed. It is observed that the tangential velocity declines with the velocity ratio parameter which is more progressive when the slip features are dominant. An increase in the solid volume fraction results a lower change in temperature profile. With increase in the curvature parameter, the heat transfer rate improved for non-convective stretching cylinder while the opposite observations are claimed for convective stretched cylinder.

Second order approximation for a quasi-incompressible Navier-Stokes Cahn-Hilliard system of two-phase flows with variable density

Phase-field model has been applied extensively and successfully for simulating two-phase flows with variable density. Several stable numerical methods have been proposed for solving such a highly nonlinear and coupled system. The key issue is to design a method such that it can preserve the (exact or slightly modified) conservative/dissipative law of energy of the fluid system at the discrete level. However, most of the existing energy stable numerical methods are restricted to only the first order accuracy in time for two-phase flow with different density. The design of a temporally second order accurate numerical method for the two-phase flows with variable density still remains challenging. In this paper, we develop several second order, robust and accurate numerical schemes for a quasi-incompressible Navier-Stokes Cahn-Hilliard model of two-phase flows with variable density which is thermodynamically consistent and was originally developed in [1] for simulating two-phase flows in complex geometries. Especially, numerical schemes proposed in this paper can preserve the mass conservation or energy dissipative law. Several numerical examples are presented to validate the robustness and accuracy of our numerical schemes.

Numerical analysis of a Reynolds Stress Model for turbulent mixing: the one-dimensional case

A mixed hyperbolic-parabolic, non conservative, Reynolds Stress Model (RSM), is studied. It is based on an underlying set of Langevin equations, and allows to describe turbulent mixing, including transient demixing effects as well as incomplete mixing. Its mathematical structure is analysed, and specific regimes, related to acoustic-like, Riemann-type, or self-similar solutions, are identified. A second-order accurate numerical scheme is proposed in arbitrary curvilinear geometry. Its accuracy and convergence behaviour are tested by comparison with analytical solutions in the different regimes. The numerical scheme can be generalized to multi-dimensional configurations, with potentially cylindrical symmetry, on unstructured meshes.

Numerical Steepest Descent Method for Hankel Type of Hypersingular Oscillatory Integrals in Electromagnetic Scattering Problems

We present a fast and accurate numerical scheme for approximating hypersingular integrals with highly oscillatory Hankel kernels. The main idea is to first change the integration path by Cauchy’s theorem, transform the original integral into an integral on a , + ∞ , and then use the generalized Gauss Laguerre integral formula to calculate the corresponding integral. This method has the advantages of high-efficiency, fast convergence speed. Numerical examples show the effect of this method.