3D Equations Research Articles

Wong-Zakai approximation and support theorem for 2D and 3D stochastic convective Brinkman-Forchheimer equations

In this work, we demonstrate the Wong-Zakai approximation results for two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations forced by Hilbert space valued Wiener noise on bounded domains. Even though the existence and uniqueness of a pathwise strong solution to SCBF equations is known, the existence of a unique solution to the approximating system is not immediate from the solvability results of SCBF equations, and we prove it by using the Faedo-Galerkin approximation technique and monotonicity arguments. Moreover, as an application of the Wong-Zakai approximation, we obtain the support of the distribution of solutions to SCBF equations.

Analysis of Particle-Resolved CFD Results for Dispersion in Packed Beds

Dispersion is the spreading of a solute while it is moved by a flowing medium. The study of dispersion in catalytic chemical reactors is fundamental to their design, since dispersion influences the reactant and product transport within the bed. In this paper, longitudinal and transverse dispersion of an inert tracer in slender packed beds of spheres and spherocylinders is studied using Computational Fluid Dynamics simulations. The focus is on the analysis of dispersion from full field data. The purpose is to develop a methodology that can later also be used to characterize dispersion from full field experimental data such as MRI measurements. Results obtained by means of particle-resolved CFD simulations are discussed. Spatial distributions and residence times are analyzed and the results are interpreted by comparison to results obtained from 1D and 2D convection-diffusion equations.

Supercloseness of linear streamline diffusion finite element method on Bakhvalov-type mesh for singularly perturbed convection-diffusion equation in 1D

For the singularly perturbed convection-diffusion equations, the uniform convergence of the finite element method on Bakhvalov-type mesh is still in its infancy. The challenge is how to handle the analysis on the last element mesh in the layer. In this manuscript, a streamline diffusion finite element method is properly defined on a Bakhvalov-type mesh. By means of delicate analysis of the terms on the last element mesh in the layer, we derive the supercloseness of almost order 2, which is consistent with the numerical experiments.

Numerical and convergence analysis of the stochastic Lagrangian averaged Navier–Stokes equations

The primary emphasis of this work is the development of a finite element based space–time discretization for solving the stochastic Lagrangian averaged Navier–Stokes (LANS-α) equations of incompressible fluid turbulence with multiplicative random forcing, under nonperiodic boundary conditions within a bounded polygonal (or polyhedral) domain of Rd, d∈{2,3}. The convergence analysis of a fully discretized numerical scheme is investigated and split into two cases according to the spacial scale α, namely we first assume α to be controlled by the step size of the space discretization so that it vanishes when passing to the limit, then we provide an alternative study when α is fixed. A preparatory analysis of uniform estimates in both α and discretization parameters is carried out. Starting out from the stochastic LANS-α model, we achieve convergence toward the continuous strong solutions of the stochastic Navier–Stokes equations in 2D when α vanishes at the limit. Additionally, convergence toward the continuous strong solutions of the stochastic LANS-α model is accomplished if α is fixed. Neither of the mentioned convergences involves the Skorokhod theorem.

Arbitrarily high-order trapezoidal rules for functions with fractional singularities in two dimensions

In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the formsIi,j=∫R2ϕ(x)xixj|x|2+αdx,0<α<2where i,j∈{1,2} and ϕ∈CcN for N≥2. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is 2p+4−α, where p∈N0 is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in n∈N dimensions when appropriate for the sake of generality. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules.

Three-dimensional magnetotelluric modeling in a mixed space-wavenumber domain

We have developed a new 3D magnetotelluric modeling scheme in a mixed space-wavenumber domain. The modeling scheme is based on using a 2D Fourier transform along two horizontal directions to solve a vector-scalar potential formula derived from Maxwell’s equations based on the primary-secondary potential separation. The derived 1D governing equations in a mixed space-wavenumber domain are solved by using the finite-element method (FEM) together with a chasing method, and then the 2D inverse Fourier transform is used to recover the final solution of the electromagnetic (EM) fields in the 3D spatial domain. An iterative scheme is applied to approximate the true solution by repeating the previous steps because the governing equations cannot be solved directly due to an unusual primary-secondary potential field separation used. Nevertheless, the new method is capable of reducing the memory requirement and computational time in the mixed domain, and the 1D governing equations are highly parallel among different wavenumbers. For each of the 1D equations, the two- or four-node Gaussian quadrature rule can be used in both horizontal directions for Gauss fast Fourier transform. It is worth mentioning that the linear matrix equation to be solved is a fixed bandwidth system, and the chasing method is more efficient and convenient than solvers with preconditioners for the 1D matrix equations. The reliability and efficiency of the newly proposed method are verified with three synthetic 3D models by comparisons with a classical integral equation solution, an adaptive FEM solution, and a nonadaptive FEM solution. The proposed algorithm will be used in electrical resistivity tomography and controlled-source EM methods in future studies.

Very fast finite element Poisson solvers on lower precision accelerator hardware: A proof of concept study for Nvidia Tesla V100

Recently, accelerator hardware in the form of graphics cards including Tensor Cores, specialized for AI, has significantly gained importance in the domain of high-performance computing. For example, NVIDIA’s Tesla V100 promises a computing power of up to 125 TFLOP/s achieved by Tensor Cores, but only if half precision floating point format is used. We describe the difficulties and discrepancy between theoretical and actual computing power if one seeks to use such hardware for numerical simulations, that is, solving partial differential equations with a matrix-based finite element method, with numerical examples. If certain requirements, namely low condition numbers and many dense matrix operations, are met, the indicated high performance can be reached without an excessive loss of accuracy. A new method to solve linear systems arising from Poisson’s equation in 2D that meets these requirements, based on “prehandling” by means of hier-archical finite elements and an additional Schur complement approach, is presented and analyzed. We provide numerical results illustrating the computational performance of this method and compare it to a commonly used (geometric) multigrid solver on standard hardware. It turns out that we can exploit nearly the full computational power of Tensor Cores and achieve a significant speed-up compared to the standard methodology without losing accuracy.

GaAs-based near-field thermophotonic devices: Approaching the idealized case with one-dimensional PN junctions

Thermophotonics (TPX) is a technology close to thermophotovoltaics (TPV), where a heated light-emitting diode (LED) is used as the active thermal emitter of the system. It allows to tune the heat flux, by means of electroluminescence, to a spectral range matching better the gap of a photovoltaic cell. The concept is extended to near-field thermophotonics (NF-TPX), where enhanced energy conversion is due to both electric control and wave tunnelling. We perform a thorough numerical analysis of a GaAs-based NF-TPX device, by coupling a near-field radiative heat transfer solver based on fluctuational electrodynamics with an algorithm based on a simplified version of the drift–diffusion equations in 1D. This allows for the investigation of the emission and absorption profiles in the LED and the photovoltaic (PV) cell, and for the scrutiny of the impact of key parameters. We also demonstrate that the performance obtained with this algorithm can approach idealized cases for improved devices. For the considered simplified architecture and 300 K temperature difference, we find a power density output of 1 W cm − 2 , underlining the potential for waste heat harvesting close to ambient temperature. • Near-field thermophotonics allows waste-heat harvesting of moderate temperatures. • GaAs is a promising material for the thermophotonic elements: LED and photovoltaic cell. • Thorough modelling of the 1D pn junction architecture is performed. • About 35 W m −2 K −1 can be produced if radiative recombinations are limited.

TCAD simulations of FDSOI devices down to deep cryogenic temperature

In this paper we demonstrate the feasibility of FDSOI device TCAD simulations down to deep cryogenic temperatures. To this purpose, Maxwell-Boltzmann carrier statistics is replaced by an analytical approximation for the Fermi-Dirac integral with 3D density of states. The device electrostatic is investigated by solving the Poisson equation in 2D, whereas the transport is simulated using the Drift-Diffusion model. We explore the impact of temperature on device performances in linear and saturation regions as well as the impact of short channel effects, accounted for various gate and spacer lengths, at room and deep cryogenic temperatures. Finally, the obtained results are compared to some experimental data, highlighting the role of TCAD simulations in providing insights in device physics and performances.

The magnetocaloric effect, thermo-magnetic and transport properties of LiH diatomic molecule

This research article investigates the effects of magnetic and Aharanov-Bohm (AB) fields on the thermo-magnetic and transport properties and magneto-caloric effect (MCE) of Lithium Hydride (LiH) modelled by the Varshni potential. So, the Schrödinger equation in 2D is solved under the influence of external fields and Varshni potential using the functional analysis approach (FAA) to obtain the energy equation and wave function, respectively. The accessible energy levels are summed to obtain the partition function, which is used to obtain the aforementioned properties. The influences of the magnetic and AB fields on the above-mentioned properties have been examined over a broad temperature region. The susceptibility has only positive values in different scenarios analysed, which implies paramagnetism. The temperature dependence of the magnetic entropy (magneto-caloric effect) changes shows a decreasing tendency. This research could find applications in molecular physics and inspire discussions and studies on MCE for several diatomic molecules.

Effective condition number for the selection of the RBF shape parameter with the fictitious point method

Based on the Uncertainty Principle of radial basis functions (RBFs), it is known that the condition number and the error cannot be both kept small at the same time. In contrast to the traditional condition number, the effective condition number provides a much better estimation of the actual condition number of the resultant matrix system. In this paper, motivated by the Uncertainty Principle of RBFs, we propose to apply the effective condition number as a numerical tool to determine a reasonably good shape parameter value in the context of the Kansa method coupled with the fictitious point method. Six examples for second and fourth order partial differential equations in 2D and 3D are presented to demonstrate the effectiveness of the proposed method.

A high order compact time/space finite difference scheme for the 2D and 3D wave equation with a damping layer

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. For unbounded domains we add a fourth order accurate sponge layer to damp the outgoing waves. We demonstrate that usually this is more efficient than lower order schemes despite being implicit and conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as multi-grid. Computations confirm the design convergence rate for the in-homogeneous, variable wave speed equation.

A 2D Model for 3D Periodic Deep-Water Waves

The paper is devoted to further development of an accelerated method for simulation of the two-dimensional surface waves at infinite depth with the use of a two-dimensional model derived with simplifications of the three-dimensional equations for potential periodic deep-water waves. A 3D full wave model (FWM) is based on a numerical solution of a 3D Poisson equation written in the surface-fitted coordinates for a nonlinear component of the velocity potential. For sufficient vertical resolution used for the Poisson equation, the 3D model provides very high accuracy. The simplified model is based on the 2D Poisson equation written for a free surface. This exact equation contains both the first and second derivatives of the velocity potential, i.e., it is unclosed. The analysis of the accurate solutions for the 3D velocity potential obtained with the 3D model shows that those variables are linearly connected to each other. This property allows us to obtain a 2D equation for the first derivative of the velocity potential (i.e., vertical velocity on surface), which gives the closed 2D formulation for a 3D problem of two-dimensional waves. The previously developed scheme was not universal since the parameters of the closure scheme had to be adjusted to the specific setting. The current paper offers a new formulation of a closing scheme based on the integral parameters of wave field. The method of closing the equations, as well as the numerical parameters, were chosen on the basis of the multiple numerical experiments with the full nonlinear wave model (FWM) and selection of a suitable closing scheme. That is why the given model can be called Heuristic Wave Model (HWM). The connection between the first and second variables is not precise; hence, the method as a whole cannot be exact. However, the derived 2D model is able to reproduce different statistical characteristics of the 2D wave field with good accuracy. The main advantage of the model developed is its high performance exceeding that of 3D model by about two decimal orders.

Variational Bayesian approximation of inverse problems using sparse precision matrices

Inverse problems involving partial differential equations (PDEs) are widely used in science and engineering. Although such problems are generally ill-posed, different regularisation approaches have been developed to ameliorate this problem. Among them is the Bayesian formulation, where a prior probability measure is placed on the quantity of interest. The resulting posterior probability measure is usually analytically intractable. The Markov Chain Monte Carlo (MCMC) method has been the go-to method for sampling from those posterior measures. MCMC is computationally infeasible for large-scale problems that arise in engineering practice. Lately, Variational Bayes (VB) has been recognised as a more computationally tractable method for Bayesian inference, approximating a Bayesian posterior distribution with a simpler trial distribution by solving an optimisation problem. In this work, we argue, through an empirical assessment, that VB methods are a flexible and efficient alternative to MCMC for this class of problems. We propose a natural choice of a family of Gaussian trial distributions parametrised by precision matrices, thus taking advantage of the inherent sparsity of the inverse problem encoded in its finite element discretisation. We utilise stochastic optimisation to efficiently estimate the variational objective and assess not only the error in the solution mean but also the ability to quantify the uncertainty of the estimate. We test this on PDEs based on the Poisson equation in 1D and 2D. A Tensorflow implementation is made publicly available on GitHub.

High-order diamond differencing schemes for the Boltzmann Fokker–Planck equation in 1D and 2D Cartesian geometries

This paper presents the implementation of high-order diamond differencing schemes, specifically the DD1 and DD2 schemes which are 4th- and 6th-order accurate respectively, to handle the spatial discretization of the Boltzmann Fokker–Planck equation in 1D and 2D Cartesian geometries. While being as computationally expensive as linear/quadratic discontinuous Galerkin schemes, the DD1/DD2 schemes are respectively an order more accurate. The energy deposition calculations presented in this work show that, besides providing correction to the oscillations and a reduced propensity to yield negative fluxes than the classical diamond difference closure relation, they tend to perform better than discontinuous Galerkin schemes, even in regions with abrupt variations of the solution due to rapid energy loss, to interface implying high-Z variations and to domain borders.

Comparative Study of Some Numerical and Semi-analytical Methods for Some 1D and 2D Dispersive KdV-type Equations

This paper aims to investigate an approximate-analytical and numerical solutions for some 1D and 2D dispersive homogeneous and non-homogeneous KdV equations by employing two reliable methods namely reduced differential transform method (RDTM) and a classical finite-difference method. RDTM provides an analytical approximate solution in the form of a convergent series. The classical finite-difference method (FDM) to solve dispersive KdV equations is employed by primarily checking Von Neumann’s stability criterion. The performance of the mentioned methods for the considered experiments are compared by computing absolute and relative errors at some spatial nodes at a given time; and to the best of our knowledge, the comparison between these two methods for the considered experiments is novel. Knowledge acquired will enable us to build methods for other related PDEs such as KdV-Burgers, stochastic KdV and fractional KdV-type equations.

A novel RBF-based meshless method for solving time-fractional transport equations in 2D and 3D arbitrary domains

A novel RBF-based meshless method for solving time-fractional transport equations in 2D and 3D arbitrary domains

A second-order well-balanced Lagrange-projection numerical scheme for shallow water Exner equations in 1D and 2D

A second-order well-balanced Lagrange-projection numerical scheme for shallow water Exner equations in 1D and 2D

Transport in Astrophysics: I. Diffusion of Solar and Galactic Cosmic Rays

Some solutions for the diffusion phenomenon as a function of time and space are reviewed. Two new solutions of the homogeneous diffusion equation in 1D and 2D are derived in the presence of an existing fixed number of particles. The initial conditions which allow deriving a power law behavior for the energy of the cosmic rays (CR) are derived. The superposition of transient diffusive phenomena on an existing power law distribution for the energy of CR allows simulating the knee, the second knee, and the ankle.

Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations

&lt;p style='text-indent:20px;'&gt;This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional torus (&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ n = 2, 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;):&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ r\geq1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. We prove that the global attractor of the above system is singleton under small forcing intensity (&lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ r\geq 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ n = 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ r\geq 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ n = 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ 2\beta\mu\geq 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ r = n = 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ 1\leq r&amp;lt;\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ 3\leq r&amp;lt;\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (&lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ 2\beta\mu\geq 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ r = 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;), when the coefficient of random perturbation converges to zero.&lt;/p&gt;