Time Step Δt Research Articles

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

MD Simulation of Water Using a Rigid Body Description Requires a Small Time Step to Ensure Equipartition.

In simulations of aqueous systems, it is common to freeze the bond vibration and angle bending modes in water to allow for a longer time step δt for integrating the equations of motion. Thus, δt = 2 fs is often used in simulating rigid models of water. We simulate the SPC/E model of water using δt from 0.5 to 3.0 fs and up to 4 fs using hydrogen mass repartitioning. In these simulations, we find that for all but δt = 0.5 fs, equipartition is not obtained between translational and rotational modes, with the rotational modes exhibiting a lower temperature than the translation modes. To probe the reasons for the lack of equipartition, we study the autocorrelation of the translational velocity of the center of mass and the angular velocity of the rigid water molecule, respectively. We find that the rotational relaxation occurs on a timescale comparable to vibrational periods, calling into question the original motivations for freezing the vibrations. Furthermore, a time step with δt ≥ 1 fs is not able to capture accurately the fast rotational relaxation, which reveals its impact as an effective slowing-down of rotational relaxation. The fluctuation-dissipation relation then leads to the conclusion that the rotational temperature should be cooler for δt greater than the reference value of 0.5 fs. Consideration of fluctuation-dissipation in equilibrium molecular dynamics simulations also emphasizes the need to capture the temporal evolution of fluctuations with fidelity and the role of δt in this regard. The time step also influences the solution thermodynamic properties: both the mean system potential energies and the excess entropy of hydration of a soft repulsive cavity are sensitive to δt.

Error analysis of an effective numerical scheme for a temporal multiscale plaque growth problem

In this work, we propose a simple numerical scheme based on a fast front-tracking approach for solving a fluid-structure interaction (FSI) problem of plaque growth in blood vessels. A rigorous error analysis is carried out for the temporal semi-discrete scheme to show that it is first-order accurate for all macro time step ΔT, micro time step Δt and scale parameter ϵ. A numerical example is presented to verify the theoretical results and demonstrate the excellent performance of the proposed multiscale algorithm.

Modeling Aquifer Responses from Flash Flood Events through Ephemeral Stream Beds: Case Studies from Saudi Arabia

The determination of groundwater recharge (GWR) with an appropriate method is among the main subjects of groundwater resource management. This research is an event-scale quantitative assessment and a development of a water balance model of an aquifer response resulting from flash flooding and infiltration from the ephemeral streambed of some representative basins in Saudi Arabia. The methodology employed in the water balance analysis of this study starts with surface runoff rather than rainfall as in the literature, since the recharge process mainly occurs from runoff on the ground surface. A conceptual model of the local surface–subsurface interaction is developed to estimate the effective aquifer response (AR). The annual cumulative groundwater recharge (GWR) depths vary from 1.22 m to 6.97 m, with an average rise of 4.24 m at the studied wells; most of this happened following the May and April flash flood events. The results show also similar profile patterns and timings for most of the water level rises and falls. An average root mean square error (RMSE) of 0.46 m is obtained from the simulations and the coefficient of determination (R2) values range from 0.57 to 0.99. The results verified that the daily simulated groundwater elevations have shown very good agreement with the observed daily groundwater elevations over a time span of one year. In this work, the key issue to interpret the GWR depth from individual well samples is whether the model on the samples fits the measured groundwater level within the experimental runoff reaches (ERRs). The time steps (Δt) of the GWR simulation process are significantly affected by the high variability in the aquifer characteristics, as shown for the hydraulic conductivity values ranging from 5.3 m/day to 438 m/day within ERRs. This study highly recommends that groundwater recharge studies be site-specific. Otherwise, the extrapolation of the results even to adjacent basins is questionable.

Expedition of melting of paraffin within heat storage system in existence of nanomaterial and curved enclosure

In this research, melting phenomena in a container with two fin lengths (L*) were examined. It was hypothesized that the homogenous combination of RT35 and Cu nanoparticles was a nano-enhanced phase change material (NEPCM). To demonstrate the effects of adding Cu, L*, and Stefan number (Ste), several situations were studied. A computational approach based on a structural mesh was used to simulate present unsteady modeling. The best values for the mesh dimensions and time step (Δt) were established after considering a variety of circumstances in an effort to bring the computational cost down to an acceptable level. To verify the assessment, empirical records were checked, and it was stated that excellent correlations were available. Higher Ste, and lower L* case is more effective in terms of melting. The value of the liquid fraction (LF) was 0.57, the highest share of the other cases, for L* = 5, φ=4%, and Ste = 0.17 after 900 s from the beginning of process, while the lowest percentage was calculated 0.16 for the case without fin (the plain tube) with the pure PCM and Ste = 0.05. When 4 % Cu is added to pure PCM in the tube with L* = 5 over 900 s, the LF increases by 15.80 % for the Ste of 0.17.

A numerical comparison of Galerkin and Collocation Isogeometric approximations of acoustic wave problems

We consider the Galerkin and Collocation Isogeometric approximations of the acoustic wave equation with absorbing boundary conditions, while the time discretization is based on second-order Newmark schemes. A numerical study investigates the properties of the two IGA methods as concerns stability thresholds, convergence errors, accuracy, computational time, and sparsity of the stiffness matrices varying the polynomial degree p, mesh size h, regularity k, and time step Δt. In order to compare the two IGA methods, we focus on two meaningful examples in the framework of wave propagation simulations: a test problem with an oscillatory exact solution having increasing wave number, and the propagation of two interfering Ricker wavelets. Numerical results show that the IGA Collocation method retains the convergence and stability properties of IGA Galerkin. In all examples considered, IGA Collocation is in general less accurate when we adopt the same choices of parameters p, k, h, and Δt. On the other hand, regarding the computational cost and the amount of memory required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the IGA Galerkin method, especially in the case of maximal regularity k=p−1 with increasing NURBS degree p.

A class of weighted energy-preserving Du Fort–Frankel difference schemes for solving sine-Gordon-type equations

Recently, invariant energy-quadratization methods (IEQMs) have been introduced by Xiaofeng Yang’s group to develop linear and energy-dissipation-preserving methods for nonlinear energy-dissipation systems. Following their work, two auxiliary functions are firstly introduced to rewrite the sine-Gordon equation (SGE) and coupled sine-Gordon equations (CSGEs) into equivalent systems, respectively. Then, two energy-preserving Du Fort–Frankel-type finite difference methods (EP-DFFT-FDMs) have been suggested for them, respectively. By using the discrete energy methods, the discrete energy conservative laws and convergence rates in the H1-norm have been derived, rigorously. It is worth mentioning that the proposed discrete energy is an approximation to the exact energy of the continuous problem. As hx=O(Δt), hy=O(Δt) and parameter λ≥1/4, the current methods are stable in the H1-norm because numerical solutions obtained by them are bounded in the H1-norm. What is more, as parameter λ≥1/4, the current methods are unconditionally stable in the L2-norm because numerical solutions obtained by them are uniformly bounded in the L2-norm. Moreover, our methods are explicit, and very easy to be implemented. However, a shortcoming of the current methods is that they are conditionally consistent. Namely, Δthx and Δthy tend to zero as time step Δt, spatial mesh sizes hx in x-direction and hy in y-direction tend to zero. Numerical findings support the correctness of theoretical analyses and the performance of the algorithms.

Adaptive mesh refinement method for the reduction of computational costs while simulating slug flow

Microchannel flow boiling has been the focus of many experimental and numerical investigations due to the high heat transfer coefficients that it can induce. However, experimental research has been limited due to the small scales involved, leading researchers to employ computational fluid dynamics (CFD) simulations to resolve the dearth of research on microchannel flow boiling. Conventional CFD methods use a fine uniform mesh to capture the small scales and gradients, such as the liquid-vapour interface. This method has a large computational cost, and as a result, most research reported in the literature has been limited to two-dimensional axisymmetric domains. An interface-tracking adaptive mesh refinement model was created in this study to overcome the limitation of high computational costs without losing accuracy. This model dynamically refined the mesh only in the regions of interest and allowed a coarser mesh in the rest of the domain. This novel approach was able to recreate previously published results with a maximum error of 6.7%, while using less than 1.6% of the mesh elements. Several simulations were conducted in ANSYS Fluent 19.1 to determine the optimal settings for this new method to maintain accuracy and reduce cell count. These settings were determined as three levels of refinement (δL = 3), four refined cells on either side of the interface (δM = 4), and was implemented every five time steps (δT = 5). Finally, a case study was conducted to illustrate the possibility of simulating two-phase flow in microchannels in three dimensions with this method.

Heat and Mass Transfer Analysis on Magneto Micropolar Fluid Flow with Heat Absorption in Induced Magnetic Field

Heat and mass transfer due to a magneto micropolar fluid flow along a semi-infinite vertical plate bounded by a porous medium are investigated in presence of induced magnetic field. In case of cooling flow, heat and mass fluxes from the plate are subjected to be constant under the action of a constant heat sink. Mathematical model related to the problem is developed from the basis of studying magnetohydrodynamics (MHD) for both lighter and heavier particles. Dimensionless model of momentum, microrotation, induction, energy and concentration equations are solved simultaneously by the explicit scheme of finite difference technique. According to the obtained stability and convergence criteria of this transient flow, very negligible time step (Δt = 0.002) compared to the existing works has been taken to perform the numerical computation. Quantities of chief physical interest of the flow as shear stress, couple stress, current density, Nusselt number and Sherwood number are also studied here. The numerically computed results are compared with published results of available research works. Interestingly an excellent agreement is found with finite difference solutions in both explicit and implicit schemes. In order to discuss the physical aspects of the problem, the flow variables for different values of associated parameters are illustrated in graphs. Finally, important findings of the study are listed as concluding remarks.

A technology demonstration of propagator matrix power method for calculation of electron velocity distribution functions in gas in long-term transient and succeeding equilibrium states under dc electric fields

A matrix-based calculation for the electron velocity distribution function (EVDF) in gas under dc electric field is demonstrated. The propagator matrix power method (PMPM) repeats squaring the propagator matrix P for its powers P 2, P 4, P 8, P 16, ⋯, up to a sufficient order of the power . P is a square matrix representing the change of EVDF due to the electron acceleration by an electric field and the electron scattering by gas molecules in a time step Δt. With an initial EVDF f(t 0) in a form of a column vector, a matrix product gives the EVDF at t = t 0 + 2 S Δt. The PMPM enables us to observe the EVDF relaxation in a logarithmic time scale with a linear increase of the matrix squaring steps S. Features of the PMPM are discussed, and theoretically possible extensions to models under crossed electric and magnetic fields and ac electric field are investigated.

Influence of molecular rebinding on the reaction rate of complex formation.

We simulated Brownian diffusion and reaction-diffusion processes to study the influence of molecular rebinding on the reaction rates of bimolecular reactions. We found that the number of rebinding events, Nreb, is proportional to the target's size and inversely proportional to the diffusion coefficient D and simulation time-step Δt. We found the proportionality constant close to π-1/2. We confirmed that Nreb is defined as a ratio of the activation-limited rate constant ka to the diffusion-limited rate constant, kD. We provide the formula describing the reactivity coefficient κ, modelling the transient-native complex transition for the activation-controlled reaction rates. We show that κ is proportional to (D/Δt)1/2. Finally, we apply our rebinding-including reaction rate model to the real reactions of photoacid dissociation and protein association. Based on literature data for both types of reactions, we found the Δt time-scale. We show that for the photodissociation of a proton, the Δt is equal to 171 ± 18 fs and the average number of rebinding events is approximately equal to 40. For proteins, Δt is of the order of 100 ps with around 20 rebinding events. In both cases the timescale is similar to the timescale of fluctuation of the solvent molecules surrounding the reactants; vibrations and bending in the case of photoacid dissociation and diffusional motion for proteins.

Modeling non-Markovian data using Markov state and Langevin models.

Markov processes provide a popular approach to construct low-dimensional dynamical models of a complex biomolecular system. By partitioning the conformational space into metastable states, protein dynamics can be approximated in terms of memory-less jumps between these states, resulting in a Markov state model (MSM). Alternatively, suitable low-dimensional collective variables may be identified to construct a data-driven Langevin equation (dLE). In both cases, the underlying Markovian approximation requires a propagation time step (or lag time) δt that is longer than the memory time τM of the system. On the other hand, δt needs to be chosen short enough to resolve the system timescale τS of interest. If these conditions are in conflict (i.e., τM > τS), one may opt for a short time step δt = τS and try to account for the residual non-Markovianity of the data by optimizing the transition matrix or the Langevin fields such that the resulting model best reproduces the observables of interest. In this work, rescaling the friction tensor of the dLE based on short-time information in order to obtain the correct long-time behavior of the system is suggested. Adopting various model problems of increasing complexity, including a double-well system, the dissociation of solvated sodium chloride, and the functional dynamics of T4 lysozyme, the virtues and shortcomings of the rescaled dLE are discussed and compared to the corresponding MSMs.

On the observability inequalities of time discrete 2‐D integro‐differential systems in square domains

AbstractIn this article, we consider an observability inequality of time discrete approximation schemes for a class of integro‐differential equations in square domains. The equation is discretized in time by the back‐ward Euler method in combination with Lubich's convolution quadrature. We derive uniform observability inequality for time discretization schemes in which the high frequency components have been filtered. In this way, the well‐known exact observability inequality of the integro‐differential systems (see Loreti and Sforza, Evolution Equations and Control Theory, 2018, Vol. 7, No. 1, pp. 61–77, Thm. 1.2) can be reproduced as the limit, as the time step Δt → 0. The time discrete observability inequality is established by means of a time‐discrete version of the classical harmonic analysis approach. Numerical experiments are presented to show the theoretical analysis.

Error estimates of a second-order decoupled scheme for the evolutionary Stokes-Darcy system

This paper presents the new error estimates for a second-order backward-differentiation scheme (BDF2) given in [3] for the evolutionary Stokes-Darcy system. We prove the unconditional optimal error estimates and the uniform in time error estimates with the modest time step (Δt≤C) for the fluid velocity and the hydraulic head in L2 norm. Besides, we also prove the error estimates for the fluid velocity and the hydraulic head in H1 norm are Δt158+h and the pressures in L2 norms are Δt74+h. Numerical tests are provided to illustrate the accuracy and validity of the decoupled scheme.

Stability analysis and error estimates of local discontinuous Galerkin methods with semi-implicit spectral deferred correction time-marching for the Allen–Cahn equation

This paper is concerned with the stability and error estimates of the local discontinuous Galerkin (LDG) method coupled with semi-implicit spectral deferred correction (SDC) time-marching up to third order accuracy for the Allen–Cahn equation. Since the SDC method is based on the first order convex splitting scheme, the implicit treatment of the nonlinear item results in a nonlinear system of equations at each step, which increases the difficulty of the theoretical analysis. For the LDG discretizations coupled with the second and third order SDC methods, we prove the unique solvability of the numerical solutions through the standard fixed point argument in finite dimensional spaces. At the same time, the iteration and integral involved in the semi-implicit SDC scheme also increase the difficulty of the theoretical analysis. Comparing to the Runge–Kutta type semi-implicit schemes which exclude the left-most endpoint, the SDC scheme in this paper includes the left-most endpoint as a quadrature node. This makes the test functions of the SDC scheme more complicated and the energy equations are more difficult to construct. We provide two different ideas to overcome the difficulty of the nonlinear terms. By choosing the test functions carefully, the energy stability and error estimates are obtained in the sense that the time step Δt only requires a positive upper bound and is independent of the mesh size h. Numerical examples are presented to illustrate our theoretical results.

Al-Au Heterogeneous Dimer-trimer Nanostructure for SERS

Background: Tunability in resonance wavelength and the enhancement of the electromagnetic field intensities around the surface are two unique properties which make metal as a plasmonic material. A theoretical investigation on the LSPR and field enhancement for heterogeneous dimer–trimer metallic nanostructure by constituting Al and Au as two different plamsonic materials has been studied. Since electrons in Al exhibit free behavior for LSPR of Au, therefore, they influence the electric field magnitude generated by Au LSPR. Methods: The electromagnetic simulations reported in this paper were performed using the FDTD Solutions (version 7.5.1), a product of Lumerical Solutions Inc., Vancouver, Canada. We adopted a cubic Yee cell of 1 nm side and a time step Δt= 1.31•10-18 s, bounded by Courant condition. Results: The extinction spectrum shows LSPR peak over UV-visible region for isotropic nanostructure which shifts to NIR region for anisotropic shape nanostructure. The spherical shape hetero dimer nanostructure shows enhancement factor ~ 3.9 X 105 whereas it increases to ~ 6.2 X 106 for anisotropic shape at 610 nm. The field distribution corresponding to the trimer nanostructure reveals a large dipolar field distribution on each of the three nanoparticles, oscillating approximately in-phase. The spherical shape Al-Au-Al shows enhancement factor ~ 8.5 X 106 at 571 nm. The anisotropic shape increase the enhancement factor to ~ 2.4 X 107 at peak wavelength 700 nm i.e. tuning the plasmon wavelength towards NIR region. Conclusion: The tunability in plasmon wavelength and field enhancement factor has been evaluated for heterogeneous nanostructure over wider spectrum range i.e. DUV-Visible-NIR using Au-Al dimer and trimer nanostructure. The isotropic shape Au-Al hetero nanostructure shows larger enhancement in the UV-visible region, whereas the anisotropic shape nanostructure contributes towards the NIR region.

L2-stability analysis of IMEX-(σ,μ)DG schemes for linear advection-diffusion equations

A fully discrete L2-stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the (σ,μ)-family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L2-stability for time steps Δt=O(d/a2), where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent (14,94)-recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stability property which is also confirmed by numerical experiments.

Meshless RBFs method for numerical solutions of two-dimensional high order fractional Sobolev equations

In this paper, meshless RBFs method is proposed to solve two-dimensional time-fractional Sobolev equations. The proposed method uses RBFs for approximation of spatial operator. Finite difference formula of O(δt2−α)(0<α≤1) is used for time-fractional derivative approximation while θ-rule (0≤θ≤1) as time stepping scheme for the advancement of solution in time. Validation of the proposed method is made by considering various test examples from literature. Simulated results are found in very good agreement with available exact solutions. A rigorous comparative analysis made with other methods testifies proposed method’s superiority in higher dimensions. Efficiency and accuracy of RBFs method are examined by varying number of nodes N in the domain of influence, time-step size δt, as well as L2, L∞ and Lrms error norms. Linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis.

An arbitrary high order series-based time-marching method with discontinuous Galerkin method for 2D convection problem

This paper presents an arbitrary high-order series-based time-marching (SBTM) method to match with the high-order discontinuous Galerkin (DG) method in spatial discretization. And the detail of this hybrid method, SBTM-DG, is presented by solving a 2D convection problem where the exact solution of the semi-discrete system derived by DG method can be obtained by SBTM method. The SBTM-DG method is explicit single-step scheme of arbitrary high order of accuracy in time and with all advantages of DG method in space. The high order of accuracy in temporal direction is obtained by multiplications between small-scale matrixes and vectors in each cells with dimension K=(k+1)(k+2)/2, where k is the degree of basic function of DG method, which makes the SBTM-DG method memory-saving and efficient. With the help of projection, the error estimation between the exact solution and the full-discrete solution is O(h)k+1+O(Δt)S, where S is the degree of polynomial of time variable. By energy method we can prove the stability. By matrix method, we obtain the stability condition where time step Δt can be larger than mesh size h. Numerical examples confirm the analysis and show the good performance of SBTM-DG method. Furthermore, the convergence orders are around 20 about k, which makes SBTM-DG method significant.

Estimating distributions of airborne contaminants released by sources with dynamic strength and dynamic location by a probabilistic model

Characteristics of contaminant release source usually contribute significant impacts to dispersion of airborne pollutants. It is necessary and also desirable to grasp detailed information of contaminant distributions under various release patterns of contaminant sources. This study developed a Markov chain based probabilistic model to simulate transport and dispersion of airborne contaminants released by sources with dynamic strengths and locations. The proposed model takes the dynamic source strengths and locations into account by replacing elements of the state vector with values in accord with specific functions, which indicate strength and location variations. The particle phase simulations need to be initialized by comprehensive data of the flow field, which can be solved by CFD tools in advance. For the first step, the developed model is validated by experiment data from a 2D turbulence duct flow with a particulate source instantaneously releasing airborne particles. Then the new model is adopted to simulate contaminant dispersion of gaseous (or particles with small diameters) contaminant released by sources with dynamic strength and location. Results indicate that the proposed model is capable of predicting the behaviors of contaminant sources under various release patterns. The effects of dynamic contaminant sources on downstream regions in unidirectional flow are limited by the distances between the target region and source location. Accuracy of the present model is significantly affected by temporal intervals (the specified time step Δt) for sources with dynamic strengths and spatial intervals (computational grid scales) for sources with dynamic locations.