Present Numerical Scheme Research Articles

Subdiffusion of heavy quarks in hot QCD matter by the fractional Langevin equation

The subdiffusion phenomena are studied for heavy quarks dynamics in the hot QCD matter. Our approach aims to provide a more realistic description of heavy quark dynamics through detailed theoretical analyses and numerical simulations, utilizing the fractional Langevin equation framework with the Caputo fractional derivative. I present numerical schemes for the fractional Langevin equation for subdiffusion and calculate the time evolution mean squared displacement and mean squared momentum of the heavy quarks. My results indicate that the mean squared displacements of the heavy quarks for the subdiffusion process deviate from a linear relationship with time. Further, I calculate the normalized momentum correlation function, kinetic energy, and momentum spread. Finally, I show the effect of subdiffusion on experimental observables, the nuclear modification factor. Published by the American Physical Society 2024

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.

Study on fracture of hyperelastic Kirchhoff-Love plates and shells by phase field method

Thin walled structures such as plates and shells are widely used in many engineering fields. To Predict its fracture behavior is of great significance for integrity design and strength evaluation of engineering structures. Numerical simulation of the fracture behavior of hyperelastic plates and shells is a challenge due to complex kinematic description, hyperelastic constitutive relationship, geometric nonlinearity and the degradation on elastic parameter caused by fracture damage. Combining Kirchhoff Love (K-L) shell theory with the fracture phase field method, and numerically discretizing the first and second order partial derivatives of displacement field and phase field by using T-splines and meeting the requirements of K-L plate and shell theory for the C1 continuity of the shape function, a model for the isogeometric analysis numerical formulation of the phase field fracture in hyperelastic K-L plates and shells is established. The fracture failure behavior of hyperelastic K-L plates and shells under the uniform load and displacement load is simulated, and the effect of the Gaussian curvature on the fracture behavior of hyperelastic K-L shells is studied. The simulation results show that the present numerical scheme can effectively capture the complex crack propagation path of plates and shells under the uniform load, and the displacement field can effectively reflect the crack distribution of materials. The thin shell with negative Gaussian curvature shows the excellent fracture performance under the internal pressure, and can withstand the greater internal pressure.

Numerical simulation of submarine landslide tsunamis based on the smoothed particle hydrodynamics model

This paper establishes a SPH (Smoothed Particle Hydrodynamics) model for simulating underwater landslides based on the mixture theory. This model requires only one layer of particles, which greatly improves the computational efficiency compared with the traditional two-layer particle simulation for a mixture theory scheme. In the numerical model, based on a mixture theory, submerged landslide flow is regarded as a mixture of water and sediment phases and is discretized into a series of SPH mixed particles employing the volume fraction of the sediment phase. Using this volume fraction, a convection–diffusion term is calculated to represent the material transport between the water phase and the sediment phase. In addition, based on this volume fraction, the SPH mixed particles at any location in the considered domain are classified into three categories: (i) pure water, (ii) low-concentration suspended sediment, and (iii) high-concentration sediment. Pure water is treated as a Newtonian fluid. High-concentration sediment is modeled as a non-Newtonian fluid, and the Herschel–Bulkley–Papanastasiou rheological model is used to describe the viscous forces. The viscosity of the low-concentration suspended sediment, which acts as a transition layer between pure water and high-concentration sediment, is derived from the Chezy relation. A comparison of the numerical and experimental results demonstrates the high accuracy of the present numerical scheme. Using this validated numerical model, underwater landslides are simulated. Specifically, the effects of landslide deformation and compaction degree on the amplitudes of the surge wave crest and trough are investigated.

Double distribution function-based lattice Boltzmann flux solver for simulation of compressible viscous flows

In this work, a double distribution function-based lattice Boltzmann flux solver (LBFS) is proposed for simulating compressible viscous flows. This approach utilizes the double distribution function compressible lattice Boltzmann model and employs Chapman–Enskog expansion analysis to connect the lattice Boltzmann equation (LBE) with the Navier–Stokes (N–S) equations. Unlike conventional computational fluid dynamics methods that compute inviscid and viscous fluxes separately, the present method simultaneously evaluates both types of fluxes at the cell interface by locally reconstructing the solution of the LBE. Recognizing the significance of considering the non-equilibrium part of distribution functions for viscous flows, a straightforward method is introduced to calculate this component. This facilitates the derivation of computational expressions for macroscopic conservative variables and fluxes in the N–S equations. To validate the accuracy and stability of the present numerical scheme, various benchmark problems, including shock tube problem, Couette flow, lid-driven cavity flow, and flow around the NACA0012 airfoil, are tested. The obtained numerical results are compared with analytical solutions or existing reference data, confirming the capability of the proposed LBFS to deliver accurate and stable numerical results for compressible flows. Moreover, this method demonstrates effectiveness in handling viscous flow problems on non-uniform grids and with curved boundaries.

Energy-conserving finite difference scheme based on velocity interpolation applicable to unsteady flows using collocated grids

The collocation method uses the Rhie–Chow scheme to find the cell interface velocity by pressure-weighted interpolation. The accuracy of this interpolation method in unsteady flows has not been fully clarified. This study constructs a finite difference scheme for incompressible fluids using a collocated grid in a general curvilinear coordinate system. The velocity at the cell interface is determined by weighted interpolation based on the pressure difference to prevent pressure oscillations. The Poisson equation for the pressure correction value is solved with the cross-derivative term omitted to improve calculation efficiency. In addition, simultaneous relaxation of velocity and pressure is applied to improve convergence. Even without the cross-derivative term, calculations can be stably performed, and convergent solutions are obtained. In unsteady inviscid flow, the conservation of kinetic energy is excellent even in a non-orthogonal grid, and the calculation result has second-order accuracy to time. In viscous analysis at a high Reynolds number, the error decreases compared with that of the Rhie–Chow interpolation method. The present numerical scheme improves calculation accuracy in unsteady flows. The possibility of applying this computational method to high Reynolds number flows is demonstrated through several analyses.

Stability and error estimates of Strang splitting method for the nonlocal ternary conservative Allen–Cahn model

The nonlocal model has attracted a great attention in materials science for describing various types of material heterogeneities and defects. In this study, we consider a nonlocal ternary conservative Allen–Cahn model, where the standard Laplace operator is intentionally replaced with a spatial convolution term that aims at describing long-range interactions among particles. A linear energy stable scheme is developed based on the operator splitting method. The mass conservation, energy stability and global convergence of the new scheme are analyzed rigorously. Numerical stability and convergence of the present numerical scheme are analyzed theoretically. Two and three dimensional numerical experiments are performed to validate the theoretical analysis and the efficiency of the method.

Cattaneo-Christov Model on Three-Dimensional Flow, Heat, and Mass Transfer of Prandtl Fluid over a Riga Plate

This paper examined the three-dimensional stretched flow, heat, and mass transports analysis of Prandtl fluid with the influences of chemical reactions, over a Riga surface. This analysis is investigated in presence of Catteneo-Cristov heat and mass fluxes. The resulting nonlinear models are simplified by appropriate similarity variables. The solutions of the reduced set of coupled equations are obtained numerically via the Chebyshev spectral collocation technique. The obtained numerical results were used to address and discuss the characteristics of flow, heat transfer, mass distribution, skin friction, Nusselt, and Sherwood numbers for various pertinent parameters. In addition, the validation of the present numerical scheme is achieved by comparing it with previous results obtained through other numerical results. It is noticed that the rate of heat and mass transfer escalate for the Prandtl parameter. Also, the thermal and mass distributions scale back with a high estimation of relaxation parameters.This paper examined the three-dimensional stretched flow, heat, and mass transports analysis of Prandtl fluid with the influences of chemical reactions, over a Riga surface. This analysis is investigated in presence of Catteneo-Cristov heat and mass fluxes. The resulting nonlinear models are simplified by appropriate similarity variables. The solutions of the reduced set of coupled equations are obtained numerically via the Chebyshev spectral collocation technique. The obtained numerical results were used to address and discuss the characteristics of flow, heat transfer, mass distribution, skin friction, Nusselt, and Sherwood numbers for various pertinent parameters. In addition, the validation of the present numerical scheme is achieved by comparing it with previous results obtained through other numerical results. It is noticed that the rate of heat and mass transfer escalate for the Prandtl parameter. Also, the thermal and mass distributions scale back with a high estimation of relaxation parameters.

Sub-Diffusion Two-Temperature Model and Accurate Numerical Scheme for Heat Conduction Induced by Ultrashort-Pulsed Laser Heating

In this study, we propose a new sub-diffusion two-temperature model and its accurate numerical method by introducing the Knudsen number (Kn) and two Caputo fractional derivatives (0<α,β<1) in time into the parabolic two-temperature model of the diffusive type. We prove that the obtained sub-diffusion two-temperature model is well posed. The numerical scheme is obtained based on the L1 approximation for the Caputo fractional derivatives and the second-order finite difference for the spatial derivatives. Using the discrete energy method, we prove the numerical scheme to be unconditionally stable and convergent with O(τmin{2−α,2−β}+h2), where τ,h are time and space steps, respectively. The accuracy and applicability of the present numerical scheme are tested in two examples. Results show that the numerical solutions are accurate, and the present model and its numerical scheme could be used as a tool by changing the values of the Knudsen number and fractional-order derivatives as well as the parameter in the boundary condition for analyzing the heat conduction in porous media, such as porous thin metal films exposed to ultrashort-pulsed lasers, where the energy transports in phonons and electrons may be ultraslow at different rates.

Improved numerical scheme for the generalized Kuramoto model

We present an improved and more accurate numerical scheme for a generalization of the Kuramoto model of coupled phase oscillators to the three-dimensional space. The present numerical scheme relies crucially on our observation that the generalized Kuramoto model corresponds to particles on the unit sphere undergoing rigid body rotations with position-dependent angular velocities. We demonstrate that our improved scheme is able to reproduce known analytic results and capture the expected behavior of the three-dimensional oscillators in various cases. On the other hand, we find that the conventional numerical method, which amounts to a direct numerical integration with the constraint that forces the particles to be on the unit sphere at each time step, may result in inaccurate and misleading behavior especially in the long time limit. We analyze in detail the origin of the discrepancy between the two methods and present the effectiveness of our method in studying the limit cycle of the Kuramoto oscillators.

Mathematical modelling of nonlinear pressure drops in arbitrarily shaped port utilizing dual boundary element method

A mathematical model is developed to analyze the wave-trapping characteristics of different types of porous and non-porous barriers in arbitrarily shaped domains utilizing the iterative Dual Boundary Element Method (DBEM) under the resonance conditions. In this analysis, Vertical, Wavy, Inverted V-shaped, and Inverted L-shaped barriers are utilized to investigate the effect of incident waves over the boundary of the harbor. The harbor is divided into an interconnected domain with variable bathymetry, and the Helmholtz equation is solved in each subdomain utilizing the prescribed boundary conditions. Further, the semi-empirical discharge equation with quadratic pressure drop is considered on the porous barrier to incorporate the effect of incident wave amplitude on the energy loss. The present numerical scheme is validated with existing studies and verified with convergence analysis. Further, the mathematical model is implemented on the Visakhapatnam port at four key locations with improved performance of different types of barriers. Moreover, the influence of the partial reflection coefficient is also investigated by considering the various combination of partially reflecting boundaries of the port. The wave trapping or absorbing characteristics of porous barriers proves an efficient tool to reduce the wave resonance inside the port.

Performance of magnetic dipole contribution on ferromagnetic non-Newtonian radiative MHD blood flow: An application of biotechnology and medical sciences

Casson flow ferromagnetic liquid blood flow over stretching region is studied numerically. The domain is influence by radiation and blood flow velocity and thermal slip conditions. Blood acts an impenetrable magneto-dynamic liquid yields governing equations. The conservative governing nonlinear partial differential equations, reduced to ODEs by the help of similarity translation technique. The transport equations were transformed into first order ODEs and the resultant system are solved with help of 4th order R-K scheme. Performing a magnetic dipole with a Casson flow across a stretched region with Brownian motion and Thermophoresis is novelty of the problem. Significant applications of the study in some spheres are metallurgy, extrusion of polymers, production in papers and rubber manufactured sheets. Electronics, analytical instruments, medicine, friction reduction, angular momentum shift, heat transmission, etc. are only few of the many uses for ferromagnetic fluids. As ferromagnetic interaction parameter value improves, the skin-friction, Sherwood and Nusselt numbers depreciates. A comparative study of the present numerical scheme for specific situations reveals a splendid correlation with earlier published work. A change in blood flow velocity magnitude has been noted due to Casson parameter. Increasing change in blood flow temperature noted due to Casson parameter. Skin-friction strengthened and Nusselt number is declined with Casson parameter. The limitation of current work is a non-invasive magnetic blood flow collection system using commercially available magnetic sensors instead of SQUID or electrodes.

A NEW NUMERICAL APPROACH OF SOLVING FRACTIONAL MOBILE-IMMOBILE TRANSPORT EQUATION USING ATANGANA-BALEANU DERIVATIVE

A numerical scheme comprising the Crank-Nicolson difference scheme in the temporal direction and cubic trigonometric <i>B</i>-spline method in the spatial direction is examined for the numerical solution of the variable coefficient time-fractional mobile-immobile solute transport equation. The time-fractional derivative is evaluated using the Atangana-Baleanu Caputo derivative. The equation has advection, dispersion, and reaction coefficients that can be influenced simultaneously by space and time variables. The present numerical scheme is unconditionally stable and second-order convergent in the temporal and spatial directions. Several test problems are solved to confirm the theoretical results.

A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term

In this paper, a novel spatial-temporal radial Trefftz collocation method (STRTCM) is proposed to solve 2D and 3D backward heat conduction equations with time-dependent source term. Unlike the traditional time-domain discretization strategies (Laplace/Fourier transformation and time-stepping methods), the proposed spatial-temporal meshless collocation method employs the derived semi-analytical spatial-temporal radial Trefftz basis function as basis function, which naturally satisfies the governing equation with zero source term in advance. Due to this property, only the final conditions and boundary conditions need to be discretized. For non-zero source term, the extended multiple reciprocity method (E-MRM) is introduced to transform the original governing equation with several specific source terms to the high-order governing equation without the source term. By deriving the high-order semi-analytical spatial-temporal radial Trefftz basis function, the STRTCM can also avoid the node discretization of governing equation. For solving the backward heat conduction equations, the present numerical scheme uses the final temperature distribution and the related boundary conditions to obtain the unknown initial temperature field. Finally, the accuracy and efficiency of the proposed method is numerically verified by four benchmark examples about backward heat conduction equations with time-dependent source term.

A robust numerical technique and its analysis for computing the price of an Asian option

An efficient numerical scheme based on exponential B-spline (EBS) functions is developed in this work for numerical solution of Asian option pricing (AOP) problem and dealing with delta values. Convergence and stability of proposed scheme are investigated. Numerical illustrations are performed to demonstrate the efficiency and feasibility of the method and to corroborate the theoretical estimate as well. We examine the effects of volatilities, maturity time and interest rate on option price and delta values. It is shown that the present method for delta values is of second order accuracy in both the spatial and temporal domains. The computed option values are compared with those obtained in Dubois and Lelièvre (2004/2005), Večeř (2001), Zvan et al. (1998) and Thompson (1999). The computational time elapsed for the present method for different values of mesh points are provided. It is shown that the present numerical scheme is accurate for small and large volatilities.

Efficient numerical calculation of frequency-domain matching method based on an analytical control surface

The matching method based on an analytical control surface has been developed to predict wave-induced motions and loads of ships with forward speed. Taking advantage of several computational techniques, this study proposes an efficient scheme for the numerical calculation of this novel method. Different from the conventional domain decomposition strategy, the control surface is analytical, and the physical quantities on it are expanded into the Fourier-Laguerre series. Therefore, the integration over the control surface associated with the forward speed Green function is performed analytically, avoiding the convergence problem caused by the singular and oscillatory behaviors of the forward speed Green function. Because the introduction of the analytical control surface may cause large computational burdens, an efficient scheme is developed by combing threaded parallelism, Chebyshev series approximation, and shared B-spline data. The excellent agreement between numerical results and experimental measurements shows that the present numerical method is reliable and practical in several applications. Furthermore, the numerical validation reveals that the present numerical scheme is affordable on the standard computing platform.

High-Order Dissipation-Preserving Methods for Nonlinear Fractional Generalized Wave Equations

In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes.

Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing

This work deals with the construction and analysis of a high‐order computational scheme for a time‐fractional Black‐Scholes model that governs the European option pricing. The time‐fractional derivative is considered in the sense of Caputo and the L1 − 2 formula is employed to approximate the Caputo temporal‐fractional derivative of order α, where α ∈ (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock‐out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock‐out option are investigated. The computational time for the proposed method is provided.

Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

A Pseudopotential Lattice Boltzmann Method for Simulation of Two-Phase Flow Transport in Porous Medium at High-Density and High–Viscosity Ratios

In this work, the capability of a multiphase lattice Boltzmann method (LBM) based on the pseudopotential Shan-Chen (S-C) model is investigated for simulation of two-phase flows through porous media at high-density and high–viscosity ratios. The accuracy and robustness of the S-C LBM are examined by the implementation of the single relaxation time (SRT) and multiple relaxation time (MRT) collision operators with integrating the forcing schemes of the shifted velocity method (SVM) and the exact difference method (EDM). Herein, two equations of state (EoS), namely, the standard Shan-Chen (SC) EoS and Carnahan-Starling (CS) EoS, are implemented to assay the performance of the numerical technique employed for simulation of two-phase flows at high-density ratios. An appropriate modification in the forcing schemes is also used to remove the thermodynamic inconsistency in the simulation of two-phase flow problems studied at low reduced temperatures. The comparative study of these improvements of the S-C LBM is performed by considering an equilibrium state of a droplet suspended in the vapor phase. The solver is validated against the analytical coexistence curve for the liquid-vapor system and the surface tension estimation from the Laplace Law. Then, according to the results obtained, a conclusion has been made to choose an efficient numerical algorithm, including an appropriate collision operator, a realistic EoS, and an accurate forcing scheme, for simulation of multiphase flow transport through a porous medium. The patterns of two-phase flow transport through the porous medium are predicted using the present numerical scheme in different flow conditions defined by the capillary number and the dynamic viscosity ratio. The results obtained for the nonwetting phase saturation, penetration structure of the invading fluid, and the displacement patterns of two-phase flow in the porous medium are comparable with those reported in the literature. The present study demonstrates that the S-C LBM with employing the MRT-EDM scheme, CS EoS, and the modified forcing scheme is efficient and accurate for estimation of the two-phase flow characteristics through the porous medium.