Second-order Scheme Research Articles

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

A second-order numerical method for nonlinear variable-order fractional diffusion equation with time delay

In this paper, a linearized numerical scheme of nonlinear variable-order fractional diffusion equation with time delay is constructed. We apply the L2−1σ formula to discretize the temporal derivative and second-order central difference scheme to discretize the spatial derivative. The proposed method is unconditionally stable and convergent with Oτ2+h2, where τ and h are the time and space steps, respectively. Numerical experiment demonstrates the effectiveness and accuracy of the numerical scheme.

Model and numerical method for soliton propagation through thermal medium based on nonlinear Schrödinger and heat transfer equations

We present a mathematical model and numerical method for simulating soliton propagation through thermal medium. The model is developed based on the nonlinear Schrödinger (NLS) equation coupled with a heat transfer equation. The well-posedness of the model and stability of the solution are analyzed. We then develop a second-order accurate finite difference scheme for solving this problem and prove the numerical solution to be bounded in l∞ norm. Furthermore, the scheme is proved to be second-order convergent in l2 norm. Numerical examples are given to verify the model and test the accuracy of the numerical scheme for simulating soliton propagations through the thermal medium.

A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficients

A second-order difference scheme for two-dimensional two-sided space distributed-order fractional diffusion equations with variable coefficients

Stability and error analysis of the SAV schemes for the inductionless MHD equations

In this paper, we consider numerical approximations for solving the inductionless magnetohydrodynamic (MHD) equations. By utilizing the scalar auxiliary variable (SAV) approach for dealing with the convective and coupling terms, we propose some first- and second-order schemes for this system. These schemes are linear, decoupled, unconditionally energy stable, and only require solving a sequence of differential equations with constant coefficients at each time step. We further derive a rigorous error analysis for the first-order scheme, establishing optimal convergence rates for the velocity, pressure, current density and electric potential in the two-dimensional case. Numerical examples are presented to verify the theoretical findings and show the performances of the schemes.

COMBINED FORCED AND NATURAL CONVECTION FROM A SINGLE TRIANGULAR CYLINDER

Unsteady laminar confined and unconfined fluid flow and mixed (forced and free) convection heat transfer around equilateral triangular cylinders are investigated numerically. The computation model is a two-dimensional domain with blockage ratios of BR=0.5, 0.25, 0.2, 0.1, 0.05, and 0.0333, with the Reynolds numbers ranging from 100 to 200. The working fluid is water (Pr = 7). The effects of aiding and opposing thermal buoyancy are incorporated into the Navier-Stokes equations using the Boussinesq approximation. The Richardson number, which is a relative measure of free convection, is varied in the range -2 ≤ Ri ≤ 2. The governing equations are solved by using the Finite Volume Method with a second-order upwind scheme used for differencing of the convection terms, and the SIMPLE algorithm is used for the velocity-pressure coupling. A discussion of the effect of the blockage ratio on the mean drag, mean rms lift coefficients, the Strouhal number, and the mean Nusselt number is also presented. The iso-vorticity contours and dimensionless temperature field are generated to interpret and understand the underlying physical mechanisms. The results reveal that, in addition to the Richardson and Reynolds numbers, the blockage rate is effective in the vortex distribution in the channel. It has been determined that the vortices formed behind the cylinder spread to the channel with a decreasing blockage rate. Especially at high Reynolds numbers, both the drag coefficient and the mean Nusselt number are significantly affected by the blockage ratio. For Ri=0, the drag coefficients for BR=0.25 in comparison to the BR=0.05 case are about 9% and 29% larger for Re= 100 and 200, respectively. For BR

Modelling study of the spread of liquid fuel spill fires using shallow water equations

In oil spill–fire accidents, the spread of burning fuel from petroleum product containers may increase accident losses and trigger further accidents. Studying the spreading process of spill fires is important to prevent accidents and mitigate losses. Herein, a spill-fire spread model was proposed to predict the spread rate and time varying burning area of liquid fuel spill fires. The model was developed using one-dimensional nonlinear shallow water equations. Fuel consumption was used as the source term in the equations. To ensure numerical stability and accuracy, the third-order TVD Runge–Kutta method and the second-order MUSCL scheme were used for time advancement and numerical space reconstruction, respectively. Numerical flux was calculated by using the HLL approximate Riemann solver. The algorithm of the shallow water equations was modified in accordance with the spread characteristics of spill fires. Using the developed spread model, the dynamic burning area and dynamic fuel thickness of a spill fire were calculated. The simulation results were verified using the experimental results on continuously released n-heptane spill fires, and the two results were in good agreement.

Development and Verification of a Higher-Order Computational Fluid Dynamics Solver

Abstract Over the past two decades, higher-order methods have gained much broader use in computational science and engineering as these schemes are often more efficient per degree-of-freedom at achieving a prescribed error tolerance than lower-order methods. During this time, higher-order variants of most discretization schemes, such as finite difference methods, finite volume methods, and finite element methods, have emerged. The finite volume method is arguably the most widely used discretization technique in production-level computational fluid dynamics solvers due to its robustness and conservation properties. However, most finite volume solvers only employ a conventional second-order scheme. To leverage the benefits of higher-order methods, the higher-order finite volume method seems the most natural for those seeking to extend their legacy solvers to higher-order. Nonetheless, ensuring higher-order accuracy is maintained is quite challenging as the implementation requirements for a higher-order scheme are much greater than those of a lower-order scheme. In this work, a methodology for verifying higher-order finite volume codes is presented. The higher-order finite volume method is outlined in detail. Order verification tests are proposed for all major components, including the treatment of curved boundaries and the higher-order solution reconstruction. System-level verification tests are performed using the weak form of the method of manufactured solutions. Several canonical verification cases are also presented for the Euler and laminar Navier–Stokes equations.

Numerical solution of the initial-boundary value problem describing separation processes in a distillation column

A modification of the numerical method of characteristics is developed to solve the initial-boundary value problem that arises when modeling the rectification process in the column. The process is described by a system of first-order hyperbolic equations. A specific peculiarity of the model is in the boundary conditions of a special type. At each of the boundaries, boundary conditions are determined from a system of ordinary differential equations, which also includes unknown values of functions on another boundary. A characteristic difference grid is constructed on the base of a linear transformation of a classical rectangular grid. Implicit second-order difference schemes are used, taking into account the features of the problem at the boundaries. The advantage of this approach is in consideration of the specifics of the propagation of perturbations in hyperbolic equations. Numerical implementation of the method was carried out. An illustrative example shows the effectiveness of the proposed modification of the characterization method. This method is a base for further solution of optimal control problems of flows in columns.

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

Wave behavior in a numerical wave tank at intermediate depth of water using stokes wave theory

The paper analyzes a second-order Stokes wave theory by CFD code and uses the volume of fluid (VOF) method in a 2D numerical wave tank (NWT) in the intermediate depth of water under different wave steepness conditions. The CFD simulation in an NWT generates a wave, validated with the already published experimental and numerical data to check the accuracy. It shows that current results are in good agreement with the published result and the difference in the results is within 2.1027%. Six tests under different wave heights for the time period [Formula: see text][Formula: see text]s have been conducted in an NWT to verify the analysis of the wave theory. Horizontal and vertical velocity, surface elevation, and the velocity vector of the particle motion are the important objectives in this analysis. The study shows that the free surface of the water is a high energy density zone. The time discretization has been achieved with the first-order implicit scheme and the second-order upwind scheme. This analysis shows that the ocean characteristic of second-order Stokes wave theory on the free surface is important to the engineering sciences and further investigation.

A conservative second order phase field model for simulation of N-phase flows

We present a novel N-phase extension to the second-order conservative phase field method, first introduced by Chiu and Lin [1]. The proposed N-phase model is in conservative form and is symmetric with respect to the phases while satisfying volume conservation. The model is reduction consistent, meaning that in the absence of M phases, the equations reduce to the equations for an N−M-phase flow. This eliminates the possibility of fictitious phases appearing. For coupling to momentum transport, we extend the two-phase mass-momentum consistent model to N-phase flows. By adopting second-order central spatial schemes, the boundedness properties of the two-phase model are inherited by the N-phase model, and the coupled mass-momentum consistent N-phase flow solver inherits the conservation properties of its two-phase version, resulting in the first N-phase flow method that analytically and discretely conserves mass, momentum and kinetic energy (in the absence of capillary and viscous effects). A novel surface tension model is proposed for modeling surface tension forces in N-phase flows. The phase field model allows for variable interface thicknesses between different phase pairs. This endows the model with the property of attaining correct equilibirum configurations at triple junction points. Specifically, we analytically demonstrate that by using interfacial thicknesses that are inversely proportional with the pairwise surface tension, the coupled system achieves steady state at the correct configuration. Using several canonical and practical numerical tests, we demonstrate the accuracy of the phase field equation, the surface tension model, and the fully coupled N-phase flow solver.

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow.

In the multiphase flow simulations based on the lattice Boltzmann equation(LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation(WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

Group Contribution Method for the Residual Entropy Scaling Model for Viscosities of Branched Alkanes

In this work it is shown how the entropy scaling paradigm introduced by Rosenfeld (Phys Rev A 15:2545–2549, 1977, https://doi.org/10.1103/PhysRevA.15.2545) can be extended to calculate the viscosities of branched alkanes by group contribution methods (GCM), making the technique more predictive. Two equations of state (EoS) requiring only a few adjustable parameters (Lee–Kesler–Plöcker and PC-SAFT) were used to calculate the thermodynamic properties of linear and branched alkanes. These EOS models were combined with first-order and second-order group contribution methods to obtain the fluid-specific scaling factor allowing the scaled viscosity values to be mapped onto the generalized correlation developed by Yang et al. (J Chem Eng Data 66:1385–1398, 2021, https://doi.org/10.1021/acs.jced.0c01009) The second-order scheme offers a more accurate estimation of the fluid-specific scaling factor, and overall the method yields an AARD of 10 % versus 8.8 % when the fluid-specific scaling factor is fit directly to the experimental data. More accurate results are obtained when using the PC-SAFT EoS, and the GCM generally out-performs other estimation schemes proposed in the literature for the fluid-specific scaling factor.

Variable-step numerical schemes and energy dissipation laws for time fractional Cahn–Hilliard model

Two temporal second-order energy stable schemes with variable time step sizes are constructed for the time fractional Cahn–Hilliard model. Nonuniform L1+ formula is utilized for the discretization of the fractional derivative, while the nonlinear term is handled by the fully-implicit scheme and the scalar auxiliary variable method, respectively. By means of the recently proposed discrete gradient structure of the temporal discretization operator, the discrete energy dissipation laws are developed via a unified framework, which are stronger than the energy boundedness results reported in the previous literature. Moreover, the discrete energy decay nature coincides with the classical analogies as the fractional order tends to one. The optimal convergence rate is verified numerically, furthermore, the coarsening dynamics simulation demonstrates the efficiency of the variable-step schemes combined with the adaptive time strategy.

Development of a Fully-Implicit Second-Order system Thermal-Hydraulics analysis code for reactor modeling and simulations

There has been continuous advancement in numerical techniques and software engineering in the field of system thermal-hydraulics over the past decade. The current state-of-the-art involves the use of fully-implicit high-resolution schemes in conjunction with the Jacobian-Free Newton-Krylov (JFNK) method for both single-phase and two-phase flows. However, the application of these advanced schemes in system thermal-hydraulics codes is not a common practice. Given the opportunity and the need to develop a new modern system thermal–hydraulic code from scratch, there is no doubt that these recent advancements should be leveraged. In this work, we introduce a new modern object-oriented system thermal-hydraulics code named RETA. RETA is characterized by several key features, including: second-order temporal and spatial discretization schemes for both incompressible and compressible flows, as well as single-phase and two-phase flows; fully-implicit solution schemes utilizing both the Newton and PJFNK methods; and an object-oriented design of major fluid components and physical models, which allows for easy extension. A series of verification and demonstration studies are conducted to verify the implementation, demonstrate the accuracy improvement, and evaluate the performance of the new code. It is concluded that results from the second-order scheme are generally more reliable. Without the concern of a code stability issue, the second-order scheme should be the preferred and default choice for a code user or an analyst.

Convergence of a second-order scheme for non-local conservation laws

In this article, we present the convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws. We combine a MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme. The resulting scheme is shown to converge to a weak solution by establishing the maximum principle, bounded variation estimates and L1Lipschitz continuity in time. Further, using a space-step dependent slope limiter, we prove its convergence to the entropy solution. We also propose a MUSCL-Hancock type second-order scheme which requires only one intermediate stage unlike the Runge-Kutta schemes and is easier to implement. The performance of the proposed second-order schemes in comparison to a first-order scheme is demonstrated through several numerical experiments.

Differential-Difference Equations with Optimal Parameters

The paper considers difference schemes with optimal parameters for solving Maxwell’s equations. Using Laguerre transforms, the numerical values of the optimal parameters are determined and differential-difference equations are constructed. Differential-difference equations are solved by the finite-difference method with iterations over small optimal parameters. Optimal second-order difference schemes for one-dimensional and two-dimensional Maxwell’s equations are considered. Optimal parameters of difference schemes are given. It is shown that the use of optimal difference schemes leads to an increase in the accuracy of solution.

A Second-Order Scheme for the Generalized Time-Fractional Burgers' Equation

Abstract A second-order numerical scheme is proposed to solve the generalized time-fractional Burgers' equation. The time-fractional derivative is considered in the Caputo sense. First, the quasi-linearization process is used to linearize the time-fractional Burgers' equation, which gives a sequence of linear partial differential equations (PDEs). The Crank–Nicolson scheme is used to discretize the sequence of PDEs in the temporal direction, followed by the central difference formulae for both the first and second-order spatial derivatives. The established error bounds (in the L2− norm) obtained through the meticulous theoretical analysis show that the method is second-order convergent in space and time. The technique is also shown to be conditionally stable. Some numerical experiments are presented to confirm the theoretical results.

Positivity-Preserving Discontinuous Galerkin Methods on Triangular Meshes for Macroscopic Pedestrian Flow Models

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes.