First-order Numerical Schemes Research Articles

On the Features of Numerical Simulation of Hydrogen Self-Ignition under High-Pressure Release

The paper is devoted to the comparative analysis of different CFD techniques used to solve the problem of high-pressure hydrogen release into the air. Three variations of a contemporary low-dissipation numerical technique (CABARET) are compared with each other and a conventional first-order numerical scheme. It is shown that low dissipation of the numerical scheme defines better resolution of the contact surface between released hydrogen and ambient air. As a result, the spatial structures of the jet and the reaction wave that arise during self-ignition are better resolved, which is useful for predicting the local effects of high-pressure hydrogen release. At the same time, the dissipation has little effect on the induction delay, so critical conditions of self-ignition can be reliably reproduced even via conventional numerical schemes. The test problem setups formulated in the paper can be used as benchmarks for compressible CFD solvers.

Enhancement of damaged-image prediction based on digital twin technology

Digital twins have revolutionized the field of image enhancement by applying their unique capabilities. A digital twin refers to a virtual replica of a physical object or system, which can be utilized to simulate and analyze real-world scenarios. In image enhancement, digital twins map entities to images, identify damaged areas, and restore them to their original state. This process involves utilizing the digital twin method to understand the underlying structure and characteristics of the image. The damaged areas can be accurately modeled and repaired using techniques like the Cahn-Hilliard equation. Additionally, neural network models are leveraged to measure the effectiveness of the image restoration process. Compared with the first-order numerical scheme, the second-order method can improve the prediction accuracy by more than 40% in some cases. Through these advancements, digital twins have significantly enhanced images’ quality, clarity, and visual appeal, contributing to various photography, healthcare, and remote sensing applications.

Unconditional energy stability and temporal convergence of first-order numerical scheme for the square phase-field crystal model

This paper presents a sixth-order nonlinear parabolic problem of the square phase-field crystal model. We first demonstrate the time-discrete backward Euler scheme with mass conservation and energy stability. Then, we prove the unconditionally optimal error estimates for the time-discrete backward Euler scheme. In the end, we present 2D and 3D numerical simulations to confirm the theoretical results.

Unconditional Energy Stability and Temporal Convergence of First-Order Numerical Scheme for the Square Phase-Field Crystal Model

Unconditional Energy Stability and Temporal Convergence of First-Order Numerical Scheme for the Square Phase-Field Crystal Model

Re-evaluating efficiency of first-order numerical schemes for two-layer shallow water systems by considering different eigenvalue solutions

The efficiency of several first-order numerical schemes for two-layer shallow water equations (SWE) are evaluated here by considering different eigenvalue solutions. This study is a continuation of our previous work (Krvavica et al., 2018) in which we have proposed an efficient implementation of a Roe solver for two-layer SWE based on analytical expressions for eigenvalues and eigenvectors. In this work, the accuracy and computational cost of numerical, analytical, and approximated eigenvalue solvers are compared when implemented in Roe, Intermediate Field CaPturing (IFCP) and Polynomial Viscosity Matrix (PVM) schemes. Several numerical tests are performed to examine the overall efficiency of numerical schemes with different eigenvalue solvers when computing two-layer shallow-water flows. The results confirm that analytical eigenvalue solutions are much faster than numerical solvers, with a computational cost closer to approximate expressions. Consequently, the Roe scheme with analytical solutions to the eigenstructure is equally efficient as the IFCP scheme. On the other hand, IFCP and PVM schemes with analytical solutions to eigenvalues are found to be equally efficient as those with approximated expressions. Analytical eigenvalues show slightly better results when dealing with larger density differences between the layers.

A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains

A First-Order Numerical Scheme for Forward-Backward Stochastic Differential Equations in Bounded Domains

Numerical investigation of turbulent swirling flows through an abrupt expansion tube

A numerical investigation of turbulent swirling flows through an abrupt expansion tube is reported. The TEFESS code, based on a staggered Finite Volume approach with the standard k-ε model and first-order numerical schemes built-in, was used to carry out all the computations. The code has been modified in the present work to incorporate the ASM and two second-order numerical schemes. The ASM, which includes the non-gradient convection terms arising from the transformation from Cartesian to cylindrical coordinates, was investigated for isothermal flows by applying it to the flow through an abrupt expansion tube with or without swirl flows. In addition, to investigate the effects of numerical diffusion on the predicted results, two second-order differencing schemes, namely, second-order upwind and the quadratic upstream interpolation, were used to compare with the first-order hybrid scheme. An abrupt expansion tube with non-swirling flow, predicted results using both the k-ε model and the ASM were in good agreement with measurements. For swirling flows, the calculated results suggested that the use of the ASM with a second-order numerical scheme leads to better agreement between the numerical results and experimental data, while the k-ε model is incapable of capturing the stabilizing effect of the swirl.

Application of high-order numerical schemes and Newton-Krylov method to two-phase drift-flux model

This paper concerns the application and solver robustness of the Newton-Krylov method in solving two-phase flow drift-flux model problems using high-order numerical schemes. In our previous studies, the Newton-Krylov method has been proven as a promising solver for two-phase flow drift-flux model problems. However, these studies were limited to use first-order numerical schemes only. Moreover, the previous approach to treating the drift-flux closure correlations was later revealed to cause deteriorated solver convergence performance, when the mesh was highly refined, and also when higher-order numerical schemes were employed. In this study, a second-order spatial discretization scheme that has been tested with two-fluid two-phase flow model was extended to solve drift-flux model problems. In order to improve solver robustness, and therefore efficiency, a new approach was proposed to treating the mean drift velocity of the gas phase as a primary nonlinear variable to the equation system. With this new approach, significant improvement in solver robustness was achieved. With highly refined mesh, the proposed treatment along with the Newton-Krylov solver were extensively tested with two-phase flow problems that cover a wide range of thermal-hydraulics conditions. Satisfactory convergence performances were observed for all test cases. Numerical verification was then performed in the form of mesh convergence studies, from which expected orders of accuracy were obtained for both the first-order and the second-order spatial discretization schemes. At last, the drift-flux model, along with numerical methods presented, were validated with three sets of flow boiling experiments that cover different flow channel geometries (round tube, rectangular tube, and rod bundle), and a wide range of test conditions (pressure, mass flux, wall heat flux, inlet subcooling and outlet void fraction).

High-Order Numerical Schemes for One-Dimensional Nonlocal Conservation Laws

This paper focuses on the numerical approximation of the solutions of non-local conservation laws in one space dimension. These equations are motivated by two distinct applications, namely a traffic flow model in which the mean velocity depends on a weighted mean of the downstream traffic density, and a sedimentation model where either the solid phase velocity or the solid-fluid relative velocity depends on the concentration in a neighborhood. In both models, the velocity is a function of a convolution product between the unknown and a kernel function with compact support. It turns out that the solutions of such equations may exhibit oscillations that are very difficult to approximate using classical first-order numerical schemes. We propose to design Discontinuous Galerkin (DG) schemes and Finite Volume WENO (FV-WENO) schemes to obtain high-order approximations. As we will see, the DG schemes give the best numerical results but their CFL condition is very restrictive. On the contrary, FV-WENO schemes can be used with larger time steps. We will see that the evaluation of the convolution terms necessitates the use of quadratic polynomials reconstructions in each cell in order to obtain the high-order accuracy with the FV-WENO approach. Simulations using DG and FV-WENO schemes are presented for both applications.

Numerical solution of the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the step

Free-surface flows are usually modelled by means of the Shallow-water Equations: this system of hyperbolic equations exhibits a source term which is proportional to the product of the water depth by the bed slope, and which takes into account the effect of gravity onto fluid mass. Recently, much attention has been paid to the case in which bottom discontinuities are present in the physical domain to be represented: in this case, it is difficult to define the non-conservative product in the distributional sense. Here, the discontinuous-bottom Shallow-water Equations with hydrostatic pressure distribution at the bed step (Bernetti et al., 2006) are discussed in the context of the theory of Dal Maso et al. (1995) [9]; finally, a first-order numerical scheme is presented, which is consistent for regular solutions, and which is able to capture contact discontinuities at bottom steps. Numerous tests are presented to show the feasibility of the scheme and its ability to converge to the exact solution in the cases of smooth as well as discontinuous bed profiles.

Some First-Order Algorithms for Total Variation Based Image Restoration

This paper deals with first-order numerical schemes for image restoration. These schemes rely on a duality-based algorithm proposed in 1979 by Bermùdez and Moreno. This is an old and forgotten algorithm that is revealed wider than recent schemes (such as the Chambolle projection algorithm) and able to improve contemporary schemes. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples. We make some comparisons with a class of efficient algorithms (proved to be optimal among first-order numerical schemes) recently introduced by Y. Nesterov.

Kinetic theory analysis of laser ablation of carbon: Importance of second-order calculation

Following some recent works that used first-order numerical schemes in modeling laser ablation, this paper compares the predictions of first- and second-order calculations for a one-dimensional kinetic theory model of laser ablation of carbon based on the moment solution of the Boltzmann equation for arbitrary strong evaporation. The model considers both conduction heat transfer in the target and gas dynamics of the ablated plume coupled through the interface boundary conditions. It was found that the difference between the first-order and the second-order calculations is limited to the parts of the flow with high gradients, such as the shock wave and the contact surface. Since the ablation rate is determined by the conditions at the solid surface, which is most of the time situated far from that area, the first-order calculation predicts ablation rate correctly.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

Sensitivity equations for hyperbolic conservation law-based flow models

The present paper focuses on the governing equations for the sensitivity of the variables to the parameters in flow models that can be described by one-dimensional scalar, hyperbolic conservation laws. The sensitivity is shown to obey a hyperbolic, scalar conservation law. The sensitivity is a conserved scalar except in the case of discontinuous flow solutions, where an extra, point source term must be added to the equations in order to enforce conservation. The propagation speed of the sensitivity waves being identical to that of the conserved variable in the original conservation law, the system of conservation laws formed by the original hyperbolic equation and the equation satisfied by the sensitivity is linearly degenerate. A consequence on the solution of the Riemann problem is that rarefaction waves for the variable of the original equation result in vacuum regions for the sensitivity. The numerical solution of the hyperbolic conservation law for the sensitivity by finite volume methods requires the implementation of a specific shock detection procedure. A set of necessary conditions is defined for the discretisation of the source term in the sensitivity equation. An application to the one-dimensional kinematic wave equation shows that the proposed numerical technique allows analytical solutions to be reproduced correctly. The computational examples show that first-order numerical schemes do not yield satisfactory numerical solutions in the neighbourhood of moving shocks and that higher-order schemes, such as the MUSCL scheme, should be used for sharp transients.

Numerical prediction of vortex flow and thermal separation in a subsonic vortex tube

This work was aimed at gaining understanding of the physical behaviours of the flow and temperature separation process in a vortex tube. To investigate the cold mass fraction’s effect on the temperature separation, the numerical calculation was carried out using an algebraic Reynolds stress model (ASM) and the standard k-ɛ model. The modelling of turbulence of compressible, complex flows used in the simulation is discussed. Emphasis is given to the derivation of the ASM for 2D axisymmetrical flows, particularly to the model constants in the algebraic Reynolds stress equations. The TEFESS code, based on a staggered Finite Volume approach with the standard k-ɛ model and first-order numerical schemes, was used to carry out all the computations. The predicted results for strongly swirling turbulent compressible flow in a vortex tube suggested that the use of the ASM leads to better agreement between the numerical results and experimental data, while the k-ɛ model cannot capture the stabilizing effect of the swirl.

Why the MUSCL–Hancock Scheme is L1-stable

The finite volume methods are one of the most popular numerical procedure to approximate the weak solutions of hyperbolic systems of conservation laws. They are developed in the framework of first-order numerical schemes. Several approaches are proposed to increase the order of accuracy. The van Leer methods are interesting ways. One of them, namely the MUSCL–Hancock scheme, is full time and space second-order accuracy. In the present work, we exhibit relevant conditions to ensure the L1-stability of the method. A CFL like condition is established, and a suitable limitation procedure for the gradient reconstruction is developed in order to satisfy the stability criterion. In addition, we show that the conservative variables are not useful within the gradient reconstruction and the procedure is extended in the framework of the primitive variables. Numerical experiments are performed to show the interest and the robustness of the method.

Berücksichtigung der Reibung in der Wirbelmethode

A new method to account the viscosity within framework of Computational Vortex Method is suggested in this paper. First-order numerical scheme is obtained by means of the splitting of problem according to the physical processes. The numerical results are compared with exact solutions and other methods.

Long-run availability of a repairable parallel system

We analyze the long-run availability of a two-unit parallel system sustained by a cold standby unit and attended by two identical repairmen. The system satisfies the usual conditions (i.i.d. random variables, perfect repair, instantaneous and perfect switch, queueing). Each operative unit has a constant failure rate and a general repair time distribution. We use Hokstad's supplementary variable method to construct a system of simultaneous partial differential equations. We present a first-order numerical scheme and an iterative algorithm for the solution of the equations. A particular but important numerical example (Weibull repair), illustrated by computer-plotted graphs, motivates the proposed approximation.

Physical Interpretation of the Adjoint Functions for Sensitivity Analysis of Atmospheric Models

The adjoint functions for an atmospheric model are the solution to a system of equations derived from a differential form of the model's equations. The adjoint functions can be used to calculate efficiently the sensitivity of one of the model's results to variations in any of the model's parameters. This paper shows that the adjoint functions themselves can be interpreted, as the sensitivity of a result to instantaneous perturbations of the model's dependent variables. This interpretation is illustrated for a radiative convective model, although the interpretation holds equally well for general circulation models. The adjoint functions are used to reveal the three time scales associated with 1) convective adjustment, 2) heat transfer between the atmosphere and space and 3) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually solving for the adjoint functions. The sensitivities given by the adjoint functions are verified by comparison with sensitivities obtained directly from recalculations. Despite sharp changes in the adjoint functions arising from convective adjustment switching on and off during a diurnal cycle, a first-order numerical scheme to solve the adjoint equations gives agreement with direct recalculations to three significant figures. For a model with N time steps, this comparison has also shown that the adjoint method is at least N/2 times more efficient than recalculation. Such an efficient method of calculating the sensitivity of a simulated synoptic state to all previous synoptic states is valuable not only to identify the time scales of various physical processes, but also to assimilate data for initialization.