Time Conservation Element Research Articles

CESE Schemes for Solar Wind Plasma MHD Dynamics

Magnetohydrodynamic (MHD) numerical simulation has emerged as a pivotal tool in space physics research, witnessing significant advancements. This methodology offers invaluable insights into diverse space physical phenomena based on solving the fundamental MHD equations. Various numerical methods are utilized to approximate the MHD equations. Among these, the space–time conservation element and solution element (CESE) method stands out as an effective computational approach. Unlike traditional numerical schemes, the CESE method significantly enhances accuracy, even at the same base point. The concurrent discretization of space and time for conserved variables inherently achieves higher-order accuracy in both dimensions, without the need for intricate higher-order time discretization processes, which are often challenging in other methods. Additionally, this scheme can be readily extended to multidimensional cases, without relying on operator splitting or direction alternation. This paper primarily delves into the remarkable progress of CESE MHD models and their applications in studying solar wind, solar eruption activities, and the Earth’s magnetosphere. We aim to illuminate potential avenues for future solar–interplanetary CESE MHD models and their applications. Furthermore, we hope that the discussions presented in this review will spark new research endeavors in this dynamic field.

Numerical modeling of laboratory-scale asteroid impact based on elastoplastic flow model and CESE method

Asteroid impacts are destructive and low-probability threats to the Earth. The numerical simulation is considered an applicable analysis tool in asteroid deflection programs. As a novel shock-capturing strategy, the space–time conservation element and solution element (CESE) method can reliably predict shock waves and mechanical behaviors under high pressure and large strain conditions. In this paper, based on an elastoplastic flow model and an updated CESE scheme, the laboratory-scale iron asteroid impacts are modeled numerically, and the multi-material boundary treatment and the interface tracing strategy are introduced. Under hypervelocity impacts of the projectile to the iron asteroid target, the construction and realization of morphologies of impact craters and the implantation of projectile material into the target are numerically calculated. Numerical results show that the crater diameter and depth increase with increasing impact velocity and with increasing temperature, which softens the target. Computational results are compared with experimental observations available in the open literature, and good agreement is found. Therefore, the CESE method is successfully extended for capturing the key features of laboratory-scale hypervelocity asteroid impacts.

Numerical analysis of heat flow in wall of detonation tube during pulse detonation cycle

The thermal ablation of the detonation tube is a key factor in pulse detonation engine (PDE) research. In this study, a symmetric two-phase detonation model and a cylinder heat conduction model are used for studying the features of transient heat transfer. The effects of the tube material and size on the heat transfer are analyzed using this model. To capture the shock wave simply and accurately, the space–time conservation element and solution element method and the finite-difference method are used for simulating the interior flow and the heat-transfer process of the detonation tube, respectively. The numerical results are consistent with the experimental results. The temperature of the PDE wall increases by 1.5 K per four detonation cycles at a frequency of 20 Hz. Then, the effects of the materials of the tube wall and the diameter and length of the tube are investigated. The inner flow greatly affects the inner-wall temperature, particularly the influence is obvious when the tube materials change. Increasing the diameter can improve the heat dissipation, and reducing the length smooths the temperature gradient along the axis. The results of this study provide guidance for the design of PDEs.

Space–Time Conservation Element and Solution Element Method and Its Applications

This paper reviews the development of the space–time conservation element and solution element (CESE) method and summarizes its applications in various research areas. The CESE method is a special finite-volume-type method that provides an alternative approach to numerical solutions of fluid-dynamic equations and conservation laws in various physical systems. Based on a unified treatment of time and space, this method solves the integral form of the governing equations by discretization of the space–time domain. Recent progress in CESE schemes mainly includes the construction of a family of upwind CESE schemes, the extended definitions of conservation elements and solution elements for arbitrary meshes, and a new approach to developing high-order CESE schemes. Selected applications of the CESE method (including high-speed aerodynamics, multifluid flows, detonations, and aeroacoustics) are presented. Features of the CESE method are described.

A space time conservation element and solution element method for solving two-species chemotaxis model

This work is related to the numerical investigation of two species chemotaxis models in both one and two dimensions. This system in its simpler form is a set of non-linear partial differential equations. First equation represents the dynamics of cell densities and the others are responsible for chemoattractant concentration. The system is known to produce delta-type singularities in finite time. Therefore, a less order accurate numerical schemes are not capable of handling such a complicated spiky solution. A Space Time Conservation Element and Solution Element (CE/SE) method is proposed for solving such systems. This method has distinct attributes which includes treatment of space and time variables in unified fashion. Furthermore, numerical diffusion is reduced on the basis that both conserved quantities and derivatives are anonymous. Due to this feature, diffusion in numerical schemes is inherently reduced. The Nessyahu–Tadmor (NT) central scheme is also implemented for validation and comparison. Several one and two dimensional case studies are carried out. Numerical results obtained through proposed numerical method are analyzed in comparison to Nessyahu–Tadmor (NT) central scheme. Moreover, effects of the different densities and concentration functions are explored to see generic applicability of the proposed method for current system of equations. It is observed that the CE/SE and NT central method has the capability to capture the delta type singularities in the solution profile. However, CE/SE resolves the solution peaks better as compare to NT central scheme.

Interfacial instability at a heavy/light interface induced by rarefaction waves

The interaction of rarefaction waves and a heavy/light interface is investigated using numerical simulations by solving the compressible Euler equations. An upwind space–time conservation element and solution element (CE/SE) scheme with second-order accuracy in both space and time is adopted. Rarefaction waves are generated by simulating the shock-tube problem. In this work, the SF6/air interface evolution under different conditions is considered. First, the gas physical parameters before and after the rarefaction waves impact the interface are calculated using one-dimensional gas dynamics theory. Then, the interaction between the rarefaction waves and a single-mode perturbation interface is investigated, and both the interface evolution and the wave patterns are obtained. Afterwards, the amplitude growth of the interface over time is compared between cases, considering the effects of the interaction period and the strength of the rarefaction waves. During the interaction of the rarefaction waves with the interface, the Rayleigh–Taylor instability induced by the rarefaction waves is well predicted by modifying the nonlinear model proposed by Zhang & Guo (J. Fluid Mech., vol. 786, 2016, pp. 47–61), considering the variable acceleration. After the rarefaction waves leave the interface, the equivalent Richtmyer–Meshkov instability is well depicted by the nonlinear model proposed by Zhang et al. (Phys. Rev. Lett., vol. 121(17), 2018, 174502), considering the growth rate transition from Rayleigh–Taylor instability to Richtmyer–Meshkov instability. The differences in the heavy/light interface amplitude growth under the rarefaction wave condition and the shock wave condition are compared. The interface perturbation is shown to be more unstable under rarefaction waves than under a shock wave.

Positivity-preserving CE/SE schemes for solving the compressible Euler and Navier–Stokes equations on hybrid unstructured meshes

We construct positivity-preserving space–time conservation element and solution element (CE/SE) schemes for solving the compressible Euler and Navier–Stokes equations on hybrid unstructured meshes consisting of triangular and rectangular elements. The schemes use an a posteriori limiter to prevent negative densities and pressures based on the premise of preserving optimal accuracy. The limiter enforces a constraint for spatial derivatives and does not change the conservative property of CE/SE schemes. Several numerical examples suggest that the proposed schemes preserve accuracy for smooth flows and strictly preserve positivity of densities and pressures for the problems involving near vacuum and very strong discontinuities.

The space–time CESE scheme for shallow water equations incorporating variable bottom topography and horizontal temperature gradients

The effects of bottom topography and horizontal temperature gradients on the shallow water flows are theoretically investigated. The considered systems of partial differential equations (PDEs) are non-strictly hyperbolic and non-conservative due to the presence of non-conservative differential terms on the right hand side. The solutions of these model equations are very challenging for a numerical scheme. Thus, our primary goal is to introduce an improved numerical scheme which can handle the non-conservative differential terms efficiently and accurately. In this paper, the space–time conservation element and solution element (CESE) method is extended to approximate these model equations. The proposed scheme has capability to overcome all difficulties posed by this nonlinear system of PDEs. The performance of the scheme is analyzed by considering several case studies of practical interest and the results of suggested scheme are compared with those of central NT scheme. The accuracy of the scheme is verified qualitatively and quantitatively.

A high-order CESE scheme with a new divergence-free method for MHD numerical simulation

In this paper, we give a high-order space–time conservation element and solution element (CESE) method with a most compact stencil for magneto-hydrodynamics (MHD) equations. This is the first study to extend the second-order CESE scheme to a high order for MHD equations. In the CESE method, the conservative variables and their spatial derivatives are regarded as the independent marching quantities, making the CESE method significantly different from the finite difference method (FDM) and finite volume method (FVM). To utilize the characteristics of the CESE method to the maximum extent possible, our proposed method based on the least-squares method fundamentally keeps the magnetic field divergence-free. The results of some test examples indicate that this new method is very efficient.

A space–time CESE scheme for shallow water magnetohydrodynamics equations with variable bottom topography

The dynamics of thin layer incompressible and electrically conducting fluids is described by shallow water magnetohydrodynamics equations (SWMHD), for which the evolution is nearly two dimensional and the magnetic equilibrium lies in the third direction. In this article, a space–time conservation element and solution element (CESE) scheme is extended for numerical investigation of one- and two-dimension shallow water magnetohydrodynamics (SWMHD) with variable bottom topography. The presence of non-conservative terms and the inclusion of variable bottom topography on the right hand side of SWMHD equations poses some major challenges for any numerical scheme. The proposed CESE scheme differs from the previous techniques because of global and local flux conservation in a space–time domain without restoring to interpolation and extrapolation. To check the validity of the scheme, a number of test problems are considered. The results are also compared with the already existing results in the literature by central-upwind scheme.

Three-dimensional detonation cellular structures in rectangular ducts using an improved CESE scheme**Project supported by the National Natural Science Foundation of China (Grant Nos. 10732010 and 10972010).

The three-dimensional premixed H2-O2 detonation propagation in rectangular ducts is simulated using an in-house parallel detonation code based on the second-order space–time conservation element and solution element (CE/SE) scheme. The simulation reproduces three typical cellular structures by setting appropriate cross-sectional size and initial perturbation in square tubes. As the cross-sectional size decreases, critical cellular structures transforming the rectangular or diagonal mode into the spinning mode are obtained and discussed in the perspective of phase variation as well as decreasing of triple point lines. Furthermore, multiple cellular structures are observed through examples with typical aspect ratios. Utilizing the visualization of detailed three-dimensional structures, their formation mechanism is further analyzed.

Numerical simulation of deflagration to detonation transition in granular HMX explosives under thermal ignition

To study the deflagration to detonation transition process in granular explosive under thermal ignition condition, the conductive burning is introduced into the classical model. The novel space–time conservation element and solution element method is applied to solve the mathematical governing equations. The transition process in granular HMX bed which is packed to 85 % of theoretical maximum density is simulated. The development of conductive burning, convective burning and detonation is analyzed. In the early stage, the combustion propagates slowly. It moves no more than 0.2 mm within 8.16 ms. After onset of convective burning, it takes only 20 μs to form a steady detonation with the velocity of 8165 m s−1. The time to detonation increases with the decrease in particle diameter.

Hyperbolicity of pressure–velocity equations for computational hydro acoustics

A set of model equations is proposed to simulate waves generated by unsteady, low-speed, nearly incompressible air and water flows. The equations include the continuity and momentum equations with pressure and velocity as the unknowns. Compressibility effect associated with waves motion is directly tracked by time-accurate calculation of pressure fluctuations. The corresponding density changes are modeled by using the bulk modulus of the medium. The three-dimensional equations are shown to be hyperbolic by analyzing eigenvalues and eigenvectors of the composite Jacobian matrix of the equations. Specifically, the matrix is shown to be diagonalizable and have a real spectrum. Moreover, an analytical form of the Riemann invariants of the one-dimensional equations is derived. To validate the model equations, the space–time Conservation Element and Solution Element (CESE) method and the SOLVCON code are employed to solve the two-dimensional equations. Aeolian tones generated by air and water flows passing a cylinder and over an open cavity are simulated. Numerical results compare well with previously reported data.

A characteristic space–time conservation element and solution element method for conservation laws II. Multidimensional extension

The characteristic space-time conservation element and solution element (CE/SE) schemes proposed by Shen et al. (2015) 15 are straightforward extended to multidimensional schemes on 2D rectangular meshes which strictly follow the space-time conservation law. The schemes for solving both scalar conservation laws and compressible Euler equations with shock waves are developed. They are accurate and robust with CFL widely ranging from 0 to nearly 1. In Euler solvers, the frequently-used Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and Roe Riemann solver are employed to calculate the upwind fluxes as examples. When standard grid-aligned Riemann solvers are employed, the carbuncle phenomena are significantly suppressed when comparing with conventional upwind schemes. If rotated Riemann solvers are employed, nearly carbuncle-free results are obtained. Several well understood numerical examples are carried out to demonstrate that the 2D characteristic CE/SE schemes can simultaneously capture shocks and details of complex flow structures very well.

Combination of space–time conservation element/solution element method and continuous prediction technique for accelerated simulation of simulated moving bed chromatography

The model equation of simulated moving bed (SMB) process is solved by a space–time conservation element and solution element (CE/SE) method, which is found to be more efficient than other methods in the two case studies. Furthermore, this CE/SE method was combined with continuous prediction technique (CPM) to further accelerate determination of the cyclic steady state in SMB operation. Comparing with conventional method of lines at the same required precisions, CE/SE with CPM was proved to save the CPU time by 77% for the pseudo-linear system of glucose–fructose separation and 96% for nonlinear enantioseparation. Finally, the combination of CE/SE and CPM was applied to the optimization study to determine the optimum switching time and flow rate. To ensure the method be convergent, the four Courant numbers in four zones need to fall between 0.1 and 1. The method for fulfill this constraint is also developed.

A characteristic space–time conservation element and solution element method for conservation laws

In this paper, an upwind space–time conservation element and solution element (CE/SE) method is developed to solve conservation laws. In the present method, the mesh quantity and spatial derivatives are the independent marching variables, which is consistent with the original CE/SE method proposed by Chang (1995) [5]. The staggered time marching strategy and the definition of conservation element (CE) also follow Chang's propositions. Nevertheless, the definition of solution element (SE) is modified from that of Chang. The numerical flux through the interface of two different conservation elements is not directly derived by a Taylor expansion in the reversed time direction as proposed by Chang, but determined by an upwind procedure. This modification does not change the local and global conservative features of the original method. Although, the time marching scheme of mesh variables is the same with the original method, the upwind fluxes are involved in the calculation of spatial derivatives, yielding a totally different approach from that of Chang's method. The upwind procedure breaks the space–time inversion invariance of the original scheme, so that the new scheme can be directly applied to capture discontinuities without spurious oscillations. In addition, the present method maintains low dissipation in a wide range of CFL number (from 10−6 to 1). Furthermore, we extend the upwind CE/SE method to solve the Euler equations by adopting three different approximate Riemann solvers including Harten, Lax and van Leer (HLL) Riemann solver, contact discontinuity restoring HLLC Riemann solver and mathematically rigorous Roe Riemann solver. Extensive numerical examples are carried out to demonstrate the robustness of the present method. The numerical results show that the new CE/SE solvers perform improved resolutions.

The space time CE/SE method for solving one-dimensional batch crystallization model with fines dissolution

This article is concerned with the numerical investigation of one-dimensional population balance models for batch crystallization process with fines dissolution. In batch crystallization, dissolution of smaller unwanted nuclei below some critical size is of vital importance as it improves the quality of product. The crystal growth rates for both size-independent and size-dependent cases are considered. A delay in recycle pipe is also included in the model. The space–time conservation element and solution element method, originally derived for non-reacting flows, is used to solve the model. This scheme has already been applied to a range of PDEs, mainly in the area of fluid mechanics. The numerical results are compared with those obtained from the Koren scheme, showing that the proposed scheme is more efficient.

Robust high-order space–time conservative schemes for solving conservation laws on hybrid meshes

In this paper, the second-order space–time conservation element and solution element (CE/SE) method proposed by Chang (1995) [3] is implemented on hybrid meshes for solving conservation laws. In addition, the present scheme has been extended to high-order versions including third and fourth order. Most methodologies of proposed schemes are consistent with that of the original CE/SE method, including: (i) a unified treatment of space and time (thereby ensuring good conservation in both space and time); (ii) a highly compact node stencil (the solution node is calculated using only the neighboring mesh nodes) regardless of the order of accuracy at the cost of storing all derivatives. A staggered time marching strategy is adopted and the solutions are updated alternatively between cell centers and vertexes. To construct explicit high-order schemes, second- and third-order derivatives are calculated by a modified finite-difference/weighted-average procedure which is different from that used to calculate the first-order derivatives. The present schemes can be implemented on a wide variety of meshes, including triangular, quadrilateral and hybrid (consisting of both triangular and quadrilateral elements). Beyond that, it can be easily extended to arbitrary-order schemes and arbitrary shape of polygonal elements by using the present methodologies. A series of common benchmark examples are used to confirm the accuracy and robustness of the proposed schemes.

A front tracking algorithm for hypervelocity impact problems with crack growth, large deformations and high strain rates

In the present paper, a front tracking algorithm, which includes a front tracking part and a specified crack growth scheme, is proposed for tracking material interfaces and describing the formation and propagation of a crack. Combined with an improved space–time Conservation Element and Solution Element (CE/SE) scheme, the algorithm can simulate high-velocity impact problems with crack growth, large deformations and high strain rates. The present front tracking algorithm shows high accuracy in the numerical test of a single vortex problem. Numerical simulations are also presented for spall fractures in a plate when impacted by a spherical projectile and perforation of a cylindrical Arne tool steel projectile impacting a plate target. The numerical results are in good agreement with the corresponding experimental observations. It is demonstrated that the present algorithm is feasible and reliable for analyzing fractures.

Study on the Mechanism of the Deflagration to Detonation Transition Process of Explosive

We present a numerical study of the mechanisms of the deflagration to detonation transition (DDT) process of explosives to assess its thermal stability. We treated the modeling system as a mixture of solid explosives and gaseous reaction products. We utilized a one-dimensional two-phase flow modeling approach with a space–time conservation element and solution element (CE/SE) method. Simulation results show that in the chemical reaction process a plug area of high density with relatively slow chemical reactions preceeds the new violent reactions and the consequent detonation. We found that steady detonation occurs at the regions where physical characteristics, such as pressure, density, temperature, and velocity, peak simultaneously. These simulation results agree well with high-temperature DDT tube experiments.