Excellent Conservation Properties Research Articles

A volume of fluid based method for consistent flux computation in large-density ratio two-phase flows and its application in investigating droplet bag breakup behavior

This article introduces a novel method for computing consistent fluxes, which enables highly robust simulations of two-phase flow problems characterized by large-density ratios. The approach is based on the geometric reconstruction volume of fluid method and utilizes a staggered grid implementation. This allows for accurate and robust simulation of phenomena like droplet bag breakup in flows with intense velocity shear and significant density differences. Through numerical experiments, it has been demonstrated that this method can reliably simulate two-phase flows with large-density ratios while preserving excellent energy conservation properties. Expanding on these findings, the researchers have developed a solver that leverages block-structured adaptive mesh to perform high-fidelity simulations of droplet bag breakup scenarios. Remarkably, this solver accurately reproduces three distinct breakup patterns: bag mode, stamen mode, and sheet-stripping mode. A comprehensive analysis has also been conducted by comparing the dimensionless maximum cross-stream radius with experimental test results. Furthermore, the study investigates the kinetic energy spectrum of fully developed two-phase turbulence under different droplet generation mechanisms and examines the distribution of droplet sizes. The numerical results validate the efficacy and reliability of this method in accurately simulating two-phase flows characterized by significant density disparities and interface momentum exchange.

The muphyII code: Multiphysics plasma simulation on large HPC systems

Collisionless astrophysical and space plasmas cover regions that typically display a separation of scales that exceeds any code's capabilities. To help address this problem, the muphyII code utilizes a hierarchy of models with different inherent scales, unified in an adaptive framework that allows stand-alone use of models as well as a model-based dynamic and adaptive domain decomposition. This requires ensuring excellent conservation properties, careful treatment of inner-domain model boundaries for interface coupling, and robust time-stepping algorithms, especially with the use of electron subcycling. This multi-physics approach is implemented in the muphyII code, tested on different scenarios of space plasma magnetic reconnection, and evaluated against space probe data and higher-fidelity simulation results from literature. Adaptive model refinement is highlighted in particular, and a hybrid model with kinetic ions, pressure-tensor fluid electrons, and Maxwell fields is appraised.

Simulations of idealised 3D atmospheric flows on terrestrial planets using LFRic-Atmosphere

Abstract. We demonstrate that LFRic-Atmosphere, a model built using the Met Office's GungHo dynamical core, is able to reproduce idealised large-scale atmospheric circulation patterns specified by several widely used benchmark recipes. This is motivated by the rapid rate of exoplanet discovery and the ever-growing need for numerical modelling and characterisation of their atmospheres. Here we present LFRic-Atmosphere's results for the idealised tests imitating circulation regimes commonly used in the exoplanet modelling community. The benchmarks include three analytic forcing cases: the standard Held–Suarez test, the Menou–Rauscher Earth-like test, and the Merlis–Schneider tidally locked Earth test. Qualitatively, LFRic-Atmosphere agrees well with other numerical models and shows excellent conservation properties in terms of total mass, angular momentum, and kinetic energy. We then use LFRic-Atmosphere with a more realistic representation of physical processes (radiation, subgrid-scale mixing, convection, clouds) by configuring it for the four TRAPPIST-1 Habitable Atmosphere Intercomparison (THAI) scenarios. This is the first application of LFRic-Atmosphere to a possible climate of a confirmed terrestrial exoplanet. LFRic-Atmosphere reproduces the THAI scenarios within the spread of the existing models across a range of key climatic variables. Our work shows that LFRic-Atmosphere performs well in the seven benchmark tests for terrestrial atmospheres, justifying its use in future exoplanet climate studies.

Influences of conservative and non-conservative Lorentz forces on energy conservation properties for incompressible magnetohydrodynamic flows

In the analysis of magnetohydrodynamic (MHD) flow, the Lorentz force significantly affects energy properties because the work generated by the Lorentz force changes the kinetic and magnetic energies. Therefore, the Lorentz force and energy conversion should be predicted accurately. Some energy conservation schemes have been proposed and validated. However, the influences of the Lorentz force discretization on conservation and conversion of energy have not yet been clarified. In this study, a conservative finite difference method is constructed for incompressible MHD flows considering the induced magnetic field. We compare the difference in energy conservation properties among three methods of calculating the Lorentz force. The Lorentz forces are calculated in conservative and non-conservative forms, and both compact and wide-range interpolations of magnetic flux density are used to calculate the non-conservative Lorentz force. The compact interpolation method proposed in this study can perform conversions between conservative and non-conservative forms of the Lorentz force even when using the finite difference method. The present numerical method improves the conservation of transport quantity. Five models were analyzed, and the accuracy and convergence of the present numerical method were verified. From the viewpoint of the conservation of the total energy in an ideal inviscid periodic MHD flow, we consider that the calculation using compact interpolation for the Lorentz force is appropriate. This method preserves the total energy even on non-uniform grids. Moreover, the divergence-free condition of the magnetic flux density is discretely satisfied even without the correction of the magnetic flux density. The present numerical method can capture the Hartmann layer in the propagation of an Alfvén wave and accurately predict the tendency of energy attenuation in the analysis of a Taylor decaying vortex under magnetic fields. Analysis of the Orszag–Tang vortex reveals energy dissipation processes and the generation of high current densities. The present numerical method has excellent energy conservation properties and can accurately predict energy conversion. Therefore, this method can contribute to understanding complex unsteady MHD flows.

A conservative fourth-order real space method for the (2+1)D Dirac equation

Modelling the time-dependent (2+1)D Dirac equation has recently gained importance since this equation effectively describes multiple condensed matter systems. To avoid the large dispersion errors of second-order real space schemes, a highly accurate method is presented here instead. The method utilises a fourth-order central difference on a staggered grid and an explicit symplectic Partitioned Runge–Kutta (PRK) time integrator. In contrast to traditional Runge–Kutta (RK) time stepping, no unphysical dissipation is introduced into the simulation. Moreover, it is demonstrated, both theoretically via Poisson maps and numerically, that the novel scheme has excellent conservation properties. Furthermore, the proposed numerical method is provably stable and the dispersion error is low and isotropic. Several interesting numerical examples are presented. Besides validating the advocated method, they also showcase its computational efficiency and low memory consumption.

EPIC: The Elliptical Parcel-In-Cell method

We present a novel approach to simulating general two-dimensional flows, which could also be applied to other areas of continuum mechanics. The approach generalises the Particle-In-Cell (PIC) method, originally used to model two-dimensional hydrodynamics, by representing fluid elements by elliptical parcels. The rotation and deformation of these parcels are calculated, and parcels split beyond a critical aspect ratio. Conversely, small parcels are eliminated by merging them with larger ones. The elliptical parcels well represent the flow deformation and have excellent conservation properties. In contrast to earlier work that combined PIC with elliptical parcels that split and merge, a vorticity-based framework is used, and accurate integration over ellipses is performed efficiently by two-point Gaussian quadrature. The small-scale mixing associated with parcel splitting and merging is shown to be strongly convergent with grid resolution. The robustness, versatility, accuracy and efficiency of the new Elliptical Parcel-In-Cell (EPIC) method is demonstrated for a variety of standard test cases, and compared with a standard pseudo-spectral method. The results indicate that EPIC is a promising, Lagrangian-based alternative to grid-based methods.

Implementation and verification of a conservative,multi‐species,gyro‐averaged, full‐f,Lenard‐Bernstein/Dougherty collision operator in the gyrokinetic codeGENE‐X

AbstractGyrokinetic simulations are among the main tools to predict turbulent transport in the core of fusion devices. Unlike the core, the plasma edge and scrape‐off layer are characterized by lower temperature and higher collisionality. To allow for realistic gyrokinetic modelling of edge and scrape‐off layer turbulence, it is crucial to include collisional effects into the simulations. In this work, we present a full‐f, gyro‐averaged, multi‐species, Lenard‐Bernstein/Dougherty (LBD) collision operator, for the use in the gyrokinetic turbulence codeGENE‐X. The operator accounts for exact particle density, momentum, and energy conservation, with collision frequencies chosen such that the momentum or temperature relaxation rates of the Boltzmann collision operator are recovered. We provide a conservative second‐order finite‐volume implementation of the operator and present a thorough verification. Due to the excellent conservation properties of the finite‐volume implementation, it is possible to use a coarser velocity space grid, save computational resources and, as a consequence, perform simulations of larger fusion devices.

A massless boundary component mode synthesis method for elastodynamic contact problems

We propose to combine the ideas of mass redistribution and component mode synthesis. More specifically, we employ the MacNeal method, which readily leads to a singular mass matrix, and an accordingly modified version of the Craig-Bampton method. Besides obtaining a massless boundary, we achieve a drastic reduction of the mathematical model order in this way compared to the parent finite element model. Contact is modeled using set-valued laws and time stepping is carried out with a semi-explit scheme. We assess the method’s computational performance by a series of benchmarks, including both frictionless and frictional contact. The results indicate that the proposed method achieves excellent energy conservation properties and superior convergence behavior. It reduces the spurious oscillations and decreases the computational effort by about 1–2 orders of magnitude compared to the current state of the art (mass-carrying component mode synthesis method). We believe that the computational performance and favorable energy conservation properties will be valuable for the prediction of vibro-impact processes and physical damping.

The Lagrangian hydrodynamics code magma2

ABSTRACTWe present the methodology and performance of the new Lagrangian hydrodynamics code magma2, a smoothed particle hydrodynamics (SPH) code that benefits from a number of non-standard enhancements. By default it uses high-order smoothing kernels and wherever gradients are needed, they are calculated via accurate matrix inversion techniques, but a more conventional formulation with kernel gradients has also been implemented for comparison purposes. We also explore a matrix inversion formulation of SPH with a symmetrization in the particle indices that is not frequently used. We find interesting advantages of this formulation in some of the tests, for example, a substantial reduction of surface tension effects for non-ideal particle setups and more accurate peak densities in Sedov blast waves. magma2 uses artificial viscosity, but enhanced by techniques that are commonly used in finite-volume schemes such as reconstruction and slope limiting. While simple to implement, this approach efficiently suppresses particle noise, but at the same time drastically reduces dissipation in locations where it is not needed and actually unwanted. We demonstrate the performance of the new code in a number of challenging benchmark tests including, for example, multidimensional vorticity creating Schulz–Rinne-type Riemann problems and more astrophysical tests such as a collision between two stars to demonstrate its robustness and excellent conservation properties.

The Rotating Rigid Body Model Based on a Non-twisting Frame

This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D’Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy–momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.

A Novel multidimensional Boltzmann neutrino transport scheme for core-collapse supernovae.

We introduce a new discrete-ordinate scheme for solving the general relativistic (GR) Boltzmann transport equation in the context of core-collapse supernovae (CCSNe). Our algorithm avoids the need to spell out the complicated advection terms in energy and angle that arise when the transport equation is formulated in spherical polar coordinates, in the comoving frame, or in a GR space-time. We instead approach the problem by calculating the advection of neutrinos across momentum space using an intuitive particle-like approach that has excellent conservation properties and fully accounts for Lorentz boosts, GR effects, and grid geometry terms. In order to avoid the need for a global implicit solution, time integration is performed using a locally implicit Lax-Wendroff scheme that correctly reproduces the diffusion limit. This will facilitate the use of our method on massively parallel distributed-memory architectures. We have verified the accuracy and stability of our scheme with a suite of test problems in spherical symmetry and axisymmetry. To demonstrate that the new algorithm works stably in CCSN simulations, we have coupled it to the GR hydrodynamics code coconut and present a first demonstration run of a [Formula: see text] progenitor with a reduced set of neutrino opacities.

A second order decoupled penalty projection method based on deferred correction for MHD in Elsässer variable

We study the deferred correction method for the magnetohydrodynamics (MHD) system written in Elsässer variables. The proposed algorithm is based on the penalty projection with grad-div stabilized Taylor Hood solutions of the Elsässer formulation. In this way, second order accuracy of the method in time is obtained through the deferred correction method with excellent mass conservation properties. For the proposed method, stability is rigorously proven and numerical experiments are presented to verify the proposed scheme and theory.

Multi-symplectic quasi-interpolation method for Hamiltonian partial differential equations

In this paper, we propose a multi-symplectic quasi-interpolation method for solving multi-symplectic Hamiltonian partial differential equations. Based on the method of lines, we first discretize the multi-symplectic PDEs using quasi-interpolation method and then employ appropriate time integrators to obtain the full-discrete system. The local conservation properties including multi-symplectic conservation laws, energy conservation laws and momentum conservation laws are discussed in detail. For illustration, we provide two concrete examples: the nonlinear wave equation and the nonlinear Schrödinger equation. The salient feature of our multi-symplectic quasi-interpolation method is that it is valid both on uniform grids and nonuniform grids. The numerical results show the good accuracy and excellent conservation properties of the proposed method.

Numerical Simulation of Conservation Laws with Moving Grid Nodes: Application to Tsunami Wave Modelling

In the present article, we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension combined with an appropriate predictor–corrector method to achieve higher resolutions. The underlying finite volume scheme is conservative, and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with large solution gradients or any other special features. No interpolation procedure is employed; thus, unnecessary solution smearing is avoided, and therefore, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according to the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. The exact well-balanced property is proven. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov–Poisson system

The Vlasov–Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approximated by a semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along a high-order accurate representation of the local characteristic lines. The time-advancing technique is based on truncated Taylor series that can be, in principle, of any order of accuracy. A variant is obtained by coupling the phase space discretization with high-order accurate Backward Differentiation Formulas (BDF). At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine the excellent features of spectral approximations with high-order time integration. The resulting method has excellent conservation properties. Indeed, it can be proven that the total number of particles, proportional to the total mass and charge, is conserved up to the machine precision. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.

Symplectic integration of PDEs using Clebsch variables

Many PDEs (Burgers' equation, KdV, Camassa-Holm, Euler's fluid equations,...) can be formulated as infinite-dimensional Lie-Poisson systems. These are Hamiltonian systems on manifolds equipped with Poisson brackets. The Poisson structure is connected to conservation properties and other geometric features of solutions to the PDE and, therefore, of great interest for numerical integration. For the example of Burgers' equations and related PDEs we use Clebsch variables to lift the original system to a collective Hamiltonian system on a symplectic manifold whose structure is related to the original Lie-Poisson structure. On the collective Hamiltonian system a symplectic integrator can be applied. Our numerical examples show excellent conservation properties and indicate that the disadvantage of an increased phase-space dimension can be outweighed by the advantage of symplectic integration.

Segmentation improvement through denoising of PET images with 3D-context modelling in wavelet domain

Positron emission tomography (PET) images have been incorporated into the radiotherapy process as a powerful tool to assist in the contouring of lesions, leading to the emergence of a broad spectrum of automatic segmentation schemes for PET images (PET-AS). However, not all proposed PET-AS algorithms take into consideration the previous steps of image preparation. PET image noise has been shown to be one of the most relevant affecting factors in segmentation tasks. This study demonstrates a nonlinear filtering method based on spatially adaptive wavelet shrinkage using three-dimensional context modelling that considers the correlation of each voxel with its neighbours. Using this noise reduction method, excellent edge conservation properties are obtained. To evaluate the influence in the segmentation schemes of this filter, it was compared with a set of Gaussian filters (the most conventional) and with two previously optimised edge-preserving filters. Five segmentation schemes were used (most commonly implemented in commercial software): fixed thresholding, adaptive thresholding, watershed, adaptive region growing and affinity propagation clustering. Segmentation results were evaluated using the Dice similarity coefficient and classification error. A simple metric was also included to improve the characterisation of the filters used for induced blurring evaluation, based on the measurement of the average edge width. The proposed noise reduction procedure improves the results of segmentation throughout the performed settings and was shown to be more stable in low-contrast and high-noise conditions. Thus, the capacity of the segmentation method is reinforced by the denoising plan used.

Variational integrators for inertial magnetohydrodynamics

Recently, an extended version of magnetohydrodynamics that incorporates electron inertia, dubbed inertial magnetohydrodynamics, has been proposed [Lingam et al., Phys. Lett. A, 379, 570–576 (2015)]. This model features a noncanonical Hamiltonian formulation with a number of conserved quantities, including the total energy and modified versions of magnetic and cross helicity. In this work, a variational integrator is presented which preserves these conservation laws to machine accuracy. As long as effects due to finite electron mass are neglected, the scheme preserves the magnetic field line topology so that unphysical reconnection is absent. Only when the effects of finite electron mass are added, magnetic reconnection takes place. The excellent conservation properties of the method are illustrated by numerical examples in 2D.

One-layer particle level set method

One-layer particle level set (OPLS) has been developed by using Lagrangian particles that are employed to correct both advection and re-initialization procedures of the level set function. In which, a level set function is utilized to smooth physical properties of the interface, while one-layer Lagrangian particles are used to track the interface directly. This method exhibits excellent mass conservation properties compared to the LS method. As a special aspect, the OPLS method enables management of merging and stretching of interface in an effective way. This capability is similar to the particle level set (PLS) method. However, the new approach of the OPLS method offers a more straightforward technique. This approach is validated with classical benchmark test cases, such as the long term advection of a circle, rotation of a slotted disk, single vertex in a box, merging and separating of circle. The results from the proposed method show good agreement with the numerical experiments published results and the OPLS method is verified to be highly reliable and accurate.

Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.