Exact Conservation Property Research Articles

A Consistent and Well‐Balanced Hybrid Weighted Essentially Non‐Oscillatory Scheme for Shallow Water Equations on Unstructured Meshes

ABSTRACTIn this article, a type of high‐order consistent and well‐balanced hybrid weighted essentially non‐oscillatory (WENO) scheme is proposed for shallow water equations with flat or non‐flat bottom on unstructured triangular meshes. The hybrid scheme presents a new consistent discretization format on the flux and the source term with the goal of obtaining a hybridization of the high‐order WENO scheme and linear scheme. According to the modified multi‐resolution analysis approach, we can select the more robust and accurate WENO reconstruction in the vicinity of discontinuities, and the less expensive linear reconstruction in the smooth regions. As a result, the suggested hybrid WENO scheme can acquire the capacity of saving the computing time cost while maintaining the excellent numerical features of the original WENO scheme. Eventually, some extensive and classical two‐dimensional numerical examples, including a tidal bore of an estuary with an irregular computation area are provided to validate the performance of this hybrid WENO scheme on triangular meshes in terms of accuracy order, exact conservation property, shock‐capturing, good resolution, and computational efficiency.

A well-balanced finite volume solver for the 2D shallow water magnetohydrodynamic equations with topography

In this paper, a second-order finite volume Non-Homogeneous Riemann Solver is used to obtain an approximate solution for the two-dimensional shallow water magnetohydrodynamic (SWMHD) equations considering non-flat bottom topography. We investigate the stability of a perturbed steady state, as well as the stability of energy in these equations after a perturbation of a steady state using a dispersive analysis. To address the elliptic constraint ∇⋅hB=0, the GLM (Generalized Lagrange Multiplier) method designed specifically for finite volume schemes, is used. The proposed solver is implemented on unstructured meshes and verifies the exact conservation property. Several numerical results are presented to validate the high accuracy of our schemes, the well-balanced, and the ability to resolve smooth and discontinuous solutions. The developed finite volume Non-Homogeneous Riemann Solver and the GLM method offer a reliable approach for solving the SWMHD equations, preserving numerical and physical equilibrium, and ensuring stability in the presence of perturbations.

A structure-preserving finite element method for the multi-phase Mullins–Sekerka problem with triple junctions

We consider a sharp interface formulation for the multi-phase Mullins–Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins–Sekerka flow, demonstrate the capabilities of the introduced method.

Eulerian discrete kinetic framework in comoving reference frame for hypersonic flows

Flow physics vary in different regimes across the full Mach number range, with our knowledge being particularly poor about the hypersonic regime. An Eulerian realization of the particles on demand method, a kinetic model formulated in the comoving reference frame, is proposed to simulate hypersonic compressible flows. The present model allows for flux evaluation in different reference frames, in this case rescaled and shifted by local macroscopic quantities, i.e. fluid speed and temperature. The resulting system of coupled hyperbolic equations is discretized in physical space with a finite volume scheme ensuring exact conservation properties. Regularization via Grad expansion is introduced to implement distribution function and flux transformation between different reference frames. It is shown that the proposed method possesses Galilean invariance at a Mach number up to $100$ . Different benchmarks including both inviscid and viscous flows are reproduced with the Mach number up to $198$ and pressure ratio up to $10^5$ . Finally, the new model is demonstrated to be capable of simulating hypersonic reactive flows, including one-dimensional and two-dimensional detonations. The developed methodology opens up possibilities for the simulation of the full range of compressible flows, without or with chemical reactions, from the subsonic to hypersonic regimes, leading to enhanced understanding of flow behaviours across the full Mach number range.

A pseudospectral implicit particle-in-cell method with exact energy and charge conservation

The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) schemes, such as the leap-frog scheme, for spatial and temporal integrations. In this study, we introduce a novel approach by employing a pseudospectral method for solving the Poisson equation and adopting a fully implicit time integration technique to achieve exact energy conservation. The pseudospectral field solver offers the advantage of spectral accuracy in obtaining field solutions. Previous investigations on implicit time integration of PIC FD equations have demonstrated the capability of enforcing precise energy exchange between the field and particles, resulting in energy-conserving schemes. We prove that the exact energy conservation property can be extended to the pseudospectral scheme. Additionally, we provide a solution to ensure a pseudospectral charge continuity equation. To validate our methodology, we implement the new scheme in a 2D electrostatic PIC code and present numerical examples that confirm the theoretical findings.

Explicit exactly energy-conserving methods for Hamiltonian systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.

Adaptive Wavelet Methods for Earth Systems Modelling

This paper reviews how dynamically adaptive wavelet methods can be designed to simulate atmosphere and ocean dynamics in both flat and spherical geometries. We highlight the special features that these models must have in order to be valid for climate modelling applications. These include exact mass conservation and various mimetic properties that ensure the solutions remain physically realistic, even in the under-resolved conditions typical of climate models. Particular attention is paid to the implementation of complex topography in adaptive models. Using wavetrisk as an example, we explain in detail how to build a semi-realistic global atmosphere or ocean model of interest to the geophysical community. We end with a discussion of the challenges that remain to developing a realistic dynamically adaptive atmosphere or ocean climate models. These include scale-aware subgrid scale parameterizations of physical processes, such as clouds. Although we focus on adaptive wavelet methods, many of the topics we discuss are relevant for adaptive mesh refinement (AMR).

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

A numerical wave tank equipped with a piston type wave-maker is presented for long-duration simulations of long waves in shallow water. Both wave maker and tank are modelled using the nonlinear shallow water equations, with motions of the numerical piston paddle accomplished via a linear mapping technique. Three approaches are used to increase computational efficiency and accuracy. First, the model satisfies the exact conservation property (C-property), a stepping stone towards properly balancing each term in the governing equation. Second, a high-order weighted essentially non-oscillatory (WENO) method is used to reduce accumulation of truncation error. Third, a cut-off algorithm is implemented to handle contaminated digits arising from round-off error. If not treated, such errors could prevent a numerical scheme from satisfying the exact C-property in long-duration simulations. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and shock-capturing ability of the present scheme. Correct implementation of the wave paddle generator is verified by comparing numerical predictions against analytical solutions of sinusoidal, solitary, and cnoidal waves. In all cases, the model gives satisfactory results for small-amplitude, low frequency waves. Error analysis is used to investigate model limitations and derive a user criterion for long wave generation by the model.

A fully implicit, scalable, conservative nonlinear relativistic Fokker–Planck 0D-2P solver for runaway electrons

Upon application of a sufficiently strong electric field, electrons break away from thermal equilibrium and approach relativistic speeds. These highly energetic ‘runaway’ electrons (∼ MeV) play a significant role in tokamak disruption physics, and therefore their accurate understanding is essential to develop reliable mitigation strategies. For this purpose, we have developed a fully implicit solver for the 0D-2P (i.e., including two momenta coordinates) relativistic nonlinear Fokker–Planck equation (rFP). As in earlier implicit rFP studies (NORSE, CQL3D), electron–ion interactions are modeled using the Lorentz operator, and synchrotron damping using the Abraham–Lorentz–Dirac reaction term. However, our implementation improves on these earlier studies by (1) ensuring exact conservation properties for electron collisions, (2) strictly preserving positivity, and (3) being scalable algorithmically and in parallel. Key to our proposed approach is an efficient multigrid preconditioner for the linearized rFP equation, a multigrid elliptic solver for the Braams–Karney potentials (Braams and Karney, 1987), and a novel adaptive technique to determine the associated boundary values. We verify the accuracy and efficiency of the proposed scheme with numerical results ranging from small electric-field electrical conductivity measurements to the accurate reproduction of runaway tail dynamics when strong electric fields are applied.

A mass conserving mixed stress formulation for the Stokes equations

AbstractWe propose stress formulation of the Stokes equations. The velocity $u$ is approximated with $H(\operatorname{div})$-conforming finite elements providing exact mass conservation. While many standard methods use $H^1$-conforming spaces for the discrete velocity $H(\operatorname{div})$-conformity fits the considered variational formulation in this work. A new stress-like variable $\sigma $ equalling the gradient of the velocity is set within a new function space $H(\operatorname{curl} \operatorname{div})$. New matrix-valued finite elements having continuous ‘normal-tangential’ components are constructed to approximate functions in $H(\operatorname{curl} \operatorname{div})$. An error analysis concludes with optimal rates of convergence for errors in $u$ (measured in a discrete $H^1$-norm), errors in $\sigma $ (measured in $L^2$) and the pressure $p$ (also measured in $L^2$). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.

Well-balanced hybrid compact-WENO scheme for shallow water equations

We investigate the performance of the high order well-balanced hybrid compact-weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for simulations of shallow water equations with source terms due to a non-flat bottom topography. The Hybrid scheme employs the nonlinear fifth order characteristic-wise WENO-Z finite difference scheme to capture high gradients and discontinuities in an essentially non-oscillatory manner, and the linear spectral-like sixth order compact finite difference scheme to resolve the fine scale structures in the smooth regions of the solution efficiently and accurately. The high order multi-resolution analysis is employed to identify the smoothness of the solution at each grid point. In this study, classical one- and two-dimensional simulations, including a long time two-dimensional dam-breaking problem with a non-flat bottom topography, are conducted to demonstrate the performance of the hybrid scheme in terms of the exact conservation property (C-property), good resolution and essentially non-oscillatory shock capturing of the smooth and discontinuous solutions respectively, and up to 2–3 times speedup factor over the well-balanced WENO-Z scheme.

A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured triangular grids. The essential ingredients to achieve this are: (i) a velocity–vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular triangular grids.

Mass-corrections for the conservative coupling of flow and transport on collocated meshes

Buoyancy-driven flow models demand a careful treatment of the mass-balance equation to avoid spurious source and sink terms in the non-linear coupling between flow and transport. In the context of finite-elements, it is therefore commonly proposed to employ sufficiently rich pressure spaces, containing piecewise constant shape functions to obtain local or even strong mass-conservation. In three-dimensional computations, this usually requires nonconforming approaches, special meshes or higher order velocities, which make these schemes prohibitively expensive for some applications and complicate the implementation into legacy code. In this paper, we therefore propose a lean and conservatively coupled scheme based on standard stabilized linear equal-order finite elements for the Stokes part and vertex-centered finite volumes for the energy equation. We show that in a weak mass-balance it is possible to recover exact conservation properties by a local flux-correction which can be computed efficiently on the control volume boundaries of the transport mesh. We discuss implementation aspects and demonstrate the effectiveness of the flux-correction by different two- and three-dimensional examples which are motivated by geophysical applications.

A well-balanced finite difference WENO scheme for shallow water flow model

In this paper, we are concerned with shallow water flow model over non-flat bottom topography by high-order schemes. Most of the numerical schemes in the literature are developed from the original mathematical model of the shallow water flow. The novel contribution of this study consists in designing a finite difference weighted essentially non-oscillatory (WENO) scheme based on the alternative formulation of the shallow water flow model, denoted as “pre-balanced’’ shallow water equations and introduced in Rogers et al. (2003) [23]. This formulation greatly simplifies the achievement of the well-balancing of the present scheme. Rigorous numerical analysis as well as extensive numerical results all verify that the current scheme preserves the exact conservation property. It is important to note that this resulting scheme also maintains the non-oscillatory property near discontinuities and keeps high-order accuracy for smooth solutions at the same time.

High-order well-balanced central WENO scheme for pre-balanced shallow water equations

In this paper, we are concerned with numerically solving shallow water equations with a source term. The equations admit steady state solutions in which the non-zero flux gradient is exactly balanced by the source term. It is difficult for numerical schemes to preserve the delicate balance between the flux gradient and the source term.By virtue of a modified governing equations as well as an original reconstruction procedure for the Runge–Kutta flux, we present a central weighted essentially non-oscillatory scheme. Extensive test cases are carried out to demonstrate the performance of the resulting scheme: exact conservation property, high-order accuracy and good resolution near discontinuities.

High order geometric methods with exact conservation properties

Conservation laws in, for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. In this paper, we develop high order interpolants, from any basis that constitutes a partition of unity, which satisfy these integral relations exactly. The resulting gradient, curl and divergence conforming spaces have the property that the conservation laws become completely independent of the basis functions. Hence, they are exactly satisfied at the coarsest level of discretization and on arbitrarily curved meshes. As an illustration we apply our approach to B-splines and compute a 2D Stokes flow inside a lid driven cavity, which displays, amongst others, a point-wise divergence-free velocity field.

An analytical particle mover for the charge- and energy-conserving, nonlinearly implicit, electrostatic particle-in-cell algorithm

We propose a 1D analytical particle mover for the recent charge- and energy-conserving electrostatic particle-in-cell (PIC) algorithm in Ref. [G. Chen, L. Chacón, D.C. Barnes, An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm, Journal of Computational Physics 230 (2011) 7018–7036]. The approach computes particle orbits exactly for a given piece-wise linear electric field. The resulting PIC algorithm maintains the exact charge and energy conservation properties of the original algorithm, but with improved performance (both in efficiency and robustness against the number of particles and timestep). We demonstrate the advantageous properties of the scheme with a challenging multiscale numerical test case, the ion acoustic wave. Using the analytical mover as a reference, we demonstrate that the choice of error estimator in the Crank–Nicolson mover has significant impact on the overall performance of the implicit PIC algorithm. The generalization of the approach to the multi-dimensional case is outlined, based on a novel and simple charge conserving interpolation scheme.

A charge- and energy-conserving implicit, electrostatic particle-in-cell algorithm on mapped computational meshes

We describe the extension of the recent charge- and energy-conserving one-dimensional electrostatic particle-in-cell algorithm in Ref. [G. Chen, L. Chacón, D.C. Barnes, An energy- and charge-conserving, implicit electrostatic particle-in-cell algorithm, Journal of Computational Physics 230 (2011) 7018–7036] to mapped (body-fitted) computational meshes. The approach maintains exact charge and energy conservation properties. Key to the algorithm is a hybrid push, where particle positions are updated in logical space, while velocities are updated in physical space. The effectiveness of the approach is demonstrated with a challenging numerical test case, the ion acoustic shock wave. The generalization of the approach to multiple dimensions is outlined.

An exact energy conservation property of the quantum lattice Boltzmann algorithm

The quantum lattice Boltzmann algorithm offers a unitary and readily parallelisable discretisation of the Dirac equation that is free of the fermion-doubling problem. The expectation of the discrete time-advance operator is an exact invariant of the algorithm. Its imaginary part determines the expectation of the Hamiltonian operator, the energy of the solution, with an accuracy that is consistent with the overall accuracy of the algorithm. In the one-dimensional case, this accuracy may be increased from first to second order using a variable transformation. The three-dimensional quantum lattice Boltzmann algorithm uses operator splitting to approximate evolution under the three-dimensional Dirac equation by a sequence of solutions of one-dimensional Dirac equations. The three-dimensional algorithm thus inherits the energy conservation property of the one-dimensional algorithm, although the implementation shown remains only first-order accurate due to the splitting error.

Smoothed particle hydrodynamics simulation of shear-induced powder migration in injection moulding

We present the application of the smoothed particle hydrodynamics (SPH) discretization scheme to Phillips' model for shear-induced particle migration in concentrated suspensions. This model provides an evolution equation for the scalar mean volume fraction of idealized spherical solid particles of equal diameter which is discretized by the SPH formalism. In order to obtain a discrete evolution equation with exact conservation properties we treat in fact the occupied volume of the solid particles as the degree of freedom for the fluid particles. We present simulation results in two- and three-dimensional channel flow. The two-dimensional results serve as a verification by a comparison to analytic solutions. The three-dimensional results are used for a comparison with experimental measurements obtained from computer tomography of injection moulded ceramic microparts. We observe the best agreement of measurements with snapshots of the transient simulation for a ratio D(c)/D(η)=0.1 of the two model parameters.