Conservative semi-Lagrangian Scheme Research Articles

An efficient energy conserving semi-Lagrangian kinetic scheme for the Vlasov-Ampère system

In this paper, we present a novel kinetic scheme termed the Energy Conserving Semi-Lagrangian (ECSL) for the Vlasov-Ampère system. The novelty of the ECSL is that it retains the efficiency of the explicit scheme while simultaneously maintaining energy conservation and unconditional stability properties of the implicit scheme without relying on nonlinear iteration. The proposed ECSL method includes two main ingredients: the conservative Semi-Lagrangian (CSL) scheme and a novel field solver. The CSL scheme is utilized for the phase space discretization of the Vlasov equation, which enables the scheme to conserve mass exactly and remove the Courant-Friedrichs-Lewy restriction. The novel field solver is proposed by coupling the Ampère equation and the moments of the Vlasov equation in a semi-implicit way, allowing for an explicit and efficient calculation of the electric field. The combination of the novel field solver and CSL ensures that ECSL scheme conserves total energy and mass on the fully discrete level, regardless of spatial resolution and time step size. Moreover, the ECSL scheme still provides reliable solution even when its spatial and temporal resolution are insufficient to fully resolve the Debye length and plasma period, making it a promising tool for multiscale and lengthy simulations. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and conservation properties of the proposed method.

A combined immersed finite element and conservative semi-Lagrangian scheme for plasma-material interactions

In this paper, the immersed finite element (IFE) coupled with the conservative semi-Lagrangian (CSL) kinetic scheme is developed for plasma-material interactions. The proposed method (IFE-CSL) enjoys respective advantages of the IFE and CSL, i.e., the flexibility and efficiency for treating complex boundary conditions, mass conservation and being free from the Courant-Friedrichs-Lewy (CFL) condition. In the current IFE-CSL, the IFE method based on structured interface-independent meshes is developed for the spatial discretization of the field equation, which provides an accurate approach with convenient implementations to solve the field problems with irregular interface. The CSL scheme combined with an immersed method termed half-way ghost-cell is proposed for the spatial discretization of the Vlasov equation, which enables the proposed method to exactly conserve mass and conveniently handle curved boundaries. Then the IFE and CSL method are coupled via the charge density depending on full kinetic or hybrid kinetic model, where the appropriate IFE solver is developed for the linear or nonlinear Poisson equation, respectively. In the current IFE-CSL, different geometries can be treated automatically for both IFE and CSL through the specific geometric information. As a result, the proposed CSL-IFE can be conveniently and efficiently applied to simulate the plasma-material interactions with structured interface-independent meshes. Finally, several numerical experiments are performed to demonstrate the capabilities of the proposed method.

A fourth-order conservative semi-Lagrangian finite volume WENO scheme without operator splitting for kinetic and fluid simulations

In this paper, we present a fourth-order conservative semi-Lagrangian (SL) finite volume (FV) weighted essentially non-oscillatory (WENO) scheme without operator splitting for two-dimensional linear transport equations with applications in kinetic models including the nonlinear Vlasov–Poisson system, the guiding center Vlasov model and the incompressible Euler equation in the vorticity-stream function formulation. To achieve fourth-order accuracy in space, two main ingredients are proposed in the SL FV formulation. Firstly, we introduce a so-called cubic-curved quadrilateral upstream cell and applying an efficient clipping method to evaluate integrals on upstream cells. Secondly, we construct a new WENO reconstruction operator, which recovers a P3 polynomial from neighboring cell averages. Mass conservation is accomplished with the mass conservative nature of the reconstruction operator and the SL formulation. A positivity-preserving limiter is applied to maintain the positivity of the numerical solution wherever appropriate. For nonlinear kinetic models, the SL scheme is coupled with a fourth-order Runge–Kutta exponential integrator for high-order temporal accuracy. Extensive benchmarks are tested to verify the designed properties.

Conservative semi-Lagrangian schemes for a general consistent BGK model for inert gas mixtures

In this work, we propose a class of high order semi-Lagrangian scheme for a general consistent BGK model for inert gas mixtures. The proposed scheme not only fulfills indifferentiability principle, but also asymptotic preserving property, which allows us to capture the behaviors of hydrodynamic limit models. We consider two hydrodynamic closure which can be derived from the BGK model at leading order: classical Euler equations for number densities, global velocity and temperature, and a multi-velocities and temperatures Euler system. Numerical simulations are performed to demonstrate indifferentiability principle and asymptotic preserving property of the proposed conservative semi-Lagrangian scheme to the Euler limits.

Boundary variation diminished conservative semi-Lagrangian method for both compressible and incompressible flows

Motivated by the enlightenment that diminishing the jump at the cell boundary can effectively reduce numerical dissipation near the critical region, and a novel constrained interpolation profile conservative semi-Lagrangian method is proposed based on a newly designed boundary variation diminishing algorithm. First, a constrained interpolation profile conservative semi-Lagrangian scheme with the piecewise tangent of hyperbola for interface capturing scheme is proposed as one candidate to represent jump-like discontinuities. Second, the constrained interpolation profile conservative semi-Lagrangian scheme with a fourth-order weighted essentially non-oscillatory limiter is used as another candidate to keep the high-order and non-oscillatory reconstruction for smooth solutions. The selection criterion of these two candidates is designed by minimizing the total variations of the first derivative at cell boundaries. A unified pressure-based projection formulation with a fractional step procedure is implemented with the proposed scheme to simulate both compressible and incompressible flows. A variety of numerical tests are studied, including linear and nonlinear scalar wave transport problems and compressible and incompressible flow problems. Results show that the proposed method can effectively eliminate numerical oscillation and diffusion, suggesting it has great potential to be applied to various types of engineering problems including both compressible and incompressible flows.

A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

A positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid. Two kinds of moments (i.e., point values (PV moment) at cell interfaces and volume integrated average (VIA moment) value) are defined within a single cell. The PV moment is updated by a conventional semi-Lagrangian method, while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation. Different from the spatial approximation used in the CSL2 (conservative semi-Lagrangian scheme with second order polynomial function) scheme, a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment. To achieve exactly positive-definite preserving, two kinds of corrections are made on the original conservative semi-Lagrangian with rational function (CSLR) scheme. The resulting scheme is inherently conservative, non-negative, and allows a Courant number larger than one. Moreover, the spatial reconstruction can be performed within a single cell, which is very efficient and economical for practical implementation. In addition, a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry, which benefitsthe implementation of the 1D CSLR solver with large Courant number. The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry. Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical non-negativity, and it has the potential to transport the tracers accurately in a real atmospheric model.

Conservative semi-Lagrangian kinetic scheme coupled with implicit finite element field solver for multidimensional Vlasov Maxwell system

In this paper, a conservative semi-Lagrangian (CSL) kinetic scheme coupled with an implicit finite element method (IFEM) is developed for multidimensional electromagnetic plasma simulations. The present method (CSL-IFEM) enjoys the respective advantages of the CSL and IFEM, including mass conservation, high-order spatial accuracy, as well as being free from the Courant-Friedrichs-Lewy (CFL) condition. In present CSL-IFEM, the CSL scheme with a general high order positivity-preserving limiter is utilized for the spatial discretization of the Vlasov equation, which enables the proposed method to remove the CFL limitation in phase space, exactly conserve mass, and preserve the positivity of the distribution function. The IFEM solver is developed for the spatial discretization of the Maxwell equation, which makes the current method free from CFL limitation induced by the speed of light and easily handles general field boundary conditions. To make the proposed CSL-IFEM more efficient in multidimensional simulations, the dimensional splitting procedure and sparse matrix technology are implemented in CSL and IFEM, respectively. Then the Vlasov solver CSL and Maxwell solver IFEM are coupled via the current density calculated from the general Ohm law. Finally, several numerical experiments, including electromagnetic wave (2d0v), particle gyromotion (0d2v), streaming Weibel instability (1d2v) and magnetic reconnection (2d3v), are performed to demonstrate the capabilities of the proposed method.

High Order Conservative Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications

In this paper, we present a new class of conservative semi-Lagrangian schemes for kinetic equations. They are based on the conservative reconstruction technique introduced in [1]. The methods are high order accurate both in space and time. Because of the semi-Lagrangian nature, the time step is not restricted by a CFL-type condition. Applications are presented to the Vlasov-Poisson system and the BGK model of rarefied gas dynamics. In the first case, operator splitting is adopted to obtain high order accuracy in time, and a conservative reconstruction that preserves the maximum and minimum of the function is used. For initially positive solutions, in particular, this guarantees exact preservation of the L1-norm. Conservative schemes for the BGK model are constructed by coupling the conservative reconstruction with a conservative treatment of the collision term. High order in time is obtained by either Runge-Kutta or BDF time discretization of the equation along characteristics. Because of L-stability and exact conservation, the resulting scheme for the BGK model is asymptotic preserving for the underlying fluid dynamic limit. Several test cases in one and two space dimensions confirm the accuracy and robustness of the methods, and the AP property of the schemes when applied to the BGK model.

Conservative semi-Lagrangian schemes for kinetic equations Part I: Reconstruction

In this paper, we propose and analyze a reconstruction technique which enables one to design high-order conservative semi-Lagrangian schemes for kinetic equations. The proposed reconstruction can be obtained by taking the sliding average of a given polynomial reconstruction of the numerical solution. A compact representation of the high order conservative reconstruction in one and two space dimension is provided, and its mathematical properties are analyzed. To demonstrate the performance of proposed technique, we consider implicit semi-Lagrangian schemes for kinetic-like equations such as the Xin-Jin model and the Broadwell model, and then solve related shock problems which arise in the relaxation limit. Applications to BGK and Vlasov-Poisson equations will be presented in the second part of the paper.

A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

<p style='text-indent:20px;'>In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [<xref ref-type="bibr" rid="b21">21</xref>], with a high order RKEI method [<xref ref-type="bibr" rid="b7">7</xref>], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.</p>

Simulation of three-dimensional chemical dissolution of limestone

The development of prediction techniques for the evolution of karstic caves is essential for geohazard prevention because limestone collapse commonly accompanies their complex dissolution process. Previous studies to understand the dissolution mechanisms focus on field-based and/or small-scale experimental approaches. Although several large-scale numerical simulations have been conducted due to improved computing capabilities, mathematical modelling and numerical simulation for the three-dimensional dissolution of limestone remains unavailable. In this study, we examine the three-dimensional dissolution phenomenon of calcium carbonate in limestone and propose mathematical and numerical models based on an advection, reaction, and diffusion system involving Darcy’s law. Additionally, we implement the models using a finite difference method involving the constrained interpolation profile with conservative semi-Lagrangian scheme, and dissolution patterns of the calcium carbonates obtained by the proposed model are presented. The simulation demonstrates that the dissolution of calcium carbonate is strongly related to the groundwater flow, with increasing pores and cavities toward the groundwater flow direction, and the dissolution rate depends on the contact area between the groundwater and limestone.

Non-stationary Antonov self-gravitating layer: analytics and numerics

ABSTRACTLarge-scale instability of gravitating systems plays a key role in collisionless relaxation and in reaching a quasi-stationary state at the early stage of evolution. Advanced high-resolution methods and permanently increasing performance of computational systems allow this phenomenon to be studied by means of computer simulations at a new level. In this paper, an approach to verification and validation of computer codes implementing high-resolution methods is proposed. The approach is based on comparisons of the simulation results with exact non-stationary solutions of the Vlasov–Poisson equations. The evolution of the gravitating layer model is considered as an example of implementation of this approach. A one-parameter family of exact models of a non-stationary gravitating layer is described, and their stability to large-scale disturbances in the linear approximation is analytically studied. Non-linear instability development is computed with the use of the fifth-order conservative semi-Lagrangian WENO scheme.

Experimental and numerical simulation of bidirectional propagation of an air cavity

This study numerically and experimentally investigates the transient flow associated with a so-called Benjamin bubble, but, in contrast to previous studies, the approach permits bidirectional flow. Two experiments investigated the bidirectional propagation of an air cavity in a square tank; these two events were also simulated using a three-dimensional numerical model. An exactly conservative semi-Lagrangian scheme was employed to solve both the advection term in the momentum equation and the advection equation of the volume of fluid function. The model was compared with previous one-dimensional Benjamin bubble experimental data and a one-dimensional Boussinesq model. For the one-dimensional Benjamin bubble experiments, the three-dimensional model showed good agreement with both published data and a one-dimensional non-hydrostatic model. After modelling the advance of a one-dimensional cavity, a bidirectional Benjamin bubble was simulated using the three-dimensional model. The numerical results are in reasonable agreement with observations of the free surface and pressurized regions and reasonably predict the shape of the air–water interface.

Conservative Multi-dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov-Poisson system using a a high order tracing mechanism proposed in [Qiu and Russo, 2016]. Such high order characteristics tracing scheme is generalized to the nonlinear guiding center Vlasov model and incompressible Euler system. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by our extensive numerical tests.

Constrained Interpolation Profile Conservative Semi-Lagrangian Scheme Based on Third-Order Polynomial Functions and Essentially Non-Oscillatory (CIP-CSL3ENO) Scheme

AbstractWe propose a fully conservative and less oscillatory multi-moment scheme for the approximation of hyperbolic conservation laws. The proposed scheme (CIP-CSL3ENO) is based on two CIP-CSL3 schemes and the essentially non-oscillatory (ENO) scheme. In this paper, we also propose an ENO indicator for the multimoment framework, which intentionally selects non-smooth stencil but can efficiently minimize numerical oscillations. The proposed scheme is validated through various benchmark problems and a comparison with an experiment of two droplets collision/separation. The CIP-CSL3ENO scheme shows approximately fourth-order accuracy for smooth solution, and captures discontinuities and smooth solutions simultaneously without numerical oscillations for solutions which include discontinuities. The numerical results of two droplets collision/separation (3D free surface flow simulation) show that the CIP-CSL3ENO scheme can be applied to various types of fluid problems not only compressible flow problems but also incompressible and 3D free surface flow problems.

Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

Abstract In today’s atmospheric numerical modeling, scalable and highly accurate numerical schemes are of particular interest. To address these issues Galerkin schemes, such as the spectral element method, have received more attention in the last decade. They also provide other state-of-the-art capabilities such as improved conservation. However, the tracer transport of hundreds of tracers, e.g., in the chemistry version of the Community Atmosphere Model, is still a performance bottleneck. Therefore, we consider two conservative semi-Lagrangian schemes. Both are designed to be multi-tracer efficient, third order accurate, and allow significantly longer time steps than explicit Eulerian formulations. We address the difficulties arising on the cubed-sphere projection and on parallel computers and show the high scalability of our approach. Additionally, we use the two schemes for the transport of passive tracers in a dynamical core and compare our results with a current spectral element tracer transport advection used by the High-Order Method Modeling Environment.

The semi-Lagrangian method on curvilinear grids

Abstract We study the semi-Lagrangian method on curvilinear grids. The classical backward semi-Lagrangian method [1] preserves constant states but is not mass conservative. Natural reconstruction of the field permits nevertheless to have at least first order in time conservation of mass, even if the spatial error is large. Interpolation is performed with classical cubic splines and also cubic Hermite interpolation with arbitrary reconstruction order of the derivatives. High odd order reconstruction of the derivatives is shown to be a good ersatz of cubic splines which do not behave very well as time step tends to zero. A conservative semi-Lagrangian scheme along the lines of [2] is then described; here conservation of mass is automatically satisfied and constant states are shown to be preserved up to first order in time.

Improvement of fast fluid dynamics with a conservative semi-Lagrangian scheme

Purpose – The purpose of this paper is to develop a simple and efficient conservative semi-Lagrangian scheme (SL) for solving advection equation in fast fluid dynamics (FFD), so FFD can provide fast indoor airflow simulations while preserving conservation for energy and species transport. Design/methodology/approach – This study thus proposed a mass-fixing type conservative SL that redistributes global surplus/deficit on the advected field after performing the standard semi-Lagrangian advection. The redistribution weights were designed to preserve the properties of conservatives and monotonicity. Findings – The effectiveness of the conservative SL was validated with several test cases, and the results show that the proposed scheme is indeed conservative with negligible impact on the accuracy of the standard solutions. The numerical tests show that the proposed scheme was indeed conservative with negligible impact on the accuracy of the flow prediction. Originality/value – The FFD with conservative SL can effectively enforce the energy and species conservation for indoor airflow and predict airflow distributions with reasonable accuracy.