semi-Lagrangian Method Research Articles

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a space-time region partitioned by time-dependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by method-of-line Runge-Kutta (RK) methods. The ELDG framework is designed as a generalization of the semi-Lagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general two-dimensional problems. No quadratic-curved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary Lagrangian-Eulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

Simplified Spectral Model of 3D Meander Flow

Most 2D (two-dimensional) models either take vertical velocity profiles as uniform, or consider secondary flow in momentum equations with presupposed velocity profiles, which weakly reflect the spatio-temporal characteristics of meander flow. To tackle meander flow in a more accurate 3D (three-dimensional) way while avoiding low computational efficiency, a new 3D model based on spectral methods is established and verified in this paper. In the present model, the vertical water flow field is expanded into polynomials. Governing equations are transformed by the Galerkin method and then advection terms are tackled with a semi-Lagrangian method. The simulated flow structures of an open channel bend are then compared with experimental results. Although a zero-equation turbulence model is used in this new 3D model, it shows reasonable flow structures, and calculation efficiency is comparable to a depth-averaged 2D model.

A comparison of semi-Lagrangian vortex method and lattice Boltzmann method for incompressible flows

The semi-Lagrangian Vortex method (VM) and the Lattice Boltzmann method (LBM) are used to investigate flows simulations in the incompressible regime. In this study, a proven version of each method is used and compared on different three dimensional benchmarks in terms of numerical accuracy, convergence, numerical diffusion and dissipation. The first comparisons are made on a convected vortex to study and compare the numerical dissipation of LBM and VM. Then the Taylor–Green vortex is investigated to compare the dissipation rate of the kinetic energy of each method. It is shown that both methods converge to the same solution but in a different way. The VM performs better than the LBM for the lowest resolution whereas LBM appears to be more accurate for the growing resolutions. These results are confirmed on 3D simulations with wall boundaries for the stiff test case of the wake behind a 3D cube at Re=290 and Re=570.

THINC scaling method that bridges VOF and level set schemes

We present a novel interface-capturing scheme, THINC-scaling, to unify the VOF (volume of fluid) and the level set methods, which have been developed as two different approaches widely used in various applications. The key to success is to maintain a high-quality THINC reconstruction function using the level set field to accurately retrieve geometrical information and the VOF field to fulfill numerical conservativeness. The interface is well defined as a surface in form of a high-order polynomial, so-called the polynomial surface of interface (PSI). The THINC reconstruction function is then used to update the VOF field via a finite volume method, and the level set field via a semi-Lagrangian method. Seeing the VOF field and the level set field as two different aspects of the THINC reconstruction function, the THINC-scaling scheme preserves at the same time the advantages of both VOF and level set methods, i.e. the mass/volume conservation of the VOF method and the geometrical faithfulness of the level set method, through a straightforward solution procedure. The THINC-scaling scheme allows to represent an interface with high-order polynomials and has algorithmic simplicity which largely eases its implementation in unstructured grids. Two and three dimensional algorithms in both structured and unstructured grids have been developed and verified. The numerical results reveal that the THINC-scaling scheme, as an interface capturing method, is able to provide high-fidelity solution comparable to other most advanced methods, and more profoundly it can resolve sub-grid filament structures if the interface is represented by a polynomial higher than second order.

Investigation of Electron Beam Characteristics Emitted From Various Gases Employing the Vlasov–Maxwell Equations With the Forward Semi-Lagrangian Method

The characteristics of the electron beams emitted from the hydrogen ( $H$ ), nitrogen ( $N$ ), neon (Ne), and argon plasma pinches have been numerically investigated and compared by solving the Vlasov–Maxwell system with the forward semi-Lagrangian approach. Several outcomes have been obtained. For examples, it was found that, in the case of 1-mbar H, the electron current density reached a value of about $1.431\times 10^{9} \,\,\text {A}\cdot \text {m}^{-2}$ due to turbulence, while it reached ${2.554} \times {10}^{7} \,\,\text {A}\cdot \text {m}^{-2}$ due to other phenomena. For 2-mbar $N$ , the calculated voltage drops across the $m = 0$ regions ranged from about 817 kV to 4 MV. Among these gases, Ne had the biggest upstream emission current of about 9.7 kA, while $N$ emitted the least electron beam current of about 4.1 kA. Computations showed that the total charge delivered by each gas beam was of order 0.1 mC. The method of modeling, presented in this article, can be efficiently used to study the dynamics of a plasma pinch. Since the estimated electron beam currents for various gases are in good agreement with the measured ones, the proposed method can be used for designing the plasma focus devices for the electron beam source purpose.

Investigation of Laser wakefield acceleration using relativistic Vlasov-‎Maxwell code

In this paper we investigate the electron laser wakefield acceleration using numerical solution of relativistic Vlasov-Maxwell equations with Semi-Lagrangian method. We found optimum pulse length to excite strong wakefields by investigation of electric fields and density variation, which caused by propagation of intense relativistic laser pulse through underdense plasma as well as average kinetic energy of electrons, for lasers with different pulse lengths.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

High order semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

Calculation of Electron Beam Emitted From Argon Plasma Pinch Using the Nonrelativistic Vlasov–Maxwell Equations

This work deals with calculation of electron beam generated in a plasma focus device. The Vlasov–Maxwell equations of the charged particles are solved using the forward semi-Lagrangian method. The traces of the electrons are computed. It is shown that electron beams are emitted by strong electric field generated by compression, turbulent, and disruption phenomena in the electrons motion. The energy spectrum of emission shows dependence on pressure and pinch current.

A Conservative Semi-Lagrangian Finite Volume Method for Convection–Diffusion Problems on Unstructured Grids

A conservative semi-Lagrangian finite volume method is presented for the numerical solution of convection–diffusion problems on unstructured grids. The new method consists of combining the modified method of characteristics with a cell-centered finite volume discretization in a fractional-step manner where the convection part and the diffusion part are treated separately. The implementation of the proposed semi-Lagrangian finite volume method differs from its Eulerian counterpart in the fact that the present method is applied at each time step along the characteristic curves rather than in the time direction. To ensure conservation of mass at each time step, we adopt the adjusted advection techniques for unstructured triangular grids. The focus is on constructing efficient solvers with large stability regions and fully conservative to solve convection-dominated flow problems. We verify the performance of our semi-Lagrangian finite volume method for a class of advection–diffusion equations with known analytical solutions. We also present numerical results for a transport problem in the Mediterranean sea.

Development of a semi-Lagrangian advection scheme for the NEMO ocean model (3.1)

Abstract. As resolutions of ocean circulation models increase, the advective Courant number – the ratio between the distance travelled by a fluid parcel in one time step and the grid size – becomes the most stringent factor limiting model time steps. Some atmospheric models have escaped this limit by using an implicit or semi-implicit semi-Lagrangian formulation of advection, which calculates materially conserved fluid properties along trajectories which follow the fluid motion and end at prescribed grid points. Unfortunately, this formulation is not straightforward in ocean contexts, where the irregular, interior boundaries imposed by the shore and bottom orography are incompatible with traditional trajectory calculations. This work describes the adaptation of the semi-Lagrangian method as an advection module for an operational ocean model. We solve the difficulties of the ocean's internal boundaries by calculating parcel trajectories using a time-exponential formulation, which ensures that all parcel trajectories remain inside the ocean domain despite strong accelerations near the boundary. Additionally, we derive this method in a way that is compatible with the leapfrog time-stepping scheme used in the NEMO-OPA (Nucleus for European Modelling of the Ocean, Océan Parallélisé) ocean model, and we present simulation results for a simplified test case of flow past a model island and for 10-year free runs of the global ocean on the quarter-degree ORCA025 grid.

A Characteristic Mapping method for the two-dimensional incompressible Euler equations

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.

An efficient trajectory tracking algorithm for the backward semi-Lagrangian method of solving the guiding center problems

In this paper, we develop an effective numerical algorithm that tracks the trajectory needed to solve guiding center models using the backward semi-Lagrangian method. In terms of numerical calculations, two appreciably fast algorithms for the departure points are designed. One is a completely explicit formula for numerical solutions of the discrete system for each Cauchy problem. This formula is characterized by numerical factors that are less than half the multiplication number used in the usual Gaussian elimination. The other finds the required departure points with an interpolation method that is at least 30% less expensive for heavy Cauchy problems. Last, we propose a method to modify the solution to improve estimation of physical quantities, such as conservation of mass, which can be lost in interpolation solutions calculated at the departure points. It turns out that the proposed method not only saves a great deal of computation time, but also preserves physical quantities such as mass and total kinetic energy much better than conventional methods. To demonstrate the numerical evidence, we use the proposed method to simulate several problems such as the incompressible Euler equation, Kelvin-Helmholtz instability, Diocotron instability and a three-dimensional guiding center model.

An explicit semi-Lagrangian, spectral method for solution of Lagrangian transport equations in Eulerian-Lagrangian formulations

An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss quadrature points from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension

Semi-Lagrangian Vlasov simulation of the interaction between femtosecond chirped and double laser pulses with a thin plasma slab

The wakefield generation and electrons acceleration in the interaction of femtosecond chirped and double laser pulses with an underdense thin plasma slab is examined by numerical solution of the relativistic Vlasov-Maxwell system of equations using the backward semi-Lagrangian method (BSL). Unlike particle-in-cell methods, the BSL method is free of numerical noise and provides an accurate representation of the distribution function, even in very low density regions of phase space where acceleration and trapping occur. The effect of density fluctuations in low density regions of phase space are important, since they affect the acceleration processes over the following interactions. Therefore, a detailed structure of the distribution function is needed to investigate this process in these regions, which only can be obtained by Vlasov codes. Excitation of wakefields by chirped, chirp free, and double pulse lasers in an underdense plasma slab are simulated using the BSL code and the results are compared. It is shown that by making use of chirped laser pulses, the mean kinetic energy of plasma electrons can be enhanced up to 40% compared with the chirp free case. The investigation of wakefield generation by double laser pulses with a time repetition equal to one plasma period, exhibits amplification of wakefield amplitude after the entrance of the second laser pulse into plasma slab.

A semi-Lagrangian discontinuous Galerkin (DG) – local DG method for solving convection-diffusion equations

In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion equations. The method generalizes our previous work on developing the SLDG method for transport equations [5], making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the ‘SLDG-LDG’ method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An L2 stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions.

Regularity of the Geodesic Flow of the Incompressible Euler Equations on a Manifold

In this paper we investigate the regularity in time of the volume-preserving geodesic flow, which is associated with the incompressible Euler equations on a compact d-dimensional Riemannian manifold with boundary. Our result, which completes the local-in-time well-posedness theory of Ebin and Marsden (Ann Math 92:102–163, 1970), states roughly that the time smoothness of geodesic curves is only limited by the smoothness of the manifold. Such regularity is measured in a broad class of ultradifferentiable functions, which includes the real analytic and Gevrey classes. A by-product of this simple and constructive proof is new ideas to design high-order semi-Lagrangian methods for integrating the incompressible Euler equations on a manifold.

Interaction of rigid body motion and rarefied gas dynamics based on the BGK model

In this paper we present simulations of moving rigid bodies immersed in a rarefied gas. The rarefied gas is simulated by solving the Bhatnager-Gross-Krook (BGK) model for the Boltzmann equation. The Newton-Euler equations are solved to simulate the rigid body motion. The force and the torque on the rigid body is computed from the surrounded gas. An explicit Euler scheme is used for the time integration of the Newton-Euler equations. The BGK model is solved by the semi-Lagrangian method suggested by Russo &amp; Filbet ^{[<span class="xref"><a href="#b22" ref-type="bibr">22</a></span>]}. Due to the motion of the rigid body, the computational domain for the rarefied gas (and the interface between the rigid body and the gas domain) changes with respect to time. To allow a simpler handling of the interface motion we have used a meshfree method for the interpolation procedure in the semi-Lagrangian scheme. We have considered a one way, as well as a two-way coupling of rigid body and gas flow. We use diffuse reflection boundary conditions on the rigid body and also on the boundary of the computational domain. In one space dimension the numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. In the two-dimensional case results are compared with DSMC simulations for the Boltzmann equation and with results obtained by other researchers. Several test problems and applications illustrate the versatility of the approach.

Numerical investigation of Landau damping in dynamical Lorentz gases

We investigate numerically the behavior of a particle, or a set of particles, which exchange momentum and energy with the environment, described as a transverse vibrational field. The large time behavior is characterized, in some specific circumstances, by means of effective friction force and Landau damping. In order to discuss these issues on numerical grounds, we set up a dedicated numerical method. The scheme couples a Finite Element Method for the wave equation, with an appropriate transparent boundary condition that preserves the dispersive effects driving the asymptotic behavior of the system, and a symplectic scheme or a Semi-Lagrangian method for the evolution of the particles. The time discretization is constructed to capture accurately the energy exchanges. The numerical simulations shed new light on the theoretical results and bring out clearly the role of the parameters of the models.