semi-Lagrangian Approach Research Articles

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

Asymptotic preserving semi-Lagrangian discontinuous Galerkin methods for multiscale kinetic transport equations

We propose a new family of asymptotic preserving (AP) discontinuous Galerkin (DG) schemes for kinetic transport equations under a diffusive scaling. They are built upon a characteristics-based model reformulation proposed in Zhang et al. (2023) [27]. The reformulated model comprises a convection-diffusion-like equation for the macroscopic density and another evolution equation for the distribution function. We employ the equations in a weak form to enjoy the advantages of a DG method. We also leverage a semi-Lagrangian (SL) approach to achieve higher efficiency than a fully implicit scheme without loss of unconditional stability. A damping technique is introduced to control spurious numerical oscillations for high-order DG methods. We establish formal AP and Fourier stability analyses for fully discrete schemes in a linear scenario. Numerical experiments, covering one-dimensional to three-dimensional in space problems, confirm the accuracy, AP property, uniformly unconditional stability, and high parallel efficiency of the proposed methods.

Field Line Universal relaXer (FLUX): A Fluxon Approach to Coronal Magnetic Field Modeling

We describe a novel method for modeling the global, steady solar wind using photospheric magnetic fields as a driving boundary condition. Prior wind models in this class include both rapid heuristic methods that use potential field extrapolation and variants thereof, trading rigor for computation speed, and detailed 3D magnetohydrodynamic (MHD) models that attempt to simulate the entire solar corona with a degree of physical rigor, but require large amounts of computation. The Field Line Universal relaXer, an open-source numerical code that implements the “fluxon” semi-Lagrangian approach to MHD modeling, provides an intermediate approach between these two general classes. In particular, the fluxon approach to MHD describes the magnetic field through discrete analogs of magnetic field lines, relaxing these structures to a stationary state of force balance. In this work we introduce a 1D solar wind solution along each field line, providing an ensemble of solutions that are interpolated back onto a uniform grid at an outer boundary surface. This provides advantages in physical rigor over heuristic semianalytic techniques, and in computational efficiency over full 3D MHD techniques. Here we describe the underlying methodology and the FLUXPipe modeling pipeline process.

Optimization-based, property-preserving algorithm for passive tracer transport

We present a new optimization-based property-preserving algorithm for passive tracer transport. The algorithm utilizes a semi-Lagrangian approach based on incremental remapping of the mass and the total tracer. However, unlike traditional semi-Lagrangian schemes, which remap the density and the tracer mixing ratio through monotone reconstruction or flux correction, we utilize an optimization-based remapping that enforces conservation and local bounds as optimization constraints. In so doing we separate accuracy considerations from preservation of physical properties to obtain a conservative, second-order accurate transport scheme that also has a notion of optimality. Moreover, we prove that the optimization-based algorithm preserves linear relationships between tracer mixing ratios. We illustrate the properties of the new algorithm using a series of standard tracer transport test problems in a plane and on a sphere.

Sweep interpolation: a cost-effective semi-Lagrangian scheme in the Global Environmental Multiscale model

Abstract. The interpolation process is the most computationally expensive step of the semi-Lagrangian (SL) approach for solving advection and is commonly used in numerical weather prediction (NWP) models. It has a significant impact on the accuracy of the solution and can potentially be the most expensive part of model integration. The sweep algorithm, which was first described by Mortezazadeh and Wang (2017), performs SL interpolation with the same computational cost as a third-order polynomial scheme but at the accuracy of the fourth order. This improvement is achieved by using two third-order backward and forward polynomial interpolation schemes in two consecutive time steps. In this paper, we present a new application of the sweep algorithm within the context of global forecasts produced with Environment Climate Change Canada's Global Environmental Multiscale (GEM) model. Results show that the SL scheme with sweep interpolation is computationally more efficient compared to a conventional SL scheme with fourth-order polynomial interpolation, especially when a very large number of passive tracers are advected. An additional advantage of this new approach is that its implementation in a chemical and weather forecast model requires minimum modifications of the interpolation weighting coefficients. An analysis of the computational performance for a set of theoretical benchmarks as well as a global ozone forecast experiment show that up to 15 % reduction in total wall clock time is achieved. Forecasting experiments using the global version of the GEM model and the new interpolation show that the sweep interpolation can perform very well in predicting ozone distribution, especially in the tropopause region, where transport processes play a significant role.

A comparison of Eulerian and semi-Lagrangian approaches for modeling stream water quality

ABSTRACT This paper describes and compares some of the advantages and limitations of Eulerian and Lagrangian approaches to water quality modeling and introduces a mixed Eulerian-Lagrangian (or semi-Lagrangian) methodology that captures the strengths of both approaches. The semi-Lagrangian modeling approach is applied to advection-dominated rivers, and flexibly ensures unconditional stability for all time step durations and grid segmentations. The semi-Lagrangian modeling approach is demonstrated by applying it to estimate the dissolved oxygen concentrations in the Sava River in Slovenia, focusing on aspects of the methodology and findings that would be of broad interest to managers of water quality in fluvial water bodies. Results of comparisons of the semi-Lagrangian model with the Eulerian-based QUAL2K model in steady and non-steady scenarios demonstrate that while both models are fully capable of producing satisfactory results when optimally configured, the semi-Lagrangian approach offers accuracy and stability without sensitivity to the interaction of time step size and computational grid segmentation scheme.

Fourier neural operator for real-time simulation of 3D dynamic urban microclimate

Global urbanization has underscored the significance of urban microclimates for human comfort, health, and building/urban energy efficiency. However, analyzing urban microclimates requires considering a complex array of outdoor parameters within computational domains at the city scale over a longer period than indoors. As a result, numerical methods like Computational Fluid Dynamics (CFD) become computationally expensive when evaluating the impact of urban microclimates. The rise of deep learning techniques has opened new opportunities for accelerating the modeling of complex nonlinear interactions and system dynamics. Recently, the Fourier Neural Operator (FNO) has been shown to be very promising in accelerating solving the Partial Differential Equations (PDEs) and modeling fluid dynamic systems. In this work, we apply the FNO network for real-time three-dimensional (3D) urban microclimate simulation. For modeling large-scale urban microclimate problems, CityFFD simulates urban microclimate features based on the semi-Lagrangian approach and fractional stepping method with the Smagorinsky large eddy simulation model. In our simulation, the 1200 sequential time steps are used as training data. We retain and analyze the data from all stages, including the spin-up period, because we wish to understand how the flow develops transiently from initial conditions, and both one-step and sequential timestep predictions are analyzed. When applied to unseen data with different wind directions, the FNO model has a 0.3% one-step prediction error and a maximum error of 5%. A real-time simulation of urban microclimates in 3D is possible with the FNO approach, which is 25 times faster than the traditional numerical solver.

Conservative semi-Lagrangian method for continuity equation with various time steps in different parts of computational domain

The system of Navier Stokes differential equations describes gas flow near rocket nozzle. To find solution of this system in general case, scientists and engineers use numerical methods. Modern numerical methods do not allow engineers to model all features of gas flow. It happened because of sophisticated physical processes and limitations of computational hardware. So, at least where are two ways to improve it: to enlarge hardware or reduce computational complicacy. The global aim of many science investigations is to develop numerical approach with reduced computational complicacy and at the same time without loss of computational accuracy. In 1959, Aksel C. Wiin-Nielsen proposed a new and effective trajectories method for problem of numerical forecasting. In 1966, K. M. Magomedov developed similar approach (method of characteristics) for numerical modelling of space (three dimensional on space) gas flow. In 1982, O. Pironneau showed a new and effective approach for two dimensional approximation of the Navier Stokes problem. It was based on method of characteristics also. Nowadays, these methods are called semi-Lagrangian or Eulerian Lagrangian methods. They use Lagrangian nature of the transport process. To apply this advantage, scientists decompose each equation of Navier Stokes system into three parts: convective part (hyperbolic type), elliptic part and part of right-hand side. Scientists use semi-Lagrangian approach to approximate the convective parts of equations. To develop and test modern algorithms from family of semi-Lagrangian methods, we use continuity equation from the system of Navier Stokes equations. Conservative versions of semi-Lagrangian approach are based on Gauss Ostrogradsky (divergence) theorem. It allows scientists to get conservation low (balance equation) for numerical solution in norm of L1 space. Aim of our investigation is to use different time steps in different parts of computation domain. It enables us to obtain at the same time three advantages: convergence of numerical solution to exact solution, reduction of computational complicacy, implementation of conservation low (balance equation) without weight coefficients. For this purpose we decompose computational domain into two parts (subdomains) and use different time steps in them. Main complication is design algorithm in the boundaries of computational subdomains. We use one dimensional (on space) problem to demonstrate ability of developing numerical method with described advantages. Generalization of considered approach for two- or even three-dimensional cases allows engineers to model gas flow more accurately and without artificial viscosity. It is essentially important in the parts of computational domain with high level of solution gradient.

Minimal Exposure Paths in Time-Varying Fields: A Semi-Lagrangian Approach

In the context of Wireless Sensor Networks (WSNs), the Minimal Exposure Path (MEP) represents an important metric to evaluate the quality of network services. Most of the current literature is concentrated on stationary sensors, while few works have addressed the existence of moving nodes. Mobile Wireless Sensor Networks (MWSNs) present a great potential for detecting invaders compared to static ones, but dealing with dynamic sensors significantly increases the coverage complexity since the overall exposure of the sensor field depends on time. Therefore, in this letter, we propose an approach to compute Minimal Exposure Paths in time-varying fields based on a control optimization method called semi-Lagrangian (SL) scheme, in such a way that an intruder will be able to penetrate the dynamic field with the lowest exposure. The SL has already been proven to reach the optimal Minimal Exposure Path (MEP) on static WSNs, but concerning dynamic nodes, the proof is much more complicated. Then, we propose a heuristics that provides convergence of the SL algorithm to a result we conjecture to be the optimal one. Results with different time-varying sensor models in obstacle-free and cluttered environments have been presented and discussed.

Comparison of wave diffraction forces on a surface-piercing body for various free-surface grid update schemes

This study analyzed the difference in wave diffraction force of a surface piercing body according to the grid update scheme of free surface nodes in the time integration process. A two-dimensional fully nonlinear numerical wave tank based on potential flow theory and boundary element method was used for numerical analysis. The MEL (mixed Eulerian–Lagrangian) method was used for free surface node treatment. This method was divided into a semi-Lagrangian approach and a full Lagrangian approach (material node approach).The Runge–Kutta 4th-order integration method was used to integrate the free surface boundary condition temporally. In the internal sub-time step of the time integration process, the frozen coefficient scheme and the full update scheme were used according to the node update scheme, respectively. The vertical diffraction forces of each frequency component for node update schemes were compared. The difference in the 2nd-order vertical forces according to wave frequency, wave steepness, time step interval, and node spacing was investigated. A more precise time-domain numerical analysis method was proposed for high-order external forces in the wave-floating body interaction.

Convergence estimates of a semi-Lagrangian scheme for the ellipsoidal BGK model for polyatomic molecules

In this paper, we propose a new semi-Lagrangian scheme for the polyatomic ellipsoidal BGK model. In order to avoid time step restrictions coming from convection term and small Knudsen number, we combine a semi-Lagrangian approach for the convection term with an implicit treatment for the relaxation term. We show how to explicitly solve the implicit step, thus obtaining an efficient and stable scheme for any Knudsen number. We also derive an explicit error estimate on the convergence of the proposed scheme for every fixed value of the Knudsen number.

A semi-Lagrangian approach for the Minimal Exposure Path Problem in Wireless Sensor Networks

A critical metric of the coverage quality in Wireless Sensor Networks (WSNs) is the Minimal Exposure Path (MEP), a path through the environment that least exposes a mobile target to the sensor nodes detection. Many approaches have been proposed in the last decades to solve this optimization problem, ranging from classic grid-based and Voronoi-based planners to meta-heuristics. However, most of them are limited to specific sensing models and obstacle-free spaces. Still, none of them guarantee an optimal solution, and the state-of-the-art is expensive in terms of execution time. Therefore, in this paper, we propose a novel method, called SL-MEP, that models the MEP as an optimal control problem and solves it by using a semi-Lagrangian (SL) scheme. This framework is shown to converge to the optimal MEP while it incorporates different homogeneous and heterogeneous sensor models and geometric constraints (obstacles). Experiments show that our method dominates the state-of-the-art, improving the results by approximately 10% with a relatively lower execution time.

On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEs

This article is devoted to the construction of a new class of semi-Lagrangian (SL) schemes with implicit-explicit (IMEX) Runge-Kutta (RK) time stepping for PDEs involving multiple space-time scales. The semi-Lagrangian (SL) approach fully couples the space and time discretization, thus making the use of RK strategies particularly difficult to be combined with. First, a simple scalar advection-diffusion equation is considered as a prototype PDE for the development of a high order formulation of the semi-Lagrangian IMEX algorithms. The advection part of the PDE is discretized explicitly at the aid of a SL technique, while an implicit discretization is employed for the diffusion terms. In this way, an unconditionally stable numerical scheme is obtained, that does not suffer any CFL-type stability restriction on the maximum admissible time step. Second, the SL-IMEX approach is extended to deal with hyperbolic systems with multiple scales, including balance laws, that involve shock waves and other discontinuities. A conservative scheme is then crucial to properly capture the wave propagation speed and thus to locate the discontinuity and the plateau exhibited by the solution. A novel SL technique is proposed, which is based on the integration of the governing equations over the space-time control volume which arises from the motion of each grid point. High order of accuracy is ensured by the usage of IMEX RK schemes combined with a Cauchy–Kowalevskaya procedure that provides a predictor solution within each space-time element. The one-dimensional shallow water equations (SWE) are chosen to validate the new conservative SL-IMEX schemes, where convection and pressure fluxes are treated explicitly and implicitly, respectively. The asymptotic-preserving (AP) property of the novel schemes is also studied considering a relaxation PDE system for the SWE. A large suite of convergence studies for both the non-conservative and the conservative version of the novel class of methods demonstrates that the formal order of accuracy is achieved and numerical evidences about the conservation property are shown. The AP property for the corresponding relaxation system is also investigated.

Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.

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.

Simulation of Electron Beam Emission in Argon–Neon Mixture Pinch Using the Nonrelativistic 6-D Valsov–Maxwell System Solved by the Forward Semi-Lagrangian Approach

The 6-D Vlasov–Maxwell equations in the cylindrical coordinates have been solved to calculate the electron beam emission from mixtures of argon and neon. The forward semi-Lagrangian approach was used for solutions. The simulation of the electron velocity field has shown that at the early stage of the pinch evolution of a 4-mbar 90% Ar–10% Ne mixture, the sausage instabilities began to develop from the top parts of the pinch due to bigger $B_{\theta }$ with respect to $B_{z}$ . In the turbulence regions, the electrons could accelerate up to speeds of $905 \times 10^{5}$ and $2387.5 \times 10^{5} \,\,{\text {m}}/{\text {s}}$ at 90.9 and 181.82 ns, respectively. It has been observed that in the high turbulence region, $E_{z}=-2.04\times {10}^{7}\,\,{\text {V}}/{\text {m}}$ , while in the weak turbulence region, $E_{z}=-1.46\times {10}^{7}\,\,\text {V}/{\text {m}}$ . Increasing the neon percentage in the mixture changed the turbulence regions and the upstream and downstream emissions.

Double-diffusive sedimentation at high Schmidt numbers: Semi-Lagrangian simulations

When particle-laden freshwater is placed above clear saltwater, the ensuing sedimentation process can take one of two forms: For small dimensionless settling velocities, it will be double diffusive in nature, whereas for large settling velocities it will be dominated by Rayleigh-Taylor instability. A high-performance semi-Lagrangian computational approach is introduced that allows for the investigation of these processes in three dimensions.

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.

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.