semi-Lagrangian Method Research Articles

A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence

While developing a new semi-Lagrangian solver, the gap between a linear Landau run in 1Dx1Dand a 5D gyrokinetic simulation in toroidal geometry is quite huge. Intermediate test cases are welcomefor testing the code. A new fully two-dimensional conservative semi-Lagrangian (CSL) method is presentedhere and is validated on 2D polar geometries. We consider here as building block, a 2D guiding-centertype equation on an annulus and apply it on two test cases. First, we revisit a 2D test case previouslydone with a PIC approach [18] and detail the boundary conditions. Second, we consider a 4D drift-kineticslab simulation (see [10]). In both cases, the new method appears to be a good alternative to deal withthis type of models since it improves the lack of mass conservation of the standard semi-Lagrangian (BSL)method.

High order semi-Lagrangian particle methods for transport equations: numerical analysis and implementation issues

This paper is devoted to the definition, analysis and implementation of semi-Lagrangian methods as they result from particle methods combined with remeshing. We give a complete consistency analysis of these methods, based on the regularity and momentum properties of the remeshing kernels, and a stability analysis of a large class of second and fourth order methods. This analysis is supplemented by numerical illustrations. We also describe a general approach to implement these methods in the context of hybrid computing and investigate their performance on GPU processors as a function of their order of accuracy.

On vector field reconstructions for semi-Lagrangian transport methods on geodesic staggered grids

We analyse several vector reconstruction methods, based on the knowledge of only specific pointwise vector components, and extend their use to non-structured polygonal C-grids on the sphere. The emphasis is on the reconstruction of the vector field at arbitrary locations on the sphere, as required by semi-Lagrangian transport schemes. This is done by first reconstructing the vector field to fixed locations, followed by interpolations with generalized barycentric coordinates. We derive a hybrid scheme, combining the efficiency of Perot's method with the accuracy of a least square scheme. This method is second order accurate, and has shown to be competitive and computationally efficient. We analysed the vector reconstruction methods within a semi-Lagrangian transport method, and demonstrated that second order accurate reconstructions are enough to fulfil the requirements for second order accurate semi-Lagrangian methods on icosahedral C-grids.

Numerical methods for kinetic equations

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

A computational framework for conservative, three-dimensional, unsplit, geometric transport with application to the volume-of-fluid (VOF) method

In this work, a novel computational framework for calculating convection fluxes is developed and employed in the context of the piecewise linear interface calculation (PLIC) volume-of-fluid (VOF) method. The scheme is three-dimensional, unsplit, discretely conservative, and bounded. The scheme leverages the idea of semi-Lagrangian transport to estimate the amount of liquid that is fluxed through each face during a time-step.The present work can be seen as an extension of the two-dimensional EMFPA method of López et al. (2004) [16] to three dimensions and an improvement of the three-dimensional FMFPA-3D method of Hernández et al. (2008) [17] with the addition of discrete conservation. In FMFPA-3D, fluxes of liquid volume fraction are calculated by transporting a cell face back in time with a semi-Lagrangian method that uses cell face edge velocities to produce a flux hexahedron with flat faces. The flux hexahedron may overlap with neighboring fluxes hindering the conservation properties of the method. The proposed method computes the fluxes by transporting the cell face back in time using a semi-Lagrangian step based on the cell face corner velocities, which results in a three-dimensional, generalized flux hexahedron that does not typically have flat faces. However, the flux volumes do not overlap and discrete conservation can be achieved. The complex flux volume is partitioned into a collection of simplices and a simple sign convention allows the calculation of the flux to be reduced to a straightforward and systematic algorithm. The proposed VOF scheme is tested on multiple benchmark cases including Zalesak's disk, two- and three-dimensional deformation tests, and the evolution of a droplet in homogeneous isotropic turbulence.

Reprint of: A conservative multi-tracer transport scheme for spectral-element spherical grids

Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT — SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.

Arbitrarily high order Convected Scheme solution of the Vlasov–Poisson system

The Convected Scheme (CS) is a ‘forward-trajectory’ semi-Lagrangian method for solution of transport equations, which has been most often applied to the kinetic description of plasmas and rarefied neutral gases. In its simplest form, the CS propagates the solution forward in time by advecting the so-called ‘moving cells’ along their characteristic trajectories, and by remapping them on the mesh at the end of the time step. The CS is conservative, positivity preserving, simple to implement, and it is not subject to time step restriction to maintain stability. Recently (Güçlü and Hitchon, 2012 [1]) a new methodology was introduced for reducing numerical diffusion, based on a modified equation analysis: the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping. While the spatial accuracy was increased from 2nd to 4th order, the new scheme retained the important properties of the original method, and was shown to be extremely simple and efficient for constant advection problems.Here the CS is applied to the solution of the Vlasov–Poisson system, which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal Coulomb interactions. The Vlasov equation is split into two constant advection equations, one in configuration space and one in velocity space, and high order time accuracy is achieved by proper composition of the operators. The splitting procedure enables us to use the constant advection solver, which we extend to arbitrarily high order of accuracy in time and space: a new improved procedure is given, which makes the calculation of the corrections straightforward. Focusing on periodic domains, we describe a spectrally accurate scheme based on the fast Fourier transform; the proposed implementation is strictly conservative and positivity preserving. The ability to correctly reproduce the system dynamics, as well as resolving small-scale features in the solution, is shown in classical 1D–1V test cases, both in the linear and the non-linear regimes.

The vmf90 program for the numerical resolution of the Vlasov equation for mean-field systems

The numerical resolution of the Vlasov equation provides complementary information with respect to analytical studies and forms an important research tool in domains such as plasma physics. The study of mean-field models for systems with long-range interactions is another field in which the Vlasov equation plays an important role. We present the vmf90 program that performs numerical simulations of the Vlasov equation for this class of mean-field models with the semi-Lagrangian method. Program summaryProgram title: vmf90Catalogue identifier: AESO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AESO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 7794No. of bytes in distributed program, including test data, etc.: 58851Distribution format: tar.gzProgramming language: Fortran 95.Computer: Single CPU computer.Operating system: No specific operating system, the program is tested under Linux and OS X.RAM: About 5 M bytesClassification: 1.5, 19.8, 19.13, 23.External routines: HDF5 for the code (tested with HDF5 v1.8.8 and above). Python, NumPy, h5py and Matplotlib for analysis.Nature of problem:Numerical resolution of the Vlasov equation for mean-field models (Hamiltonian Mean-Field model and Single Wave model).Solution method:The equation is solved with the semi-Lagrangian method and cubic spline interpolation.Running time:The examples provided with the program take 1 m 30 for the Hamiltonian-Mean Field model and 10 m for the Single Wave model, on an Intel Core i7 CPU @ 3.33 GHz. Increasing the number of grid points or the number of time steps increases the running time.

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.

A Semi-Lagrangian Method for 3-D Fokker Planck Equations for Stochastic Dynamical Systems on the Sphere

In this paper, we consider stochastic dynamical systems on the sphere and the associated Fokker---Planck equations. A semi-Lagrangian method combined with a Finite Volume discretization of the sphere is presented to solve the Fokker---Planck equation. The method is applied to a typical problem in fiber dynamics and textile production. The numerical results are compared to explicit solutions and Monte-Carlo solutions.

Determining the effective resolution of advection schemes. Part I: Dispersion analysis

The effective resolution of a numerical scheme describes the smallest spatial scale (largest wavenumber) that is completely resolved by that scheme. Using dispersion relation analysis allows the effective resolution of a numerical scheme for the advection equation to be calculated. The advection equation is a fundamental building block of dynamical cores of atmospheric and ocean models, and this analysis provides an indication of the effective resolution of the numerical methods used by dynamical cores. Using a variety of finite-difference schemes, the effect on effective resolution of using explicit diffusion and hyper-diffusion terms is examined. The choice of order-of-accuracy, and the time-stepping of the numerical scheme is also investigated with regard to effective resolution. Finally, we apply this analysis to methods that are commonly used in dynamical cores of atmospheric general circulation models, namely semi-Lagrangian and finite-volume methods.

Semi-Lagrangian Methods for Parabolic Problems in Divergence Form

Semi-Lagrangian methods have traditionally been developed in the framework of hyperbolic equations, but several extensions of the semi-Lagrangian approach to diffusion and advection--diffusion problems have been proposed recently. These extensions are mostly based on probabilistic arguments and share the common feature of treating second-order operators in trace form, which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics. We propose here some basic ideas for treating second-order operators in divergence form. A general framework for constructing consistent schemes in one space dimension is presented, and a specific case of nonconservative discretization is discussed in detail and analyzed. Finally, an extension to (possibly nonlinear) problems in an arbitrary number of dimensions is proposed. Although the resulting discretization approach is only of first order in time, numerical results in a number of test cases highlight the advantages of these methods for applications to computational fluid dynamics and their superiority over to more standard low-order time discretization approaches for both the advection dominated and stiff diffusion regimes.

A Computational Realization of a Semi-Lagrangian Method for Solving the Advection Equation

A parallel implementation of a method of the semi-Lagrangian type for the advection equation on a hybrid architecture computation system is discussed. The difference scheme with variable stencil is constructed on the base of an integral equality between the neighboring time levels. The proposed approach allows one to avoid the Courant-Friedrichs-Lewy restriction on the relation between time step and mesh size. The theoretical results are confirmed by numerical experiments. Performance of a sequential algorithm and several parallel implementations with the OpenMP and CUDA technologies in the C language has been studied.

Vlasov on GPU (VOG project)

This work concerns the numerical simulation of the Vlasov-Poisson equation using semi-Lagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.

Comparison Between the Mean-Variance Optimal and the Mean-Quadratic-Variation Optimal Trading Strategies

ABSTRACTWe compare optimal liquidation policies in continuous time in the presence of trading impact using numerical solutions of Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs). In particular, we compare the time-consistent mean-quadratic-variation strategy with the time-inconsistent (pre-commitment) mean-variance strategy. We show that the two different risk measures lead to very different strategies and liquidation profiles. In terms of the optimal trading velocities, the mean-quadratic-variation strategy is much less sensitive to changes in asset price and varies more smoothly. In terms of the liquidation profiles, the mean-variance strategy is much more variable, although the mean liquidation profiles for the two strategies are surprisingly similar. On a numerical note, we show that using an interpolation scheme along a parametric curve in conjunction with the semi-Lagrangian method results in significantly better accuracy than standard axis-aligned linear interpolation. We also demonstrate how a scaled computational grid can improve solution accuracy.

Combustion Waves on the Point Set Surface

This paper introduces a combustion model of heat transfer and fuel consumption for the propagation of a fire front on a point cloud surface. The heat transfer includes the heat advection by the airflow as well as diffusion, chemical reaction, and heat loss to generate complex, but controllable heat flows with a designed airflow velocity. For the stable heat advection, we solve a semi-Lagrangian method on point samples using discrete exponential maps to trace the position from which the wind blows while preserving the geodesic distance. We also propose angular Voronoi weights for a discrete Laplace-Beltrami operator that shows better isotropic diffusion on the inhomogeneous distribution of point clouds than the cotangent or moving least-squares schemes. We demonstrate a diversity of burning scenarios by incorporating factors affecting the fire spreading such as buoyancy and object geometries in the airflow velocity fields, or by synthesizing patterns.

A conservative multi-tracer transport scheme for spectral-element spherical grids

Atmospheric models used for practical climate simulation must be capable handling the transport of hundreds of tracers. For computational efficiency conservative multi-tracer semi-Lagrangian type transport schemes are appropriate. Global models based on high-order Galerkin approach employ highly non-uniform spectral-element grids, and semi-Lagrangian transport is a challenge on those grids. A conservative semi-Lagrangian scheme (SPELT — SPectral-Element Lagrangian Transport) employing a multi-moment compact reconstruction procedure is developed for non-uniform quadrilateral grids. The scheme is based on a characteristic semi-Lagrangian method that avoids complex and expensive upstream area computations. The SPELT scheme has been implemented in the High-Order Method Modeling Environment (HOMME), which is based on a cubed-sphere grid with spectral-element spatial discretization. Additionally, we show the (strong) scalability and multi-tracer efficiency using several benchmark tests. The SPELT solution can be made monotonic (positivity preserving) by combining the flux-corrected transport algorithm, which is demonstrated on a uniform resolution grid. In particular, SPELT can be efficiently used for non-uniform grids and provides accurate and stable results for high-resolution meshes.

Towards an ultra efficient kinetic scheme. Part II: The high order case

In a recent paper we presented a new ultra efficient numerical method for solving kinetic equations of the Boltzmann type (Dimarco and Loubere, 2013) [17]. The key idea, on which the method relies, is to solve the collision part on a grid and then to solve exactly the transport part by following the characteristics backward in time. On the contrary to classical semi-Lagrangian methods one does not need to reconstruct the distribution function at each time step. This allows to tremendously reduce the computational cost and to perform efficient numerical simulations of kinetic equations up to the six-dimensional case without parallelization. However, the main drawback of the method developed was the loss of spatial accuracy close to the fluid limit. In the present work, we modify the scheme in such a way that it is able to preserve the high order spatial accuracy for extremely rarefied and fluid regimes. In particular, in the fluid limit, the method automatically degenerates into a high order method for the compressible Euler equations. Numerical examples are presented which validate the method, show the higher accuracy with respect to the previous approach and measure its efficiency with respect to well-known schemes (Direct Simulation Monte Carlo, Finite Volume, MUSCL, WENO).

A mass-conserving and multi-tracer efficient transport scheme in the online integrated Enviro-HIRLAM model

Abstract. In this paper a new advection scheme for the online coupled chemical–weather prediction model Enviro-HIRLAM is presented. The new scheme is based on the locally mass-conserving semi-Lagrangian method (LMCSL), where the original two-dimensional scheme has been extended to a fully three-dimensional version. This means that the three-dimensional semi-implicit semi-Lagrangian scheme which is currently used in Enviro-HIRLAM is largely unchanged. The HIRLAM model is a computationally efficient hydrostatic operational short-term numerical weather prediction model, which is used as the base for the online integrated Enviro-HIRLAM. The new scheme is shown to be efficient, mass conserving, and shape preserving, while only requiring minor alterations to the original code. It still retains the stability at long time steps, which the semi-Lagrangian schemes are known for, while handling the emissions of chemical species accurately. Several mass-conserving filters have been tested to assess the optimal balance of accuracy vs. efficiency.

Subspace fluid re-simulation

We present a new subspace integration method that is capable of efficiently adding and subtracting dynamics from an existing high-resolution fluid simulation. We show how to analyze the results of an existing high-resolution simulation, discover an efficient reduced approximation, and use it to quickly "re-simulate" novel variations of the original dynamics. Prior subspace methods have had difficulty re-simulating the original input dynamics because they lack efficient means of handling semi-Lagrangian advection methods. We show that multi-dimensional cubature schemes can be applied to this and other advection methods, such as MacCormack advection. The remaining pressure and diffusion stages can be written as a single matrix-vector multiply, so as with previous subspace methods, no matrix inversion is needed at runtime. We additionally propose a novel importance sampling-based fitting algorithm that asymptotically accelerates the precomputation stage, and show that the Iterated Orthogonal Projection method can be used to elegantly incorporate moving internal boundaries into a subspace simulation. In addition to efficiently producing variations of the original input, our method can produce novel, abstract fluid motions that we have not seen from any other solver.