Abstract:
We present an extension of the multi-moment advection scheme (Minoshima et al., 2011, J. Comput. Phys.) to the three-dimensional case, for full electromagnetic Vlasov simulations of magnetized plasma. The scheme treats not only point values of a profile but also its zeroth to second order piecewise moments as dependent variables, and advances them on the basis of their governing equations. Similar to the two-dimensional scheme, the three-dimensional scheme can accurately solve the solid body rotation problem of a gaussian profile with little numerical dispersion or diffusion. This is a very important property for Vlasov simulations of magnetized plasma. We apply the scheme to electromagnetic Vlasov simulations. Propagation of linear waves and nonlinear evolution of the electron temperature anisotropy instability are successfully simulated with a good accuracy of the energy conservation.

Abstract:
Explicit numerical computations of super-fast differentially rotating disks are subject to the time-step constraint imposed by the Courant condition. When the bulk orbital velocity largely exceeds any other wave speed the time step is considerably reduced and a large number of steps may be necessary to complete the computation. We present a robust numerical scheme to overcome the Courant limitation by extending the algorithm previously known as FARGO (Fast Advection in Rotating Gaseous Objects) to the equations of magnetohydrodynamics (MHD). The proposed scheme conserves total angular momentum and energy to machine precision and works in Cartesian, cylindrical, or spherical coordinates. The algorithm is implemented in the PLUTO code for astrophysical gasdynamics and is suitable for local or global simulations of accretion or proto-planetary disk models. By decomposing the total velocity into an average azimuthal contribution and a residual term, the algorithm solves the MHD equations through a linear transport step in the orbital direction and a standard nonlinear solver applied to the MHD equations written in terms of the residual velocity. Since the former step is not subject to any stability restriction, the Courant condition is computed only in terms of the residual velocity, leading to substantially larger time steps. The magnetic field is advanced in time using the constrained transport method in order to preserve the divergence-free condition. Conservation of total energy and angular momentum is enforced at the discrete level by properly expressing the source terms in terms of upwind fluxes available during the standard solver. Our results show that applications of the proposed orbital-advection scheme to problems of astrophysical relevance provides, at reduced numerical cost, equally accurate and less dissipative results than standard time-marching schemes.

Abstract:
We present a forward semi-Lagrangian numerical method for systems of transport equations able to advect smooth and discontinuous fields with high-order accuracy. The numerical scheme is composed of an integration of the transport equations along the trajectory of material elements in a moving grid and a reconstruction of the fields in a reference regular mesh using a non-linear mapping and adaptive moment-preserving interpolations. The non-linear mapping allows for the arbitrary deformation of material elements. Additionally, interpolations can represent discontinuous fields using adaptive-order interpolation near jumps detected with a slope-limiter function. Due to the large number of operations during the interpolations, a serial implementation of this scheme is computationally expensive. The scheme has been accelerated in many-core parallel architectures using a thread per grid node and parallel data gathers. We present a series of tests that prove the scheme to be an attractive option for simulating advection equations in multi-dimensions with high accuracy.

Abstract:
A new scheme for numerical integration of the 1D2V relativistic Vlasov-Maxwell system is proposed. Assuming that all particles in a cell of the phase space move with the same velocity as that of the particle located at the center of the cell at the beginning of each time step, we successfully integrate the system with no artificial loss of particles. Furthermore, splitting the equations into advection and interaction parts, the method conserves the sum of the kinetic energy of particles and the electromagnetic energy. Three test problems, the gyration of particles, the Weibel instability, and the wakefield acceleration, are solved by using our scheme. We confirm that our scheme can reproduce analytical results of the problems. Though we deal with the 1D2V relativistic Vlasov-Maxwell system, our method can be applied to the 2D3V and 3D3V cases.

Abstract:
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 by advecting the `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 not subject to time step restriction to maintain stability. Recently [Y. G\"u\c{c}l\"u and W.N.G. Hitchon, 2012] 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 simple and efficient for constant advection problems. Here the CS is applied to the solution of the Vlasov-Poisson system: 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: 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.

Abstract:
By using the moment algebra of the Vlasov kinetic equation, we characterize the integrable Bloch-Iserles system on symmetric matrices (arXiv:math-ph/0512093) as a geodesic flow on the Jacobi group. We analyze the corresponding Lie-Poisson structure by presenting a momentum map, which both untangles the bracket structure and produces particle-type solutions that are inherited from the Vlasov-like interpretation. Moreover, we show how the Vlasov moments associated to Bloch-Iserles dynamics correspond to particular subgroup inclusions into a group central extension (first discovered in arXiv:math/0410100), which in turn underlies Vlasov kinetic theory. In the most general case of Bloch-Iserles dynamics, a generalization of the Jacobi group also emerges naturally.

Abstract:
Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles. Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These solutions are shown to form one of the legs of a dual pair of momentum maps. The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

Abstract:
According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.

Abstract:
We present a new Vlasov code for collisionless plasmas in the nonrelativistic regime. A Darwin approximation is used for suppressing electromagnetic vacuum modes. The spatial integration is based on an extension of the flux-conservative scheme, introduced by Filbet et al. [J. Comp. Phys. Vol. 172 (2001) 166]. Performance and accuracy is demonstrated by comparing it to a standard finite differences scheme for two test cases, including a Harris sheet magnetic reconnection scenario. This comparison suggests that the presented scheme is a promising alternative to finite difference schemes.

Abstract:
A high order splitting scheme for the advection diffusion equation of pollutants is proposed in this paper. The multidimensional advection diffusion equation is splitted into several one dimensional equations that are solved by the scheme. Only three spatial grid points are needed in each direction and the scheme has fourth order spatial accuracy. Several typically pure advection and advection diffusion problems are simulated. Numerical results show that the accuracy of the scheme is much higher than that of the classical schemes and the scheme can be efficiently solved with little programming effort.