In simulations of compressible flows, the conservative finite difference method (FDM) based on the nonlinear upwind schemes, e.g. WENO5, might violate free-stream preserving (FP), due to the loss of the geometric conservation law (GCL) identity when applied on the curvilinear grids. Although some techniques on FP have been proposed previously, no general rule is given for this issue. In this paper, by rearranging the upwind dissipation of the nonlinear schemes as a combination of sub-stencil reconstructions (taking WENO5 as an example), it can be proved that the upwind dissipation diminishes under the uniform flow condition if the metrics yield an identical value under the same schemes with these reconstructions, making the free-stream condition be preserved. According to this sufficient condition, the novel FP metrics are constructed for WENO5 and WENO7. By this means the original forms of these WENO schemes can be kept. In addition, the accuracy of these schemes can be retained as well with a simple accuracy compensation by replacing the central part fluxes with a high-order one. Various validations indicate that the present FP schemes retain the great capability to resolve the smooth regions accurately and capture the discontinuities robustly.
Magneto-hydrodynamics is one of the foremost models in plasma physics with applications in inertial confinement fusion, astrophysics and elsewhere. Advanced numerical methods are needed to get an insight into the complex physical phenomena. The classical Lagrangian methods are typically limited to the low orders of convergence and suffer from violation of the divergence-free condition for magnetic field or conservation of the invariants. This paper is the first part of a new series about high-order non-ideal magneto-hydrodynamics, where a multi-dimensional conservative Lagrangian method based …
The hybrid kinetic-ion fluid-electron plasma model is widely used to study challenging multi-scale problems in space and laboratory plasma physics. Here, a novel conservative scheme for this model employing implicit particle-in-cell techniques is extended to arbitrary coordinate systems via curvilinear maps from logical to physical space. The scheme features a fully non-linear electromagnetic formulation with a multi-rate time advance - including sub-cycling and orbit-averaging for the kinetic ions. By careful choice of compatible particle-based kinetic-ion and mesh-based fluid-electron discre…
In this paper, we develop a new free-stream preserving (FP) method for high-order upwind conservative finite-difference (FD) schemes on the curvilinear grids. This FP method is constrcuted by subtracting a reference cell-face flow state from each cell-center value in the local stencil of the original upwind conservative FD schemes, which effectively leads to a reformulated dissipation. It is convenient to implement this method, as it does not require to modify the original forms of the upwind schemes. In addition, the proposed method removes the constraint in the traditional FP conservative FD…
Applying high-order finite-difference schemes, like the extensively used linear-upwind or WENO schemes, to curvilinear grids can be problematic. The geometrically induced error from grid Jacobian and metrics evaluation can pollute the flow field, and degrade the accuracy or cause the simulation failure even when uniform flow imposed, i.e. free-stream preserving problem. In order to address this issue, a method for general linear-upwind and WENO schemes preserving free-stream on stationary curvilinear grids is proposed. Following Lax-Friedrichs splitting, this method rewrites the numerical flux…
In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton-Jacobi (H-J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme [10] for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme …
Efficient simulation of plasmas in various contexts often involves the use of meshes that conform to the intrinsic geometry of the system under consideration. We present here a description of a new magnetohydrodynamic code, Gamera (Grid Agnostic MHD for Extended Research Applications), designed to combine geometric flexibility with high-order spatial reconstruction and constrained transport to maintain the divergence-free magnetic field. Gamera carries on the legacy of its predecessor, the LFM (Lyon-Fedder-Mobarry), a research code whose use in space physics has spanned three decades. At the t…
Structure-Preserving High-Order Methods for the Compressible Euler Equations in Potential Temperature Formulation for Atmospheric Flows
Institute of Mathematics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学数学研究所）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；Institute for Atmospheric Physics, Johannes Gutenberg University Mainz（美因茨约翰内斯·古腾堡大学大气物理研究所）
We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with the Poisson equation, which is a {classical mean-field} primary model for collisional plasmas. Two subproblems, i.e. the Vlasov-Poisson problem and homogeneous Landau problem, are obtained through time-splitting methods, and treated separately by the Runge-Kutta Discontinuous Galerkin method and a conservative spectral method, respectively. To ensure conservation when projecting between the two different computing grids, a special conservation routine is designed to link the s…
The lattice Boltzmann (LB) method has gained much success in a variety of fields involving fluid flow and/or heat transfer. In this method, the bounce-back scheme is a popular boundary scheme for treating nonslip boundaries. However, this scheme leads to staircase-shaped boundaries for curved walls. Therefore many curved boundary schemes have been proposed, but mostly suffer from mass leakage at the curved boundaries. Several correction schemes have been suggested for simulating single-phase flows, but very few discussions or studies have been made for two-phase LB simulations with curved boun…
For more than half a century, most of the plasma scientists have encountered a violation of the conservation laws of charge, momentum, and energy whenever they have numerically solve the first-principle equations of kinetic plasmas, such as the relativistic Vlasov--Maxwell system. This fatal problem is brought by the fact that both the Vlasov and Maxwell equations are indirectly associated with the conservation laws by means of some mathematical manipulations. Here we propose a quadratic conservative scheme, which can strictly maintain the conservation laws by discretizing the relativistic Vla…
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this step, a one-dimensional curvilinear stencil is assembled for each edge to perform one-dimensional fast non-uniform WENO interpolation derived in this paper, which is much easier and faster than multi-dimensional interpolation. The second step is to compute the divergence of fluxes at each vertex from the fluxes at nearby edges and vertices by least square ap…
An implicit, conservative and asymptotic-preserving electrostatic particle-in-cell algorithm for arbitrarily magnetized plasmas in uniform magnetic fields
We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass …
A nonhydrostatic dynamical core has been developed by using the multi-moment finite volume method that ensures the rigorous numerical conservation. To represent the spherical geometry free of polar problems, the cubed-sphere grid is adopted. A fourth-order multi-moment discretization formulation is applied to solve the governing equations cast in the local curvilinear coordinates on each patch of cubed sphere through a gnomonic projection. In vertical direction, the height-based terrain-following grid is used to deal with the topography and a conservative finite difference scheme is adopted fo…
An adaptive numerical method for free surface flows passing rigidly mounted obstacles
Institute of Numerical Mathematics of the Russian Academy of Sciences（俄罗斯科学院计算数学研究所）；Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences（俄罗斯科学院凯尔迪什应用数学研究所）；Department of Mathematics, University of Houston（休斯顿大学数学系）；School of Earth, Energy & Environmental Sciences, Stanford University（斯坦福大学地球、能源与环境科学学院）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）
A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each vertex is computed directly from fluxes nearby by means of high order least-square. In order to capture discontinuities, numerical flux of high order accuracy is calculated on each edge and serves as supporting data of the least-square computation of the divergence. The high accuracy of the numerical flux depends on the high order WENO interpolation on each ed…
