An efficient cell-centered diffusion scheme for quadrilateral grids

A new algorithm for solution of diffusion equations in two dimensions on structured quadrilateral grids is proposed. The algorithm is based on a semi-implicit method for the time discretization and has a nine-point local stencil in space. Our scheme is fast, quite accurate and demonstrates good spatial convergence. The presented numerical tests show that it is well suited for hydrocodes with cell-centered principal variables.     

An unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations

Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations in more than one space dimension must either confront the challenge of controlling errors in the discrete divergence of the magnetic field, or else be faced with nonlinear numerical instabilities. One approach for controlling the discrete divergence is through a so-called constrained transport method, which is based on first predicting a magnetic field through a standard finite volume solver, and then correcting this field through the appropriate use of a magnetic vector potential. In this work we develop a constrained transport method for the 3D ideal MHD equations that is based on a high-resolution wave propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme developed by Rossmanith [J.A. Rossmanith, An unstaggered, high-resolution constrained transport method for magnetohydrodynamic flows, SIAM J. Sci. Comput. 28 (2006) 1766], and is based on the high-resolution wave propagation method of Langseth and LeVeque [J.O. Langseth, R.J. LeVeque, A wave propagation method for threedimensional hyperbolic conservation laws, J. Comput. Phys. 165 (2000) 126]. In particular, in our extension we take great care to maintain the three most important properties of the 2D scheme: (1) all quantities, including all components of the magnetic field and magnetic potential, are treated as cell-centered; (2) we develop a high-resolution wave propagation scheme for evolving the magnetic potential; and (3) we develop a wave limiting approach that is applied during the vector potential evolution, which controls unphysical oscillations in the magnetic field. One of the key numerical difficulties that is novel to 3D is that the transport equation that must be solved for the magnetic vector potential is only weakly hyperbolic. In presenting our numerical algorithm we describe how to numerically handle this problem of weak hyperbolicity, as well as how to choose an appropriate gauge condition. The resulting scheme is applied to several numerical test cases.     

A fourth-order divergence-free method for MHD flows

This paper extends our previous third-order method [S. Li, High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys. 227 (2008) 7368–7393] to the fourth-order. Central finite-volume schemes on overlapping grid are used for both the volume-averaged variables and the face-averaged magnetic field. The magnetic field at the cell boundaries falls within the dual grid and is naturally continuous so that our method eliminates the instability triggered by the discontinuity in the normal component of the magnetic field. Our fourth-order scheme has much smaller numerical dissipation than the third-order scheme. The divergence-free condition of the magnetic field is preserved by our fourth-order divergence-free reconstruction and the constrained transport method. Numerical examples show that the divergence-free condition is essential to the accuracy of the method when a limiter is used in the reconstruction. The high-order, low-dissipation, and divergence-free properties of this method make it an ideal tool for direct magneto-hydrodynamic turbulence simulations.     

A phase-space particles motion scheme for electron kineticsimulation

A phase-space particle motion scheme (PSPMS) for electron kinetic simulation is proposed. This scheme is based on the convenient representation of electron motion in phase space using the particles method. PSPMS offers the possibility of calculating the electron energy distribution function (EEDF) much faster than by conventional methods. PSPMS electron calculation is shown to match convective scheme and Monte Carlo simulations of swarms in the uniform electric field and the cathode fall (CF). PSPMS may be used for a self-consistent model of low-pressure glow discharge construction     

An unsplit staggered mesh scheme for multidimensional magnetohydrodynamics

We introduce an unsplit staggered mesh scheme (USM) for multidimensional magnetohydrodynamics (MHD) that uses a constrained transport (CT) method with high-order Godunov fluxes and incorporates a new data reconstruction–evolution algorithm for second-order MHD interface states. In this new algorithm, the USM scheme includes so-called “multidimensional MHD terms”, proportional to $?·\mathbf{B}$, in a dimensionally-unsplit way in a single update. This data reconstruction–evolution step, extended from the corner transport upwind (CTU) approach of Colella, maintains in-plane dynamics very well, as shown by the advection of a very weak magnetic field loop in 2D. This data reconstruction–evolution algorithm is also of advantage in its consistency and simplicity when extended to 3D. The scheme maintains the $?·\mathbf{B}=0$ constraint by solving a set of discrete induction equations using the standard CT approach, where the accuracy of the computed electric field directly influences the quality of the magnetic field solution. We address the lack of proper dissipative behavior in the simple electric field averaging scheme and present a new modified electric field construction (MEC) that includes multidimensional derivative information and enhances solution accuracy. A series of comparison studies demonstrates the excellent performance of the full USM–MEC scheme for many stringent multidimensional MHD test problems chosen from the literature. The scheme is implemented and currently freely available in the University of Chicago ASC FLASH Center’s FLASH3 release.     

A nominally second-order accurate finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids

In this paper, we describe a second-order accurate cell-centered finite volume method for solving anisotropic diffusion on two-dimensional unstructured grids. The resulting numerical scheme, named CCLAD (Cell-Centered LAgrangian Diffusion), is characterized by a local stencil and cell-centered unknowns. It is devoted to the resolution of diffusion equation on distorted grids in the context of Lagrangian hydrodynamics wherein a strong coupling occurs between gas dynamics and diffusion. The space discretization relies on the introduction of two half-edge normal fluxes and two half-edge temperatures per cell interface using the partition of each cell into sub-cells. For each cell, the two half-edge normal fluxes attached to a node are expressed in terms of the half-edge temperatures impinging at this node and the cell-centered temperature. This local flux approximation can be derived through the use of either a sub-cell variational formulation or a finite difference approximation, leading to the two variants CCLADS and CCLADNS. The elimination of the half-edge temperatures is performed locally at each node by solving a small linear system which is obtained by enforcing the continuity condition of the normal heat flux across sub-cell interface impinging at the node. The accuracy and the robustness of the present scheme is assessed by means of various numerical test cases.     

A Monte Carlo scheme for tracking filling fronts

A Monte Carlo (MC) scheme for tracking filling fronts is developed. The test problem of tracking fluid injected into a thin mold is considered. The MC scheme iteratively redistributes the volume entering the mold among the cells of a spatial discretization. The transition probabilities used in the MC scheme, which determine how the fluid volume is redistributed, are derived from a discrete representation of the governing steady-state pressure equation. Analysis shows that the MC steps are equivalent to an iterative solution of the discrete equations. Further, it is shown that the MC scheme can be reconfigured into the form of a standard Lattice Boltzmann Method (LBM). Results show that the proposed MC scheme is accurate, does not require an explicit field calculation of the fluid pressure field, and, when compared with existing numerical filling algorithms, exhibits computation times over 1000 times faster.     

Stabilization of cell-centered compressible Lagrangian methods using subzonal entropy

This work deals with the stabilization of early stages of Lagrangian calculations of compressible gas dynamics in the context of cell-centered discretization. We describe a totally conservative algorithm based on an entropy analysis of the distortion of a Lagrangian mesh. It prevents the tangling of the mesh, while remaining consistent and conservative in mass, momentum, and total energy. The method described can be applied to any cell-centered Lagrangian scheme. In this article, we detail the extension to the cell-centered Glace scheme published in Carré et al. (2009) [G. Carré, S. Del Pino, B. Després, E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, J. Comput. Phys. 228 (2009) 5160–5183]. Numerical tests are proposed to assess the accuracy and robustness.     

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

We present a new cell-centered multi-material arbitrary Lagrangian–Eulerian (ALE) scheme to solve the compressible gas dynamics equations on two-dimensional unstructured grid. Our ALE method is of the explicit time-marching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cell-centered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cell-centered remap phase which consists of interpolating conservatively the Lagrangian solution onto the rezoned grid. The multi-material modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the Cauchy–Green tensor over the Lagrangian phase. Regarding the cell-centered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept face-based method whereas the second one is a cell-intersection-based method. Our multi-material ALE methodology is assessed through several demanding two-dimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.