Mimetic finite difference schemes for transport operators with divergence-free advective field and applications to plasma physics

TL;DR

This paper addresses discretizing transport operators with divergence-free advective fields on staggered grids to preserve discrete energy and the divergence theorem. It develops a mimetic finite difference scheme that rewrites the parallel gradient as a weighted average of the advective and divergence forms, with energy conservation guaranteed when $\alpha+\beta=1$. The method is validated on a 3D wave model and applied to electrostatic shear Alfvén waves in tokamak-like magnetic fields, demonstrating energy stability and convergence. The approach is well-suited to anisotropic plasma problems and can improve numerical fidelity in fluid/plasma codes.

Abstract

In wave propagation problems, finite difference methods implemented on staggered grids are commonly used to avoid checkerboard patterns and to improve accuracy in the approximation of short-wavelength components of the solutions. In this study, we develop a mimetic finite difference (MFD) method on staggered grids for transport operators with divergence-free advective field that is proven to be energy-preserving in wave problems. This method mimics some characteristics of the summation-by-parts (SBP) operators framework, in particular it preserves the divergence theorem at the discrete level. Its design is intended to be versatile and applicable to wave problems characterized by a divergence-free velocity. As an application, we consider the electrostatic shear Alfvén waves (SAWs), appearing in the modeling of plasmas. These waves are solved in a magnetic field configuration recalling that of a tokamak device. The study of the generalized eigenvalue problem associated with the SAWs shows the energy conservation of the discretization scheme, demonstrating the stability of the numerical solution.

Figures (10)

• Figure 1: Sketch of the two grids in a two dimensional setting where the red line represents the physical domain, the green dashed line the $p$-grid with the green points as nodes and the blue dashed line describes the $q$-grid with blue stars as nodes.
• Figure 2: Schematic 1D solution of the problem Eq. \ref{['eqn:model_decoup']} in a $x$-$t$ space with the application of the boundary conditions. In this case, the inflow of $r_1$ is $x_0$ and the outflow is $x_{N}$. For $r_2$ the inflow is $x_{N}$ where the information arrives from the outflow of $r_1$ and the outflow is $x_0$.
• Figure 3: The quiver plot of the $x$-$y$ component of the vector field $\textbf{b}$ with a space grid $N_x \times N_y \times N_z=16\times16\times16$. The $z$ component is independent of the position and is always equal to -1.
• Figure 4: Convergence study of the operator $\nabla_{\parallel}|_{pq}$ in Fig. \ref{['fig:convergence_gradpar_v2n']} and operator $\nabla_{\parallel} |_{qp}$ in Fig. \ref{['fig:convergence_gradpar_n2v']}, with $\textbf{b}$ given in Eq. \ref{['eqn:psi']} and uniform spatial resolution such that $h = \frac{\Delta x}{\Delta x_0} \times \frac{\Delta y}{\Delta y_0}\times \frac{\Delta z}{\Delta z_0}$ where $\Delta x_0$, $\Delta y_0$ and $\Delta z_0$ are the grid spacings for $N_x\times N_y\times N_y= 64\times 64\times 64$.
• Figure 5: Spectrum of the eigenvalue problem Eq. \ref{['eqn:eigen_probl']} for different values of the parameter $\alpha$ and $\beta$. On the right, the real part of the eigenvalues are equal to zero up to machine precision.

Theorems & Definitions (6)

• Remark 1
• Theorem 1
• proof
• Remark 2
• Corollary 1.1
• proof