An anisotropic nonlinear stabilization for finite element approximation of Vlasov-Poisson equations

TL;DR

This work develops a high-order continuous finite element method for the Vlasov-Poisson equations using $\mathbb{Q}_k$-space spatial discretization and explicit time integration, augmented by a novel anisotropic residual-based artificial viscosity to suppress numerical oscillations. The nonlinear RV framework combines an $L^2$-projected residual with a localized, anisotropic viscosity that activates only near non-smooth regions, enabling fourth-order accuracy in smooth regimes while stabilizing shocks. The method conserves mass and achieves robust performance on classic 1D1V benchmarks (Landau damping, two-stream, bump-on-tail), with results showing high-order convergence for smooth data and strong stabilization in nonlinear regimes; anisotropic stabilization outperforms isotropic approaches under large mesh anisotropy. The approach provides a matrix-free, high-order alternative to PIC and traditional FE methods, with potential extensions to higher dimensions and Vlasov–Maxwell systems, albeit with remaining challenges in positivity preservation and energy conservation.

Abstract

We introduce a high-order finite element method for approximating the Vlasov-Poisson equations. This approach employs continuous Lagrange polynomials in space and explicit Runge-Kutta schemes for time discretization. To stabilize the numerical oscillations inherent in the scheme, a new anisotropic nonlinear artificial viscosity method is introduced. Numerical results demonstrate that this method achieves optimal convergence order with respect to both the polynomial space and time integration. The method is validated using classic benchmark problems for the Vlasov-Poisson equations, including Landau damping, two-stream instability, and bump-on-tail instability in a two-dimensional phase space.

Figures (12)

• Figure 1: Quadrangulation of the domain around the point ${\pmb{x}}_{i,j}:=(x_i, v_j)$.
• Figure 2: Local index for the nodes in ${\mathcal{V}}_x$, ${\mathcal{V}}_v$, and ${\mathcal{V}}_h$, when the polynomial degree is 1, 2, and 3.
• Figure 3: Landau damping. Linear damping, evolution of $E_e$ for $\alpha=0.01$. The red line represents the analytical damping rate. The RV solutions using ${\mathbb Q}_1$ elements with varying degrees of freedom (a) and different polynomial degrees with the same degree of freedom distribution (b).
• Figure 4: Landau damping. Strong damping, the evolution of $E_e$ for $\alpha = 0.5$. The distribution of nodes is $N_x\times N_v=49\times97$. Comparison of the Galerkin solution and stabilized RV solutions for different polynomial spaces.
• Figure 5: Two-stream instability: convergence orders of the methods. The RV method does not decrease the accuracy of high-order solutions. Run until $t=5$ and then back to $t=10$.

Theorems & Definitions (10)

• Proposition 1: Conservation
• proof
• remark 1
• Proposition 2
• proof
• remark 2: Isotropic viscosity
• Proposition 3
• proof
• remark 3
• remark 4: Extending the method to higher dimensions