An optimization-based positivity-preserving limiter in semi-implicit discontinuous Galerkin schemes solving Fokker-Planck equations

TL;DR

The paper addresses enforcing positivity in high-order semi-implicit DG schemes for the Rosenbluth–Fokker–Planck equation by developing an optimization-based two-stage postprocessing that preserves mass and high-order accuracy. The first stage solves a constrained L^2 minimization to enforce bounds on cell averages via Douglas–Rachford splitting, while the second stage applies a Zhang–Shu type limiter to fix pointwise undershoot/overshoot. The approach achieves conservation, positivity, and SPD-consistent diffusion in challenging implicit simulations, with O(N) per-iteration cost and good parallelizability demonstrated through multiple FP-related test cases. This contributes a robust, scalable method for high-fidelity kinetic simulations where positivity is critical for stability and physical fidelity in fully nonlinear or self-consistent settings.

Abstract

For high-order accurate schemes such as discontinuous Galerkin (DG) methods solving Fokker-Planck equations, it is desired to efficiently enforce positivity without losing conservation and high-order accuracy, especially for implicit time discretizations. We consider an optimization-based positivity-preserving limiter for enforcing positivity of cell averages of DG solutions in a semi-implicit time discretization scheme, so that the point values can be easily enforced to be positive by a simple scaling limiter on the DG polynomial in each cell. The optimization can be efficiently solved by a first-order splitting method with nearly optimal parameters, which has an $\mathcal{O}(N)$ computational complexity and is flexible for parallel computation. Numerical tests are shown on some representative examples to demonstrate the performance of the proposed method.

Figures (10)

• Figure 1: Schematic postprocessing procedure for 1D piecewise linear polynomial. The $m$ and $M$ are desired lower and upper bounds. Left: the original DG solution (red line) with out-of-bound cell averages (red dashed line). Middle: applying an optimization based limiter to modify cell averages. The modified cell averages (blue dashed line) are in $[m,M]$ but the modified DG solution (blue line) may still contains out-of-bound point values. Right: in each cell, apply a limiter to each DG polynomial to eliminate undershoot and/or overshoot, which gives a bound-preserving solution (green line).
• Figure 2: Plot the distribution function $f$ and its value along the diagonal $\{x=y\}$ at time $t^\mathrm{end} = 20$. From left to right: simulation results associated with $\mathds{P}^2$ and $\mathds{P}^3$ scheme of mesh resolution $\Delta x = 1/128$.
• Figure 3: Non-identity diffusion test. Snapshot of covariances trajectories. From top to bottom: simulation results associated with $\mathds{P}^2$ and $\mathds{P}^3$ schemes. The black dashed line denotes the analytic solution.
• Figure 4: Plotting the entries in the coefficient matrix $\mathbfsf{D}$ and their ratios on a $128$-by-$128$ grid. The first three sub-figures from left to right: the $D_{b, 00}^M$, $D_{b, 11}^M$, and $D_{b, 01}^M$. The last two sub-figures from left to right: the ratio of $D_{b, 01}^M$ to $D_{b, 00}^M$ and $D_{b, 11}^M$.
• Figure 5: The $\mathds{P}^2$ scheme. Snapshot of the discrete distribution function at time $t^\mathrm{end} = 20$. From left to right: simulation results associated with the inverse collision time-scale $\varepsilon^{-1} = 10^1$, $10^2$, and $10^3$.

Theorems & Definitions (6)

• Remark 1
• Lemma 1
• proof
• Remark 2
• Remark 3
• Remark 4