Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlinear nonlocal Fokker-Planck equations

Abstract

This paper is concerned with structure-preserving numerical approximations for a class of nonlinear nonlocal Fokker-Planck equations, which admit a gradient flow structure and find application in diverse contexts. The solutions, representing density distributions, must be non-negative and satisfy a specific energy dissipation law. We design an arbitrary high-order discontinuous Galerkin (DG) method tailored for these model problems. Both semi-discrete and fully discrete schemes are shown to admit the energy dissipation law for non-negative numerical solutions. To ensure the preservation of positivity in cell averages at all time steps, we introduce a local flux correction applied to the DDG diffusive flux. Subsequently, a hybrid algorithm is presented, utilizing a positivity-preserving limiter, to generate positive and energy-dissipating solutions. Numerical examples are provided to showcase the high resolution of the numerical solutions and the verified properties of the DG schemes.

Figures (8)

• Figure 1: (A) displays the $L^2$ error of the porous media equation in Example 2 for $k=3$. (B) illustrates the numerical solution $\rho_h$ with $N = 64, k=3$ at $t=0$ and $32$, represented by green crosses and red circles, respectively. The equilibrium $\rho_\infty$ specified in \ref{['porousmedium_infty']} is shown by the blue solid curve.
• Figure 2: $N= 128, k = 3$. Consider the final time $t = 10$ in Example 3. (A) displays the cell average. (B) illustrates the numerical solution $\rho_h$ with a lift $\delta = 10^{-12}$. Both preserve positivity perfectly. (C) shows the time evolution of the energy $E_h$ and the difference between the numerical solution $\rho_h$ and the equilibrium $\rho_{\infty}$ in $L^2$ and $L^\infty$ norms.
• Figure 3: $N = 128, k = 2$. Consider the final time $t = 30$ in Example 4.
• Figure 4: $N = 128, k = 2$. The time evolution of the energy $E_h$ in Example 4 for two initial data \ref{['ini_a_2']} and \ref{['ini_a_3']} in green dashed curve and blue solid curve, respectively.
• Figure 5: Consider $N = 128$ and $k = 2$ for the final time $t = 600$ in Example 5. The numerical solution $\rho_h$ and the flux $q_h(t,x)$ are depicted by the blue solid curve and red dashed curve, respectively. Figure (D) provides a zoomed-in view of the numerical flux $q_h(t,x)$ within the circled spacial domain $[0.2,1.4]$. It is noticeable that the flux reaches a constant value.

Theorems & Definitions (12)

• Theorem 2.1
• proof
• Remark 2.1
• Theorem 3.1
• proof
• Remark 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof