A fourth-order active flux method for parabolic problems with application to porous medium equation

TL;DR

The paper develops a fourth-order active flux method for parabolic problems, notably the porous medium equation $u_t=\nabla\cdot(A\nabla u)$, by formulating a degenerate first-order system with auxiliary variables to represent derivatives. The method preserves conservation of cell averages while updating interface point values with a fourth-order central finite-difference stencil and uses SSP-RK time integration, yielding a maximum stability CFL around $0.27$ in 1D (and $0.15$ in 2D). A positivity-preserving limiter blends high-order AF fluxes with low-order PP fluxes to maintain nonnegativity of the PME solution, with post-processing ensuring nonnegative point values. Comprehensive 1D and 2D tests confirm fourth-order accuracy, stable time stepping, and robust nonnegativity, underscoring the method’s potential for high-fidelity parabolic simulations and PME applications.

Abstract

The active flux (AF) method is a compact high-order finite volume method originally proposed for solving hyperbolic conservation laws, in which cell averages and point values at cell interfaces are evolved simultaneously. This paper develops a fourth-order AF method for one- and two-dimensional parabolic problems, employing the explicit strong-stability-preserving Runge-Kutta (SSP-RK) method for time integration. The proposed method is built on a degenerate first-order system with auxiliary variables representing the derivatives of the primal variable, similar to local discontinuous Galerkin (LDG) methods, which avoids introducing pseudo-time or performing iterations within a physical time step in the existing hyperbolic formulations. The evolution of cell averages follows the standard finite volume method, ensuring conservation, while the point values of both the primal and auxiliary variables are updated using fourth-order central finite difference operators. A discrete Fourier analysis confirms the fourth-order accuracy in 1D. With the third-order SSP-RK method, the maximum CFL number for stability is $0.27$ in 1D, as obtained by von Neumann analysis, larger than that of LDG methods. The proposed method is further applied to the porous medium equation, and positivity-preserving limitings are incorporated to guarantee the non-negativity of the numerical solutions. Several numerical experiments validate the theoretical results and efficacy of the method.

Figures (14)

• Figure 1: The modulus of the two eigenvalues of $\bm{\mathcal{G}}$. Left: $C_{\texttt{CFL}} = 0.27$, right: $C_{\texttt{CFL}} = 0.28$.
• Figure 2: The DoFs of $u$: cell average (circle), face-centered values (squares), values at corners (dots). Note that the cell-centered point value $u_{i,j}$ (cross) is used in constructing the scheme, but does not belong to the DoFs.
• Figure 3: The stability region of the AF scheme using the SSP-RK3 with different values of $\theta$ (dot means stable). From left to right: $\theta = 0$, $\pi/12$, $\pi/6$, $\pi/4$.
• Figure 4: Example \ref{['ex:accuracy']}. The errors and convergence rates in 1D (left) and 2D (right).
• Figure 5: Example \ref{['ex:2d_ring']}. The numerical solutions obtained by our AF scheme at $T=0$, $0.1$, $0.2$, $0.5$, $1.0$, $5.0$ (from left to right, top to bottom).

Theorems & Definitions (22)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.1