A compact fourth-order doubly conservative active flux method with maximum-principle-preserving limiting for degenerate convection--diffusion equations

Abstract

The active flux (AF) method is a compact high-order finite volume method originally proposed for solving hyperbolic conservation laws, that evolves cell averages together with point values shared by neighboring cells. This paper develops a compact fourth-order AF method for scalar degenerate convection--diffusion equations on Cartesian meshes. Fourth-order accuracy is achieved without introducing additional unknowns relative to the standard third-order AF method. For convection, a downwind point value is incorporated into the biased point-value stencil. For diffusion, the proposed method directly discretizes the diffusion potential using compact fourth-order operators, thereby avoiding auxiliary gradient variables, hyperbolic reformulations, pseudo-time stepping, or inner iterations in the existing high-order AF methods. To deal with degeneracy, the diffusion contribution to the cell-average evolution is written in a doubly conservative form, while the point values are evolved using compact finite-difference operators. To enforce the discrete maximum principle, a monolithic convex limiting is constructed, which simultaneously limits convection fluxes and diffusion potentials by blending them with their low-order maximum-principle-preserving counterparts, thus the conservation for convection and double conservation for diffusion are maintained. The point values are projected onto the admissible interval, and an optional smoothness-indicator-based blending is employed to control oscillations. For the 1D linear problem, Fourier analysis establishes fourth-order spatial accuracy, while von Neumann analysis yields larger stable CFL limits than the fourth-order $P^3$ local discontinuous Galerkin method for the explicit Runge--Kutta methods considered. Numerical experiments demonstrate the fourth-order convergence, preservation of bounds, and accurate resolution of shocks and sharp wave fronts.