An Assessment of Solvers for Algebraically Stabilized Discretizations of Convection-Diffusion-Reaction Equations

TL;DR

This work analyzes solver efficiency for algebraically stabilized discretizations of 3D convection-diffusion-reaction equations, comparing flux-corrected transport with Zalesak's limiter against monolithic convex limiting under CN and SSP time stepping. By deriving low- and high-order schemes and implementing fixed-point nonlinear solvers with various linear algebra backends in PETSc, it benchmarks CPU time, convergence, and accuracy on new 3D test problems for both time-dependent and stationary cases. Key findings show that the choice of limiter and solver (notably FGMRES with appropriate preconditioners) profoundly affects performance, with implicit schemes often outperforming explicit ones in 3D. The results provide practical guidance on selecting limiter–solver pairings to balance accuracy and computational cost in large-scale AFC simulations, and contribute new 3D benchmarks for the field.

Abstract

We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and $\mathbb{P}_1$ or $\mathbb{Q}_1$ finite elements. Time integration is performed using the Crank-Nicolson method or an explicit strong stability preserving Runge-Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time-dependent and stationary convection-diffusion-reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

Figures (10)

• Figure 1: Example \ref{['ex:rotating_shapes']}: Initial condition.
• Figure 2: Example \ref{['ex:rotating_shapes']}: $u_0(\boldsymbol x)$ and the resulting $u(T, \boldsymbol x)$ for $\boldsymbol x$ in the plane given by $z=0.5$.
• Figure 3: Example \ref{['ex:rotating_shapes']}: Time evolution of the numerical solution and the true solution at the point $(0.5, 0.25, 0.5)^T$ (i.e., the initial center of the cube).
• Figure 4: Example \ref{['ex:concentration_species2']}: Time evolution of $u$ at $(1,7/16,9/16)^\text{T}$. The MC limiter produces essentially the same result for both time discretizations.
• Figure 5: Example \ref{['ex:non_constant_convection2']}: Isosurface for $u=0.05$ of the solution computed using the MU limiter for the level 5, and sketch of the coarsest grid (level 0).

Theorems & Definitions (2)

• Remark 1
• Remark 2