Lax-Wendroff Flux Reconstruction for advection-diffusion equations with error-based time stepping

TL;DR

The paper develops a novel extension of Lax-Wendroff Flux Reconstruction to second-order parabolic PDEs on curvilinear meshes by integrating the BR1 first-order reduction and an embedded error-based time-stepping strategy. Time advancement relies on time-averaged fluxes, enabling a single-stage, high-order update that preserves free-stream states on curved geometries. The method demonstrates optimal convergence for scalar and Navier–Stokes problems and validates against canonical CFD benchmarks, including lid-driven cavity, transonic airfoil, and flow past a cylinder, with credible agreement to reference data. This work broadens the applicability of high-order FR/DG methods to diffusion-dominated and curved-domain problems, offering potential gains in accuracy and efficiency on modern hardware.

Abstract

This work introduces an extension of the high order, single stage Lax-Wendroff Flux Reconstruction (LWFR) of Babbar et al., JCP (2022) to solve second order time-dependent partial differential equations in conservative form on curvilinear meshes. The method uses BR1 scheme to reduce the system to first order so that the earlier LWFR scheme can be applied. The work makes use of the embedded error-based time stepping introduced in Babbar, Chandrashekar (2024) which becomes particularly relevant in the absence of CFL stability limit for parabolic equations. The scheme is verified to show optimal order convergence and validated with transonic flow over airfoil and unsteady flow over cylinder.

Figures (8)

• Figure 1: Illustration of reference map, solution point projections, reference and physical normals
• Figure 2: Convergence analysis for scalar advection-diffusion equation with $\boldsymbol{a} = (1.5,1)$ and coefficient (a) $\nu = 5 \times 10^{-2}$ (b) $\nu = 10^{-12}$
• Figure 3: Convergence analysis with non-periodic boundary conditions. (a) Eriksson-Johnson test (Section 4 of Ellis2016) and (b) Navier-Stokes equations with manufactured solution
• Figure 4: Velocity profiles of lid driven cavity test. (a) $v_y$ cut at $y=0.5$ (b) $v_x$ cut at $x=0.5$.
• Figure 5: Transonic flow over a NACA-0012 airfoil with $M_\infty = 0.8$ solved on a mesh with 728 elements using solution polynomial degree $N=4$. (a) Mesh (b) Mach number contour.

Theorems & Definitions (2)

• definition 1: Covariant basis
• definition 2: Contravariant basis