García-Archilla et al. integrate POD-ROM for semilinear reaction-diffusion with BDF-q (q≤5) time stepping, proving and numerically confirming optimal order-q temporal convergence

The authors prove that a fully discrete POD-ROM for a semilinear reaction-diffusion equation using BDF-q (1≤q≤5) time stepping achieves optimal order-q convergence in time, with pointwise-in-time error bounds. Prior POD error analysis mostly treated implicit Euler or at most second-order time integrators; this work extends high-order BDF temporal error analysis to reduced-order models. A priori error bounds are given through a stability theorem (Theorem 1), a consistency theorem (Theorem 2), and convergence theorems (Theorems 3 and 4), with bounds combining eigenvalue tail, temporal error, and spatial discretization error; numerical experiments on the Brusselator system report slopes of about -1.17, -1.96, -2.98, -3.98, -4.93 for q=1 to 5.

The authors show that any BDF-q scheme can be written as a linear combination of first-order difference quotients, and use this to include difference quotients in the snapshot set and thereby control the projection error of the time derivative onto the reduced-order space. This algebraic identity had previously been used only for BDF2; the paper extends it to q≤5, enabling the reduced-order error analysis of higher-order time schemes. The consistency proof gives the explicit decomposition ∂q u_h^n = (1/Δt)Σ α_j D u_h^{n-j} and uses the relation between snapshot difference quotients and the eigenvalue tail (equation 18) to obtain bound (53).

The authors construct the POD basis using H1_0 projections, making the error bound optimal in terms of the eigenvalue tail. Compared with L2 projections, H1_0 projections yield bounds optimal in terms of the eigenvalue tail; the paper notes the analysis also works for L2 projections, where the difference-quotient linear-combination step is not needed. The final bound (55) contains the eigenvalue tail Σ_{k>r} λ_k, the temporal term (Δt)^{2q}, and the spatial term h^{k+1}; the authors describe it as optimal in eigenvalue tail, temporal error, and spatial discretization error.

Numerical experiments show that as the reduced dimension r grows, the error becomes dominated by the time integrator, so higher-order BDF is needed to reach optimal accuracy without increasing the number of time steps. This gives concrete guidance for choosing the time-scheme order in practice: second order for r=26, third order for r=32, fourth order for r=42, and fifth order for r=50. Maximum L2 errors for q=1 to 5 are compared at fixed Δt=T/1024 (Table 3), and Newton iteration counts are reported between 2 and 4 with an average of 2.97.

The result is aimed at numerical-analysis researchers and scientific-computing practitioners using POD-ROM for semilinear reaction-diffusion-type parabolic problems, in settings where snapshots come from sufficiently accurate offline finite element approximations and the time step satisfies condition (24). It gives high-order BDF time schemes a predictable order-q convergence guarantee in reduced-order models, supporting fewer time steps at a given accuracy; the framework can also inform high-order temporal discretization of other parabolic reduced-order models.

The analysis assumes a globally Lipschitz nonlinearity and snapshots free of temporal error; in practice offline snapshot temporal error adds to the error bound, and its magnitude needs case-by-case assessment. The BDF6 analysis is explicitly left for separate treatment, and q>5 is not covered. Numerical validation is concentrated on the Brusselator system with specific parameters (ν=0.002, quadratic finite elements, 51200 degrees of freedom), so behavior for other equations and parameters remains open. In addition, the time step needed to reach the theoretical rate decreases as q grows, so the practical trade-off between order and step size deserves further observation.