Nonuniform Temporal Mesh Methods

• Nonuniform temporal mesh is a time discretization strategy that uses variable subinterval sizes to precisely capture singular behaviors and heterogeneous dynamics.
• Graded and adaptive construction methods adjust time-steps based on solution features, irregular data sampling, and physical constraints to optimize numerical performance.
• These meshes improve stability and accuracy in finite difference schemes, fractional time discretizations, and neural surrogate models while reducing computational costs.

A nonuniform temporal mesh refers to a subdivision of a time interval into subintervals whose sizes vary, as opposed to the classical uniform time mesh of constant step size. Nonuniform temporal meshes arise in a broad spectrum of computational methodologies, including the numerical solution of time-dependent PDEs (with or without fractional derivatives), large-scale spatio-temporal modeling, surrogate modeling using neural architectures, and k-space pseudospectral solvers for wave propagation. The use of nonuniform meshes is frequently motivated by solution singularities, heterogeneous physics, adaptivity requirements, or the need to handle irregular or sparse data sampling in time.

1. Construction Paradigms for Nonuniform Temporal Meshes

Nonuniform temporal grids are most commonly constructed via either a graded deterministically-defined mesh, an adaptive mesh informed by solution features, or as an arbitrary collection of time instances as dictated by irregular data sampling or physical model constraints.

Graded Meshes

A standard graded mesh is given by

$$t_n = T \left(\frac{n}{N}\right)^{r},\qquad n=0,\ldots,N,$$

where $r\geq 1$ is a grading parameter. For $r=1$, the mesh is uniform; $r>1$ clusters nodes close to $t=0$, which is essential in resolving initial singularities in subdiffusive and related problems (Qiu et al., 11 Jan 2026, Liao et al., 2018).

Adaptive/Arbitrary Meshes

Adaptive approaches select the next time-step $\tau_\ell = t_{\ell+1} - t_\ell$ based on real-time solution monitors. For singularity-resolving applications, strategies include:

• Arc-length or curvature monitors on solution extrema or derivatives.
• Source-term monitors, adjusting $\tau_\ell$ proportional to the inverse of the perceived stiffness or reactivity of the system (Padgett et al., 2019).

Irregular meshes also arise naturally in data-driven contexts; for example, with training/observation times $\{t_i\}$ sampled nonuniformly—potentially even randomly—according to experimental or sensor schedules (Pan et al., 2022).

2. Principal Numerical Schemes on Nonuniform Temporal Meshes

A wide range of schemes have been extended to or analyzed on nonuniform time grids; three representative methodologies are outlined below.

Finite Difference Schemes for PDEs

Crank–Nicolson-type schemes can be cast on arbitrary time grids, provided discrete updates are adapted to local time-steps and stability/positivity conditions on these steps are respected (such as local CFL-type bounds) (Padgett et al., 2019). The resulting method maintains second-order local accuracy and, with suitable assumptions and global controls, preserves positivity, monotonicity, and spectral-norm stability.

Fractional Time Discretizations

Both the L1 and Alikhanov-type (L2-1$\sigma$) discretizations of Caputo derivatives have nonuniform extensions. These formulas utilize integral weights and tailored offset points to accommodate graded meshes and preserve convergence order, especially near singularities, as in the error estimates

$$E_N = O\left(N^{-\min\{2-\alpha(0),\,r\alpha(0)\}}\right)$$

for the nonuniform L1 method under proper mesh grading parameter $r$ (Qiu et al., 11 Jan 2026) and

$$\|u(t_n) - u_h^n\|_{L^2} = O(\tau^{\min(\gamma\sigma, 2)} + h^2)$$

for the Alikhanov L2-1$\sigma$ formula with grading parameter $\gamma$ (Liao et al., 2018).

Mesh-agnostic Neural Surrogate Models

Modern deep learning paradigms treat time as a continuous input coordinate: e.g., Neural Implicit Flow (NIF) uses a hypernetwork (“ParameterNet”) that ingests $(t, \mu)$ for any $t\in\mathbb{R}^+$ and produces, via a second MLP (ShapeNet), a spatial map $x\mapsto u(x, t, \mu)$. No change in training or inference protocol is required to handle irregular, missing, or adaptively chosen temporal samples (Pan et al., 2022).

3. Stability, Accuracy, and Error Analysis

Formal analyses reveal that nonuniform temporal meshes can deliver comparable accuracy and stability to their uniform-mesh counterparts, provided step selection respects problem-specific constraints.

• Stability: For fractional-order equations, unconditional $L^2$ stability is achieved by leveraging discrete fractional Grönwall inequalities and positivity/monotonicity of kernel weights (Liao et al., 2018, Qiu et al., 11 Jan 2026).
• Local and Global Error: Second-order global consistency in time is observed under proper choice of mesh grading parameters ($r$ or $\gamma$). In the presence of solution singularities at $t=0$, mesh grading (e.g., $r \ge (2-\alpha(0))/\alpha(0)$) restores full second-order accuracy (Qiu et al., 11 Jan 2026, Liao et al., 2018).
• Nonlinearity and Singularity Resolution: In highly nonlinear or singular regimes (e.g., quenching in the Kawarada problem), adaptivity ensures that extremely small time-steps are enforced precisely when required for accuracy, while global schemes remain monotone and positive if local step-size bounds are respected (Padgett et al., 2019).

4. Nonuniform Meshes in Pseudospectral and Wave Propagation Methods

High-resolution simulation of acoustic or wave propagation in heterogeneous domains benefits substantially from nonuniform temporal discretization. In the k-space pseudospectral time-domain (k-PSTD) method, nonuniform time-stepping is incorporated via analytically derived k-space correction factors $\kappa_1$, $\kappa_2$ to preserve stability and reduce dispersion (King et al., 10 Jul 2025).

• Small time-steps are used in spatial regions or epochs where accuracy is critical (e.g., at interfaces, within heterogeneous phantoms), while coarser steps are used elsewhere.
• The nonuniform update equations retain spectral spatial accuracy and formal second-order temporal accuracy, with unconditional stability governed by the proper definition of k-space corrections.
• Best-practices include matching the reference speed for k-space corrections to regions where step changes occur, and enforcing a mild global CFL bound to limit temporal aliasing.

5. Mesh-Agnostic Learning and Surrogate Models

Neural surrogates such as Neural Implicit Flow (NIF) have demonstrated direct compatibility with nonuniform temporal sampling. Time $t$ is treated purely as a continuous input, and empirical loss is calculated over all available (potentially irregular) time observations. NIF achieves interpolation and extrapolation capability across arbitrary temporal instances without explicit mesh regularization or post-hoc interpolation schemes (Pan et al., 2022). Empirical studies report that NIF achieves lower RMSEs and greater data efficiency compared to baseline models, even under severely irregular temporal sampling.

6. Practical Considerations: Adaptivity, Efficiency, and Implementation

The choice of temporal mesh has significant computational and statistical implications:

• Efficiency: By clustering time-steps where solution evolves rapidly and spacing them further where it is smooth, overall computational cost can be sharply reduced with no loss of accuracy in critical regions (King et al., 10 Jul 2025, Padgett et al., 2019).
• Sparse or Irregular Data: In data-driven modeling, especially with sparse sensor or observational data, nonuniform temporal meshes are not a numerical design choice but a physical necessity, requiring algorithms that are natively robust to arbitrary temporal configurations (Pan et al., 2022).
• Parameter Selection: Explicit formulae link grading parameters $(r, \gamma)$ to anticipated singularity order, ensuring theoretically optimal convergence rates (Qiu et al., 11 Jan 2026, Liao et al., 2018).
• Positivity/Monotonicity: Step-size constraints (CFL-like bounds) must be enforced at each step in nonlinear or degenerate contexts to guarantee qualitative solution properties (Padgett et al., 2019).

7. Summary Table: Numerical Schemes on Nonuniform Temporal Meshes

Scheme/Context	Mesh Strategy	Key Theoretical Results
Crank-Nicolson (Kawarada)	Arbitrary/adaptive	Local error $O(\tau^3)$, monotonicity, positivity, stability (Padgett et al., 2019)
L1/Alikhanov (Fractional PDEs)	Graded ($t_n \sim n^r$)	$O(N^{-\min\{2-\alpha_0, r\alpha_0\}})$ (Qiu et al., 11 Jan 2026), $O(\tau^{\min(\gamma\sigma,2)})$ (Liao et al., 2018)
k-PSTD (Acoustics)	Piecewise, region-based	Spectral accuracy, 2-term k-space correction, unconditional stability (King et al., 10 Jul 2025)
Neural Surrogate (NIF)	Arbitrary/irregular data	Mesh-agnostic loss, continuous inference (Pan et al., 2022)

All evidence indicates that, with appropriate calibration, nonuniform temporal meshes not only accommodate the complexities imposed by physical singularities, experimental design, or domain heterogeneity, but also yield significant improvements in computational efficiency, adaptive resolution, and modeling power across both classical and modern (deep learning-based) spatio-temporal problem settings.