Symmetric Graded Time Mesh in Fractional Problems

• Symmetric graded time mesh is a nonuniform temporal partition that clusters nodes near both endpoints of delay windows to capture repeated singular behavior.
• It employs a piecewise construction with a grading parameter controlling refinement strength to improve accuracy in fractional subdiffusion with delays.
• The design, distinct from one-sided graded meshes, enables optimal error estimation and stability in numerical solutions of delay-subdiffusion equations.

Searching arXiv for recent and foundational papers on symmetric graded time meshes and closely related graded temporal meshes in fractional problems. A symmetric graded time mesh is a nonuniform temporal partition in which node density is increased toward both ends of a relevant time window, rather than only toward the initial time. In the cited arXiv literature, the term is introduced explicitly for a constant-delay subdiffusion equation on $[-\tau,K\tau]$, where weak temporal regularity occurs near $t=0$ and $t=\tau$, and then repeats across delay windows. In that setting, the mesh is constructed so that, within each interval $((i-1)\tau,i\tau]$, points are clustered near both $(i-1)\tau$ and $i\tau$; when the grading parameter is set to $r=1$, the construction degenerates to a uniform time mesh (Bu et al., 16 Sep 2025). This notion should be distinguished from the more common one-sided graded temporal meshes used for initial-time singularities in Caputo subdiffusion problems (Kumari et al., 2023, Kopteva et al., 15 Jun 2025, Furati et al., 2019).

1. Canonical construction

In the delay-subdiffusion formulation that explicitly introduces the term, the time interval is $[-\tau,K\tau]$, and the mesh is defined blockwise with respect to each delay window. The nodes satisfy

$t_{-2N}=-\tau,$

and for each block $i=0,1,\dots,K$,

$t=0$0

Here $t=0$1 is an integer, $t=0$2 is the grading parameter, $t=0$3 are the temporal nodes, $t=0$4 are the subinterval lengths, $t=0$5, and $t=0$6 (Bu et al., 16 Sep 2025).

The symmetry is local to each delay block. Within $t=0$7, the mesh grades toward both endpoints, so the node pattern is adapted to singular behavior associated with the left endpoint $t=0$8 and the right endpoint $t=0$9. The grading parameter controls the strength of endpoint clustering. The characteristic behavior

$t=\tau$0

together with

$t=\tau$1

makes explicit that larger $t=\tau$2 yields stronger refinement near the singular locations. For $t=\tau$3, the power law disappears and the mesh becomes uniform (Bu et al., 16 Sep 2025).

2. Singular structure and the rationale for symmetry

The mesh is introduced for a subdiffusion equation with constant time delay and a Riemann–Liouville fractional derivative. In the transformed form, the equation is written as

$t=\tau$4

where $t=\tau$5 (Bu et al., 16 Sep 2025).

The motivation for symmetric grading is the repeated loss of temporal regularity at delay-induced singular locations. The recorded temporal regularity is

$t=\tau$6

and

$t=\tau$7

These estimates show singular behavior near $t=\tau$8 and near $t=\tau$9, and by repetition at later delay points. The mesh is therefore aligned with the repeated singular structure induced by the delay, rather than only with the initial layer (Bu et al., 16 Sep 2025).

A direct consequence is that the construction differs qualitatively from standard graded meshes for non-delay fractional diffusion. Those meshes are refined only near $((i-1)\tau,i\tau]$0, because the singularity they target is an initial-time singularity. Symmetric grading is thus not a generic synonym for graded temporal refinement; it is a response to a two-sided or repeated endpoint-singularity structure.

3. Fractional discretization on the mesh

On the symmetric graded time mesh, the Caputo derivative is approximated by the L1 formula on a nonuniform grid: $((i-1)\tau,i\tau]$1 with weights

$((i-1)\tau,i\tau]$2

The delayed Riemann–Liouville integral term is discretized by the fractional right rectangular rule,

$((i-1)\tau,i\tau]$3

using the same coefficients $((i-1)\tau,i\tau]$4 (Bu et al., 16 Sep 2025).

With piecewise linear finite elements $((i-1)\tau,i\tau]$5, the fully discrete method is

$((i-1)\tau,i\tau]$6

where

$((i-1)\tau,i\tau]$7

and $((i-1)\tau,i\tau]$8 is initialized from the history function $((i-1)\tau,i\tau]$9 (Bu et al., 16 Sep 2025).

The discretization is therefore coupled to the mesh at two levels. First, the L1 coefficients depend on the nonuniform nodal distribution. Second, the delayed integral inherits the same time-grid geometry. This coupling is central to the local truncation analysis, because the weak regularity occurs exactly near the endpoints toward which the mesh is graded.

4. Truncation error, stability, and convergence

The local truncation analysis on the symmetric graded mesh is carried out separately for the Riemann–Liouville integral and for the L1 approximation of the Caputo derivative. For the integral term,

$(i-1)\tau$0

For the L1 formula, an auxiliary quantity $(i-1)\tau$1 is introduced to measure time regularity across different mesh regions, and the resulting local truncation error satisfies

$(i-1)\tau$2

These estimates are tailored to the two singular regimes: the initial interval and the intervals following the first delay shift (Bu et al., 16 Sep 2025).

The analysis then bifurcates according to whether the mesh is uniform or graded. For the uniform case $(i-1)\tau$3, a discrete sequence $(i-1)\tau$4 is introduced, satisfying

$(i-1)\tau$5

together with

$(i-1)\tau$6

Using these relations, the paper establishes unconditional stability and the local time error estimate

$(i-1)\tau$7

This yields order $(i-1)\tau$8 near $(i-1)\tau$9 and order $i\tau$0 after the first delay interval (Bu et al., 16 Sep 2025).

For the symmetric graded mesh case $i\tau$1, the coefficients depend on the time level, and the analysis employs a discrete fractional Gronwall inequality with recursively defined sequences $i\tau$2. Applying this machinery to the finite element error gives the global estimate

$i\tau$3

The global rate becomes first-order in time when $i\tau$4. In the paper’s interpretation, the symmetric graded mesh is precisely the mechanism that recovers this globally optimal time error estimate under low regularity at $i\tau$5 and $i\tau$6 (Bu et al., 16 Sep 2025).

5. Relation to one-sided graded temporal meshes

Several closely related papers analyze graded temporal meshes for fractional diffusion, but they do not use symmetric grading. Instead, they employ one-sided meshes refined near $i\tau$7, because the singularity to be resolved is an initial-time singularity rather than a repeated endpoint singularity.

Paper	Time mesh	Symmetry status
(Kumari et al., 2023)	$i\tau$8	not symmetric
(Kopteva et al., 15 Jun 2025)	$i\tau$9, $r=1$0	not symmetric
(Furati et al., 2019)	$r=1$1	not symmetric

In the high-order Caputo approximation of time-fractional diffusion equations with an initial-time singularity, the temporal mesh is

$r=1$2

with nodes clustered near $r=1$3 when $r=1$4. The truncation error rate is $r=1$5, the optimal grading parameter is

$r=1$6

and the best possible temporal order $r=1$7 is recovered when $r=1$8 (Kumari et al., 2023). The mesh is explicitly described as a graded, forward-in-time mesh, not a symmetric one.

For semilinear and quasilinear fractional subdiffusion equations, the graded-mesh condition is

$r=1$9

with a standard example

$[-\tau,K\tau]$0

The resulting L1 analysis yields sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading, but no symmetry hypothesis is introduced anywhere in the theory (Kopteva et al., 15 Jun 2025).

In the inverse-source problem for a two-parameter anomalous diffusion model, the mesh

$[-\tau,K\tau]$1

is introduced because the recovered source term is singular or weakly regular near $[-\tau,K\tau]$2. Under the graded mesh, the discontinuous collocation method satisfies

$[-\tau,K\tau]$3

and choosing

$[-\tau,K\tau]$4

recovers the optimal second-order rate $[-\tau,K\tau]$5 (Furati et al., 2019).

Taken together, these results show that symmetric graded time meshes are not a generic replacement for one-sided graded meshes. They are a specialized construction for problems in which singular behavior appears at more than one distinguished temporal location.

6. Terminological extensions and distinctions

The phrase “symmetric” appears in other temporal-discretization literatures, but with different meanings. On arbitrary time scales $[-\tau,K\tau]$6, the symmetric derivative is defined through the forward and backward jump operators $[-\tau,K\tau]$7 and $[-\tau,K\tau]$8. At dense points it reduces to

$[-\tau,K\tau]$9

while at non-dense points, under continuity,

$t_{-2N}=-\tau,$0

This framework is centered and compatible with nonuniform spacing, but it is a calculus on time scales rather than a delay-adapted graded-mesh construction (Cruz et al., 2012).

A distinct use of symmetry arises in a geometric variational discretization of second-order initial value problems. There, the independent variable is an auxiliary world-line parameter $t_{-2N}=-\tau,$1, not physical time $t_{-2N}=-\tau,$2. An equidistant grid is imposed in $t_{-2N}=-\tau,$3, while $t_{-2N}=-\tau,$4 is solved for as part of the discrete dynamics. The resulting physical time mesh is non-equidistant and is interpreted as automatic AMR guided by the system symmetries; the associated continuum Noether charge $t_{-2N}=-\tau,$5 is exactly preserved in the interior of the simulated time interval by the discrete scheme (Rothkopf et al., 2023).

A plausible implication is that “symmetric graded time mesh” should be reserved for a prescribed temporal partition that grades toward multiple singular endpoints, rather than being used loosely for any centered difference formula or any symmetry-preserving nonuniform time discretization. In the literature cited here, the explicit, technical meaning is the delay-window construction on $t_{-2N}=-\tau,$6 designed to resolve low regularity near $t_{-2N}=-\tau,$7, whereas the dominant graded-mesh tradition in fractional diffusion remains one-sided and focused on the initial layer (Bu et al., 16 Sep 2025).