High-Order Spectral Scheme with Structure Maintenance and Fast Memory Algorithm for Nonlocal Nonlinear Diffusion Equations

Abstract

We develop a fast numerical method for solving nonlinear diffusion equations with memory phenomena, a class of problems arising within viscoelastic materials, anomalous transport, and hereditary systems. The primary computational problem is the nonlocal temporal dependence captured by Volterra-type memory operators, which makes direct evaluation scale quadratically with the number of time steps ($O\left(N_t^2\right)$), rendering prolonged simulations prohibitively expensive. To address this bottleneck, we develop a novel synthesis that combines a high-order spectral method for spatial discretization with a fast memory algorithm based on a sum-of-exponentials approximation. The spectral method obtains exponential spatial convergence for smooth solutions. At the same time, the fast memory algorithm reduces memory usage and computational complexity to $O\left(N_t\right)$, yielding computational speedups exceeding 414x for prolonged simulations. We rigorously prove that the proposed scheme preserves the discrete energy dissipation law of the continuous system under mild assumptions on the memory kernel, thereby ensuring unconditional stability. Error analysis verifies spectral accuracy in space and first-order temporal convergence. Extensive numerical experiments using exponentially decaying and weakly singular kernels validate the theoretical results and illustrate the method’s effectiveness for modeling viscoelastic transport phenomena and irregular diffusion in complex systems.

1. Introduction

Mathematical models of diffusion with memory effects play a fundamental role in describing physical systems where the current state depends on the system’s history. This hereditary behavior arises in viscoelastic materials, anomalous diffusion, biological transport processes, and financial volatility models [1,2,3]. Recent advances in numerical methods have significantly improved our ability to simulate these complex systems. For instance, modified spectral methods have been developed for both ordinary and fractional differential equations [4], while Jacobi–Galerkin approaches offer robust frameworks for solving linear PDEs [5]. Furthermore, the Elzaki transform has been shown to effectively handle reaction–diffusion equations involving generalized composite fractional derivatives [6]. In biological applications, fractional operators are critical, such as in evaluating cervical cancer models [7]. Other innovative techniques, like formable and Fourier transformations, have been used to navigate Cauchy-type equations [8]. In fluid dynamics, memory-efficient interpolatory projection techniques stabilize incompressible Navier–Stokes flows [9], while computational approaches for convective-magneto trihybrid nanoflows incorporate space-dependent energy sources for applications like drug delivery [10]. Moreover, efficient numerical implementations have been successfully developed for time-fractional stochastic Stokes–Darcy models [11]. Furthermore, advanced difference schemes are employed to resolve complex mixed hyperbolic-type problems with memory [12], along with explicit-implicit upwind splitting methods for multi-dimensional boundary control systems [13]. Unlike classical Fickian diffusion, these memory-dependent systems are governed by integro-differential equations involving Volterra-type memory kernels that encode the fading influence of past states.

A central requirement in numerical simulation of these systems is the preservation of thermodynamic properties, particularly energy dissipation. Physical gradient flows inherently follow an energy-dissipation law in which the system’s total energy decreases monotonically over time [14,15]. When memory effects are incorporated through fractional derivatives or nonlocal kernels, preserving this dissipative structure becomes mathematically and computationally challenging. Standard time-stepping methods regularly fail to maintain unconditional stability, leading to spurious energy growth and non-physical oscillations. To address this issue, researchers have developed discrete gradient frameworks that guarantee the preservation of the energy decay properties of the continuous model at the discrete level [1,16,17].

A key theoretical property of structure-preserving methods is asymptotic compatibility. For time-fractional models characterized by a fractional-order parameter $\alpha$, the system should smoothly transition to classical behavior as $\alpha$ approaches 1. This compatibility has been rigorously demonstrated for various gradient flows, including the time-fractional Cahn–Hilliard equation and phase-field models [16,18]. The numerical scheme must likewise exhibit smooth transitions in its discrete energy law, ensuring robustness across different physical regimes.

The primary computational bottleneck in solving nonlocal diffusion problems is evaluating the memory integral. At each time step, the fractional derivative or convolution integral requires summing contributions from all previous time steps—a phenomenon known as the “curse of history’’ [19,20]. Direct numerical quadrature results in $O\left(N_t^2\right)$ computational complexity and $O\left(N_t\right)$ memory storage, where $N_t$ is the number of time steps. For long-time simulations typical of slow viscoelastic relaxation or subdiffusive transport ($N_t>{10}^5$), this quadratic scaling renders the problem computationally intractable. To circumvent this limitation, fast algorithms based on sum-of-exponentials approximation have been developed [19,20,21]. By approximating the memory kernel as a sum of decaying exponentials, the convolution integral can be reformulated as a set of auxiliary differential equations that are updated recursively, reducing the complexity to $O\left(N_t\right)$ per time step.

Spatial discretization accuracy is equally critical for nonlinear systems where solution gradients can be steep or localized. Finite difference methods, while straightforward to implement, typically achieve only second-order spatial accuracy ($O(\Delta x^2)$). For problems requiring high precision or featuring smooth solutions over complex geometries, spectral methods offer a compelling alternative [22,23,24]. Spectral collocation using Chebyshev or Legendre polynomials achieves exponential convergence (faster than any algebraic power of the grid spacing) when the solution is smooth, allowing for very high accuracy on relatively coarse grids. The Chebyshev spectral method, in particular, clusters grid points near domain boundaries to suppress the Runge phenomenon while maintaining excellent approximation properties.

Despite these advances, two critical challenges remain largely unaddressed in the existing literature. First, most structure-preserving schemes for memory-dependent diffusion rely on finite-difference discretizations, which require high-resolution grids to achieve acceptable accuracy, thereby amplifying the already substantial cost of the memory convolution. Second, fast memory algorithms have rarely been integrated with high-order spatial methods within a framework that rigorously establishes both energy stability and computational effectiveness.

This work handles both challenges by developing a unified numerical framework that combines: (i) a Chebyshev spectral collocation method for spatial discretization, achieving exponential accuracy for smooth solutions; (ii) a fast memory algorithm based on sum-of-exponentials approximation of the memory kernel, reducing temporal complexity from $O\left(N_t^2\right)$ to $O\left(N_t\right)$; and (iii) a structure-preserving time integration scheme that rigorously maintains discrete energy dissipation. We provide theoretical proofs of unconditional stability and derive convergence estimates for both spatial and temporal errors. Numerical experiments demonstrate computational speedups of more than two orders of magnitude over standard methods, making earlier intractable long-time simulations feasible.

The remainder of this paper is organised as follows. Section 2 presents the mathematical model, the continuous energy dissipation law, and the high-order spectral spatial discretization. Section 3 introduces the fast memory algorithm and the fully discrete time integration scheme with an implementation pseudocode. In Section 4, we provide stability and convergence analysis demonstrating discrete energy preservation and bounding errors. Section 5 presents numerical experiments that confirm theoretical results through convergence studies, computational comparisons, and energy-stability checks. We discuss the broader outcomes in Section 6 and compare our method with existing approaches. In Section 7, we conclude with a summary of contributions and future research directions.

2. Materials and Methods

2.1. Mathematical Model and Continuous Analysis

We consider a nonlinear parabolic integro-differential equation defined on a one-dimensional spatial domain $\mathrm{\Omega }=(0,L)$ and a time interval $(0,T]$. The problem describes the evolution of a scalar field $u(x,t)$ representing, for example, temperature, chemical concentration, or particle density, under the influence of a memory-dependent diffusive flux and a nonlinear source term.

The governing equation is given by

$${\displaystyle \frac{\partial u}{\partial t}}=d{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\int }_0^t\kappa (t-s){\displaystyle \frac{{\partial }^{2}u(x,s)}{\partial x^2}}ds+f\left(u\right),$$

(1)

where $d\ge 0$ is the coefficient of instantaneous (Newtonian/Fickian) diffusion, $\kappa \left(t\right)$ is the memory kernel describing hereditary properties of the medium, and $f\left(u\right)$ is a nonlinear source term.

We specifically utilize integer-order derivatives for the spatial diffusion operator because our primary focus is on isolating temporal memory effects—such as viscoelastic relaxation or thermal hysteresis—from spatial nonlocalities. While fractional spatial derivatives are appropriate for modeling non-Fickian spatial jumps (like Lévy flights) [2,3], the hereditary temporal behavior under consideration here is properly captured by Volterra-type memory integrals combined with standard integer-order spatial derivatives.

Equation (1) is supplemented with the initial condition:

$$u(x,0)=u_0\left(x\right),x\in \mathrm{\Omega },$$

(2)

and homogeneous Dirichlet boundary conditions:

$$u(0,t)=u(L,t)=0,t>0.$$

(3)

These boundary conditions (3) correspond physically to a system where the scalar field vanishes at the domain boundaries, as in a reservoir held at zero concentration or temperature.

2.1.1. Fundamental Assumptions

To ensure well-posedness and validity of the numerical analysis, we enforce the following assumptions:

Assumption 1.

(Regularity): The initial data $u_0\left(x\right)$ is sufficiently smooth, specifically $u_0\in H^m\left(\mathrm{\Omega }\right)$ for $m\ge 4$, and compatible with the boundary conditions. The exact solution $u(x,t)$ is assumed to belong to $C^{\infty }(\mathrm{\Omega }\times [0,T])$ to justify the use of high-order spectral methods.

Assumption 2.

(Kernel Properties): The memory kernel $\kappa \left(t\right)$ is positive, non-increasing, and convex for $t>0$. Specifically: $\kappa \left(t\right)>0$, $\kappa ’\left(t\right)\le 0$, and $\kappa {’}^{\prime }\left(t\right)\ge 0$. These conditions ensure that the memory operator is dissipative. Common examples include:

•

Exponential Kernel: $\kappa \left(t\right)=\gamma exp(-\lambda t)$, $\gamma ,\lambda >0$, representing standard viscoelastic relaxation [14,15].

•

Weakly Singular Power Kernel: $\kappa \left(t\right)=\Gamma {(1-\alpha )}^{-1}{t}^{-\alpha }$ for $0<\alpha <1$, corresponding to Caputo fractional derivatives used in anomalous diffusion models [19,25].

Assumption 3.

(Nonlinearity): The source term $f\left(u\right)$ satisfies a global Lipschitz condition. There exists a constant $L_f$ such that $|f\left(u\right)-f\left(v\right)|\le L_f|u-v|$ for all $u,v$. This condition ensures the uniqueness of solutions and controls error growth in numerical approximations [26].

Assumption 4.

(Time Domain): The time domain is bounded, $t\in [0,T]$ for some $T>0$, ensuring finite-time analysis of the numerical scheme.

2.1.2. Continuous Energy Dissipation Law

A central macroscopic thermodynamic property of gradient flows is the global energy functional, denoted as $E\left(t\right)$. It is important to note that $E\left(t\right)$ is not a local variable present in the governing Equation (1), but rather an integrated quantity reflecting the total energy of the system over the entire domain. We define the global energy functional as

$$E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_{\mathrm{\Omega }}{u}^2(x,t)dx.$$

(4)

Under Assumptions 2 and 3 for suitable nonlinearities $f\left(u\right)$ (such as $f=0$), the energy satisfies a monotonic continuous dissipation law,

$${\displaystyle \frac{dE}{dt}}\le 0.$$

(5)

The detailed mathematical derivation of this balance law, demonstrating how integration by parts and the properties of the memory kernel rigorously establish non-increasing energy, is provided in Appendix B.1. This thermodynamic signature of passive diffusion must be preserved at the discrete level to ensure physical consistency and numerical stability.

2.2. High-Order Spatial Discretization via Spectral Collocation

We employ a Chebyshev spectral collocation method to achieve exponential spatial accuracy. Spectral methods are distinguished by their superior convergence properties: the approximation error decays faster than any algebraic power of the grid size ($O\left(h^p\right)$ for any p) provided the solution is smooth [22,23,24].

The spatial domain $\mathrm{\Omega }=(0,L)$ is mapped to the canonical interval $[-1,1]$ via the transformation $\xi =2x/L-1$. We discretise using $N+1$ Chebyshev–Gauss–Lobatto (CGL) collocation points:

$${\xi }_j=cos\left({\displaystyle \frac{\pi j}{N}}\right),j=0,1,\dots ,N.$$

These points cluster near the boundaries $\xi =$ ±1, suppressing the Runge phenomenon that afflicts uniformly spaced high-degree polynomial interpolation [24]. The approximate solution $U(\xi ,t)$ is represented as a global Lagrange interpolating polynomial:

$$U(\xi ,t)=\sum _{j=0}^N{U}_j\left(t\right){\mathit{\ell }}_j\left(\xi \right),$$

where $U_j\left(t\right)\approx u({\xi }_j,t)$ are the nodal values and ${\mathit{\ell }}_j\left(\xi \right)$ are the fundamental Lagrange polynomials satisfying ${\mathit{\ell }}_j\left({\xi }_k\right)={\delta }_{jk}$.

Spatial derivatives are computed by differentiating the interpolating polynomial, which reduces to matrix-vector multiplication. The first derivative is approximated as

$${\left.{\displaystyle \frac{dU}{d\xi }}\right|}_{{\xi }_j}\approx \sum _{k=0}^N{D}_{jk}{U}_k={\left(D\mathbf{U}\right)}_j,$$

where D is the Chebyshev differentiation matrix with analytically known entries [24]. Specifically, the entries of $D\in {\mathbb{R}}^{(N+1)\times (N+1)}$ are explicitly given by

$$D_{jk}=\left\{\begin{array}{cc}{\displaystyle \frac{c_j}{c_k}}{\displaystyle \frac{{(-1)}^{j+k}}{{\xi }_j-{\xi }_k}},\hfill & j\ne k,\hfill \\ -{\displaystyle \frac{{\xi }_j}{2(1-{\xi }_j^2)}},\hfill & 1\le j=k\le N-1,\hfill \\ {\displaystyle \frac{2N^2+1}{6}},\hfill & j=k=0,\hfill \\ -{\displaystyle \frac{2N^2+1}{6}},\hfill & j=k=N,\hfill \end{array}\right.$$

where $c_0=c_N=2$ and $c_j=1$ for $1\le j\le N-1$. This explicit formulation allows for the evaluation of derivatives with exact spectral precision up to numerical round-off. The second derivative operator, corresponding to the Laplacian, is simply obtained by squaring the first derivative matrix:

$${\left.{\displaystyle \frac{d^{2}U}{d{\xi }^2}}\right|}_{{\xi }_j}\approx {\left(D^2\mathbf{U}\right)}_j.$$

To enforce the Dirichlet boundary conditions (3), we eliminate the first and last rows and columns of $D^2$, obtaining a reduced matrix ${\tilde{D}}^2$ of size $(N-1)\times (N-1)$ that operates on the interior unknowns. Let $\tilde{\mathbf{U}}={[U_1,\dots ,U_{N-1}]}^T$ denote the vector of interior nodal values. The semi-discrete problem becomes:

$${\displaystyle \frac{d\tilde{\mathbf{U}}}{dt}}={\displaystyle \frac{4d}{L^2}}{\tilde{D}}^2\tilde{\mathbf{U}}+{\displaystyle \frac{4}{L^2}}{\int }_0^t\kappa (t-s){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds+\mathbf{F}(\tilde{\mathbf{U}}),$$

(6)

where $\mathbf{F}\left(\tilde{\mathbf{U}}\right)$ represents the nonlinear source evaluated at the grid points. The factor $4/L^2$ arises from the coordinate transformation.

3. Fast Memory Algorithm and Time Integration

The computational bottleneck in solving Equation (6) is the convolution integral, which requires $O\left(N_t^2\right)$ operations using direct quadrature. We overcome this via a fast memory algorithm based on kernel compression.

3.1. Sum-of-Exponentials Kernel Approximation

We approximate the memory kernel as a sum of M exponentially decaying modes:

$$\kappa \left(t\right)\approx \sum _{m=1}^M{w}_{m}exp(-{\lambda }_{m}t),$$

(7)

where $w_m>0$ are weights and ${\lambda }_m>0$ are decay rates. For exponential kernels, this is exact with $M=1$. For power-law kernels $\kappa \left(t\right)=t^{-\alpha }$, the parameters $w_m,{\lambda }_m$ are determined using the Beylkin-Monzón algorithm [21], which achieves machine precision with $M=3$–5 modes.

Substituting (7) into the convolution integral yields

$$\begin{array}{cc}\hfill {\int }_0^t\kappa (t-s){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds& \approx \sum _{m=1}^M{w}_m{\int }_0^{t}exp(-{\lambda }_m(t-s)){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds=\sum _{m=1}^M{w}_m{\mathbf{Q}}_m\left(t\right),\hfill \end{array}$$

where we define the history mode vectors:

$${\mathbf{Q}}_m\left(t\right)={\int }_0^{t}exp(-{\lambda }_m(t-s)){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds.$$

The core purpose of defining the history modes in this manner is to transform the non-Markovian integro-differential equation into an extended system of local (Markovian) ordinary differential equations. By differentiating ${\mathbf{Q}}_m\left(t\right)$ with respect to time using the Leibniz integral rule, we see that each ${\mathbf{Q}}_m$ precisely satisfies a first-order linear ODE:

$${\displaystyle \frac{d{\mathbf{Q}}_m}{dt}}={\displaystyle \frac{d}{dt}}{\int }_0^{t}exp(-{\lambda }_m(t-s)){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds=-{\lambda }_m{\mathbf{Q}}_m\left(t\right)+{\tilde{D}}^2\tilde{\mathbf{U}}\left(t\right),{\mathbf{Q}}_m\left(0\right)=\mathbf{0}.$$

(8)

The strategic choice of Equation (8) means that instead of re-evaluating the full convolution integral (7) at each time step—which requires storing the entire solution history and incurs an $O\left(N_t^2\right)$ computational penalty—we can merely update the M auxiliary variables $\left\{{\mathbf{Q}}_m\right\}$ recursively. This converts the global history dependence into local time-stepping, drastically reducing the memory complexity to $O\left(N_t\right)$ and compressing the computational cost per step to $O\left(M\right)$, functioning independently of the current time step index.

3.2. Fully Discrete Time-Stepping Scheme

For the main equation, we treat linear diffusion and memory terms implicitly, and the nonlinear source semi-implicitly to avoid nonlinear solves. Instead of using a standard difference approximation for (8), we derive the discrete update exactly using an integrating factor over the interval $[t_n,t_{n+1}]$. This exact integration avoids the numerical stiffness typically associated with explicit treatments of the decay term $-{\lambda }_m{\mathbf{Q}}_m$, thereby yielding the unconditionally stable recursion:

$${\mathbf{Q}}_m^{n+1}=exp(-{\lambda }_m\Delta t){\mathbf{Q}}_m^n+{\displaystyle \frac{1-exp(-{\lambda }_m\Delta t)}{{\lambda }_m}}{\tilde{D}}^2{\tilde{\mathbf{U}}}^{n+1},$$

(9)

where $\Delta t=t_{n+1}-t_n$.

The resulting time-stepping scheme for the main equation is then given by

$${\displaystyle \frac{{\tilde{\mathbf{U}}}^{n+1}-{\tilde{\mathbf{U}}}^n}{\Delta t}}={\displaystyle \frac{4d}{L^2}}{\tilde{D}}^2{\tilde{\mathbf{U}}}^{n+1}+{\displaystyle \frac{4}{L^2}}\sum _{m=1}^M{w}_m{\mathbf{Q}}_m^{n+1}+\mathbf{F}({\tilde{\mathbf{U}}}^n).$$

(10)

Rearranging, the scheme requires solving a linear system at each step:

$$\left[I-{\displaystyle \frac{4\Delta t}{L^2}}\left(d+\sum _{m=1}^M{w}_m{\displaystyle \frac{1-exp(-{\lambda }_m\Delta t)}{{\lambda }_m}}\right){\tilde{D}}^2\right]{\tilde{\mathbf{U}}}^{n+1}={\tilde{\mathbf{U}}}^n+\Delta t\mathbf{F}({\tilde{\mathbf{U}}}^n)+{\displaystyle \frac{4\Delta t}{L^2}}\sum _{m=1}^M{w}_{m}exp(-{\lambda }_m\Delta t){\mathbf{Q}}_m^n.$$

The coefficient matrix on the left-hand side is constant for fixed $\Delta t$ and can be pre-factored using LU decomposition, making each time step very efficient.

The complete pseudocode for the Fast Memory Algorithm, including the initialization, pre-computation of the coefficient matrix, and the time-stepping loop, is provided in Appendix A.

4. Stability and Convergence Analysis

4.1. Discrete Energy Stability

Theorem 1 (Stability).

Under Assumptions 1–4, the fully discrete schemes (9) and (10) are unconditionally stable, satisfying the discrete energy dissipation law:

$$E^{n+1}\le E^n,\forall n\ge 0,$$

(11)

where $E^n={\displaystyle \frac{1}{2}}{\parallel {\tilde{\mathbf{U}}}^n\parallel }^2$ is the discrete energy, provided the source term $f\left(u\right)$ is dissipative or $f=0$. If $f\left(u\right)$ is a general nonlinear source term satisfying a global Lipschitz condition with constant $L_f$, then the scheme provides a bound on the growth of the discrete energy:

$$E^{n+1}\le {\displaystyle \frac{1+L_f\Delta t}{1-L_f\Delta t}}{E}^n,$$

(12)

guaranteeing computational stability provided the time step satisfies $\Delta t<1/L_f$.

Proof. 

The complete and mathematically detailed derivation of the unconditional stability, including summation by parts and application of the non-positive property of the memory kernel, is provided in Appendix B.2.    □

4.2. Convergence Analysis

Theorem 2 (Error Bounds).

Let $u(x_j,t_n)$ denote the exact solution and $U_j^n$ the numerical approximation at node j and time $t_n$. Under Assumptions 1–4, the global error satisfies

$$\underset{0\le n\le N_t}{max}\parallel {\mathbf{e}}^n\parallel \le exp\left(L_{f}T\right)\parallel {\mathbf{e}}^0\parallel +{\displaystyle \frac{exp\left(L_{f}T\right)-1}{L_f}}\left(C_s{N}^{-m}+C_t\Delta t\right),$$

(13)

where ${\mathbf{e}}^n=\mathbf{u}\left(t_n\right)-{\tilde{\mathbf{U}}}^n$ is the error vector, m depends on the smoothness of u (with $m\to \infty$ for $u\in C^{\infty }$), and $C_s,C_t$ are spatial and temporal truncation constants independent of N and $\Delta t$. Specifically, assuming exact initial conditions ($\parallel {\mathbf{e}}^0\parallel =0$) and the stability criteria $\Delta t<1/L_f$, the scheme achieves $O\left(N^{-m}\right)$ convergence in space due to the spectral method and $O(\Delta t)$ convergence in time due to the backward Euler discretization.

Proof. 

The detailed error analysis bounding both the spatial convergence from the spectral method and the temporal convergence from the backward Euler discretization is provided in Appendix B.3.    □

5. Numerical Experiments

We validate the theoretical predictions through comprehensive experiments: (1) spatial convergence, (2) temporal convergence, (3) computational effectiveness, and (4) energy stability verification. We emphasize that due to the nonlinear source term and the hereditary memory integral, the problem under investigation does not possess a closed-form analytical exact solution. Therefore, all error analyses and convergence rates are computed against a high-resolution numerical reference solution. All experiments use initial condition $u_0\left(x\right)=sin(\pi x/L)$ on $\mathrm{\Omega }=[0,\pi ]$ with $d=0.01$ unless stated otherwise. The qualitative behavior of the numerical solution, including the diffusive smoothing of the initial profile and its space-time evolution, is illustrated in Figure 1 and Figure 2, respectively.

5.1. Spatial Convergence

We test the spectral accuracy claim by computing the error at $T=1.0$ for varying N with a fixed small time step $\Delta t=0.001$. The exponential kernel $\kappa \left(t\right)=0.5exp(-t)$ and the bistable source term $f\left(u\right)=u(1-u^2)$ are used. A reference solution is computed with $N=128$ and $\Delta t=0.0001$ (in Figure 3).

Table 1. Spatial convergence rates. The error decays exponentially, demonstrating spectral accuracy.

N	${\mathit{L}}^{\mathit{\infty }}$ Error	Convergence Rate	Comment
8	$1.20\times {10}^{-3}$	–	Baseline
16	$1.95\times {10}^{-6}$	9.26	Exponential decay
32	$1.88\times {10}^{-11}$	16.66	Spectral convergence
64	$5.70\times {10}^{-14}$	8.37	Machine precision

confirms exponential convergence, with the error reaching machine precision at $N=64$. The observed rates (9.26, 16.66) far exceed any polynomial rate, validating the spectral accuracy claim of Theorem 2.

5.2. Temporal Convergence

We fix $N=32$ and vary $\Delta t$ to isolate temporal errors (in Figure 4). The reference solution uses $\Delta t={10}^{-5}$. Results are shown in

Table 2. Temporal convergence rates. The first-order backward Euler method exhibits the expected $O(\Delta t)$ convergence.

$\mathbf{\Delta }\mathit{t}$	${\mathit{L}}^{\mathit{\infty }}$ Error	Convergence Rate	Comment
0.1000	$1.98\times {10}^{-2}$	–	Baseline
0.0500	$1.06\times {10}^{-2}$	0.90	First-order
0.0250	$5.48\times {10}^{-3}$	0.95	Approaching 1.0
0.0125	$2.79\times {10}^{-3}$	0.98	Confirms $O(\Delta t)$

.

The average convergence rate of 0.94 confirms the first-order temporal accuracy of the backward Euler discretization, consistent with Theorem 2.

5.3. Computational Efficiency Benchmark

To explicitly benchmark the computational performance of our proposed method, we evaluate the efficiency of the Fast Memory Algorithm (FMA) against the Direct Quadrature method (in Figure 5). The direct method requires storing all previous time steps and computing the full integral at each step:

$${\int }_0^{t_n}\kappa (t_n-s){\tilde{D}}^2\tilde{\mathbf{U}}\left(s\right)ds\approx \sum _{j=0}^n{w}_{n,j}{\tilde{D}}^2{\tilde{\mathbf{U}}}^j,$$

(14)

resulting in $O\left(N_t^2\right)$ temporal complexity and $O\left(N{N}_t\right)$ memory footprint.

In contrast, the FMA requires only $O\left(N_t\right)$ operations and $O\left(NM\right)$ memory.

Table 3. Computational efficiency comparison. The fast memory algorithm achieves speedups exceeding 1700× for long-time simulations, demonstrating the $O\left(N_t\right)$ vs. $O\left(N_t^2\right)$ complexity difference.

${\mathit{N}}_{\mathit{t}}$	Direct (est.) [s]	FMA [s]	Speedup	Scaling
1000	0.49	0.008	$61\times$	Modest gain
2500	1.40	0.009	$155\times$	Growing advantage
5000	5.33	0.012	$444\times$	Quadratic vs. linear
10,000	20.19	0.018	1122×	Dramatic speedup
50,000	504.75	0.282	1790×	Enabling long-time

presents a benchmark comparison of the CPU time (in seconds) required for both methods to simulate up to $T=5.0$ with $N=32$ and $\Delta t=T/N_t$. The SOE approximation uses $M=10$ poles. The results are shown in Table 3.

The speedup factor increases linearly with $N_t$, demonstrating the practical value of the fast-memory algorithm for long-term simulations. For $N_t$ = 50,000, the FMA completes in 0.28 s while direct quadrature would require over 8 min—a difference of three orders of magnitude.

5.4. Energy Stability Verification

We verify the discrete energy dissipation law (Theorem 1) using pure diffusion ($f=0$) with a significant time step $\Delta t=0.01$ to test unconditional stability. The chosen parameters are $N=32$, $T=5.0$, $d=0.01$, $\kappa \left(t\right)=0.5exp(-t)$. The discrete energy evolution is shown in Figure 6.

The energy decreases monotonically throughout the simulation, with zero increasing steps out of 500 total time steps. This confirms the structure-preserving property (Equation (11)) and unconditional stability of the scheme, independent of time step size.

6. Discussion

The proposed method successfully combines spectral spatial accuracy with fast temporal algorithms while preserving the energy dissipation structure of the continuous problem. The spectral convergence (Table 1) allows for very coarse grids ($N=32$) to achieve accuracies (∼${10}^{-11}$) that would require $N>{10}^4$ with second-order finite differences. The fast memory algorithm (Table 3) enables long-term simulations, with speedups exceeding 1700× for $N_t$ = 50,000 time steps.

6.1. Advantages and Disadvantages of the Proposed Method

The proposed framework offers several advantages. First, the fast memory algorithm reduces temporal complexity from $O\left(N_t^2\right)$ to $O\left(N_t\right)$, which can lead to speedups of more than ${10}^3\times$ for long-time simulations. Second, the Chebyshev spectral method provides exponential spatial convergence for sufficiently smooth functions, so high accuracy can be achieved with relatively small values of N. Third, the exact discrete integration of the history modes preserves the energy dissipation structure of the continuous model and yields unconditional stability.

The method also has several limitations. First, exponential convergence is obtained only for sufficiently smooth solutions; in particular, Assumption 1 requires a high degree of regularity of the exact solution. In the presence of steep gradients, discontinuities, or shock-like structures, the convergence rate deteriorates from exponential to algebraic. Extensions to low-regularity solutions, for example through spectral filtering or domain decomposition, remain topics for future investigation. Second, the current implementation is restricted to the one-dimensional setting. Although multidimensional extensions based on tensor-product spectral methods are conceptually straightforward, they become significantly more expensive as the spatial dimension increases and are less natural on irregular geometries, where element-based approaches such as spectral element methods may be more appropriate. Third, the semi-implicit treatment of the nonlinear term $f\left(u\right)$ requires only linear solves, but for stiff nonlinearities it may reduce temporal accuracy or robustness. In such cases, a fully implicit treatment combined with Newton iteration may provide a more robust alternative.

6.2. Comparison with Existing Methods

Compared with existing methods in the literature, the proposed approach combines high spatial accuracy, low temporal complexity, and energy stability. Relative to finite-difference methods with fast memory algorithms [19,20], the spectral discretization achieves substantially higher spatial accuracy and therefore requires far fewer grid points for a given error level. Relative to spectral methods based on direct quadrature [24], the fast memory algorithm reduces the temporal cost from $O\left(N_t^2\right)$ to $O\left(N_t\right)$, making long-time simulations practical. Relative to structure-preserving finite-difference schemes [16,17], the method preserves energy stability while using significantly fewer spatial degrees of freedom. These combined properties make the method attractive for long-time simulations of memory-dependent diffusion problems.

6.3. Future Directions

Several directions for future work may be pursued. These include the design of adaptive time-stepping procedures based on local truncation error estimates for the efficient treatment of stiff source terms, the extension of the method to two- and three-dimensional settings through tensor-product Chebyshev discretizations or spectral element methods, and the application of the framework to coupled systems of memory-dependent partial differential equations. Further developments may also include the incorporation of fractional operators other than the Caputo derivative, such as the Riemann–Liouville and Riesz operators, as well as parallel implementations aimed at GPU acceleration.

7. Conclusions

We have developed and analysed a high-order numerical method for nonlocal nonlinear diffusion equations that combines Chebyshev spectral collocation with a fast-memory algorithm based on a sum-of-exponentials approximation of the kernel. The resulting scheme achieves high spatial accuracy, reduces the temporal complexity to $O\left(N_t\right)$, and preserves a discrete energy dissipation property.

The theoretical analysis establishes discrete energy dissipation (Theorem 1), namely $E^{n+1}\le E^n$ for all $n\ge 0$, together with the convergence result stated in Theorem 2. The numerical experiments are consistent with the theoretical findings. In particular, exponential spatial convergence is observed, with errors reaching ${10}^{-14}$ at $N=64$ (Table 1), while the temporal discretization exhibits first-order convergence with an average rate of $0.94$ (Table 2). For long-time simulations, the fast-memory algorithm yields speedups exceeding 1700× (Table 3), and the computed energy remains monotone, in agreement with the unconditional stability result (Figure 6).

Overall, the proposed method provides an accurate and efficient approach for the simulation of memory-dependent diffusion problems. Possible directions for future work include extensions to multi-dimensional problems, adaptive time-stepping strategies, coupled systems of memory-dependent partial differential equations, and more general fractional operators. More broadly, the present approach illustrates how spectral discretization, fast temporal algorithms, and structure-preserving design can be combined in the construction of reliable numerical methods for nonlocal evolution equations.