Parareal in time and spectral in space fast L1 quasilinear subdiffusion solver

TL;DR

This work tackles nonlinear subdiffusion modeled by $\\partial^\\alpha_t u = \\nabla \\cdot(D(x,t,u)\\nabla u) + f(x,t,u)$ with Caputo memory. It develops a fully discrete solver that couples parareal in time with an $L1$ discretization in time and a spectral Galerkin discretization in space, proving that the parareal iterations converge in a finite number of steps to the exact fine‑grid $L1$(spectral) solution with rates independent of $\\alpha$. Theoretical results establish $L^2$‑Lipschitz properties for coarse and fine propagators and a discrete Gronwall framework, ensuring finite‑step convergence and stability under realistic regularity assumptions. Numerically, the spectral spatial discretization yields exponential accuracy in space, while parareal provides clock speedups that scale nearly linearly with the workload, achieving up to 14–15× speedups for large fine grids and reducing memory demands. The methodology offers a practical and scalable tool for high‑fidelity simulations of memory‑driven diffusion processes, with potential extensions to higher‑order temporal schemes and GPU acceleration.

Abstract

We consider the initial-boundary value problem for a quasilinear time-fractional diffusion equation, and develop a fully discrete solver combining the parareal algorithm in time with a L1 finite-difference approximation of the Caputo derivative and a spectral Galerkin discretization in space. Our main contribution is the first rigorous convergence proof for the parareal-L1 scheme in this nonlinear subdiffusive setting. By constructing suitable energy norms and exploiting the orthogonality of the spectral basis, we establish that the parareal iterations converge exactly to the fully serial L1-spectral solution in a finite number of steps, with rates independent of the fractional exponent. The spectral spatial discretization yields exponential accuracy in space, while the parareal structure induces a clock speedup proportional to the number of processors, making the overall method highly efficient. Numerical experiments for both subdiffusive and classical diffusion problems confirm our theoretical estimates and demonstrate up to an order of magnitude reduction in computational time compared to the conventional sequential solver. We observe that the speedup of the parareal method increases linearly with the fine integrator degrees of freedom.

Figures (6)

• Figure 1: The coarse and the fine time grids.
• Figure 2: The parareal algorithm.
• Figure 3: Logarithm of sums involving binomials and their bounds. $\bullet$ is the double sum in \ref{['eqn:DoubleSumEst']} and $\blacksquare$ is its estimate. $\blacklozenge$ is the single sum in \ref{['eqn:SingleSumEst']} and $\blacktriangle$ is its estimate. Here, $n=10$, $b = 1.1$, $c = 1.001$.
• Figure 4: The difference $\|U^k_{N} - U^{fine}_N\|$ with respect to the iteration count $k$. Here, $N_t = 2^8$, $N = 2^4$, $M = 2^5$, $\alpha = 0.5$.
• Figure 5: The runtime and the speedup for computations with the fine sequential solver and the parareal algorithm. $N = 2^4$, $\alpha = 0.5$.

Theorems & Definitions (17)

• Proposition 1
• proof
• Remark 1
• Remark 2
• Proposition 2
• proof
• Remark 3
• Remark 4
• Lemma 1
• proof