Adaptive Hyperbolic-cross-space Mapped Jacobi Method on Unbounded Domains with Applications to Solving Multidimensional Spatiotemporal Integrodifferential Equations

TL;DR

This work advances numerical analysis for multidimensional spatiotemporal integrodifferential equations on unbounded domains by introducing the Adaptive Hyperbolic-Cross-Space Mapped Jacobi (AHMJ) method. It builds sparse hyperbolic-cross spectral expansions using mapped Jacobi bases and employs adaptive scaling, displacement, and expansion-order control guided by hyperbolic-cross indicators, enabling efficient representation of algebraically decaying solutions. The authors provide a rigorous error decomposition into mapped Jacobi, IRK time-stepping, and adaptive-technique contributions, with a uniform upper bound that confirms controllable accuracy. Numerical experiments in one, two, three, and four dimensions demonstrate that AHMJ achieves high accuracy with far fewer basis functions than dense or non-adaptive approaches, outperforming adaptive Hermite methods for algebraic or algebraically-decaying solutions and offering a robust framework for unbounded-domain problems.

Abstract

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

Figures (7)

• Figure 1: (a, b, c) The basis functions that are used to our proposed hyperbolic-cross-space frequency indicators $\mathcal{F}_x, \mathcal{F}_y$, and $\mathcal{F}_p$ defined in Eqs \ref{['scaling_hyper']} and \ref{['p_hyper']}. The red dots are the indices of the mapped Jacobi basis functions used in the calculation of the numerators of Eq. \ref{['scaling_hyper']} and \ref{['p_hyper']}. The red and yellow dots are the indices of the mapped Jacobi basis functions used in the calculation of the denominators of Eqs. \ref{['scaling_hyper']} and \ref{['p_hyper']}. (d, e, f) The basis functions used to calculate the direct-truncation-strategy frequency indicators $\tilde{\mathcal{F}}_x, \tilde{\mathcal{F}}_y$, and $\tilde{\mathcal{F}}_p$ defined in Eqs \ref{['c_scale']} and \ref{['cp']}. The red dots are the indices of the mapped Jacobi basis functions used in the calculation of the numerators of Eq. \ref{['c_scale']}, and \ref{['cp']}. The red and yellow dots are the indices of the mapped Jacobi basis functions used in the calculation of the denominators of Eq. \ref{['c_scale']}, and \ref{['cp']}. Here, we take $N=20, \gamma=-1$ for the hyperbolic space $V_{N, \gamma}^{\boldsymbol{\beta}, \boldsymbol{x}_0}$.
• Figure 2: (a) The errors of the non-adaptive mapped Jacobi method, the scaled-only mapped Jacobi method, and the AHMJ method as well as the adaptive Hermite method. (b) The frequency indicator of the adaptive Hermite method, the frequency indicators of the non-adaptive mapped Jacobi method, the scaled-only mapped Jacobi method, and the AHMJ method (c) The scaling factor $\beta$ of the scaled-only mapped Jacobi method, and the AHMJ method as well as the adaptive Hermite method. (d) The expansion order of the AHMJ method and the adaptive Hermite method.
• Figure 3: (a) The analytical solution, which translates rightward over time. (b) The errors of the AHMJ, the non-adaptive mapped Jacobi, the adaptive Hermite, and the non-adaptive Hermite methods. (c) The displacement of the basis function for the AHMJ method as well as the displacement for the adaptive Hermite method. (d) The exterior-error indicators of the AHMJ, the non-adaptive mapped Jacobi, the adaptive Hermite, and the non-adaptive Hermite methods. The exterior-error indicator of the AHMJ method and the exterior-error indicator of the adaptive Hermite method are well controlled.
• Figure 4: (a) The errors of the non-adaptive mapped Jacobi, the ADMJ, and the AHMJ methods. (b) The frequency indicators of the non-adaptive, the ADMJ, and the AHMJ methods. (c) The scaling factors of the ADMJ method and the AHMJ method. (d) The displacements $x_0, y_0$ of the ADMJ method and the AHMJ method. Here, the reference displacement is the center of the analytical solution Eq. \ref{['exam:2']}$(x(t), y(t)) = (\cos(\frac{\pi}{3})t, \sin(\frac{\pi}{3})t)$. (e, f) The left and right exterior-error indicators (Eq. \ref{['exte']}) of the ADMJ method and the AHMJ method.
• Figure 5: (a) The errors of the non-adaptive, scaled-only mapped Jacobi method, the ADMJ method, and the AHMJ method. (b) The scaling factors $\beta_x, \beta_y$, and $\beta_z$ of the scaled-only mapped Jacobi method, ADMJ method, and the AHMJ method, respectively. (c) The expansion order $N$ is generated by the ADMJ method and generated by the AHMJ method. Our proposed AHMJ method can maintain the error small over time without using a too large number of basis functions, while the previous ADMJ method terminates prematurely because the number of basis functions increases too fast, leading to memory overflow.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 2.1
• Proposition 1
• Lemma 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3: Restated Theorem \ref{['th:1-1']}
• Lemma A.1
• proof
• Lemma A.2