Fast and robust method for screened Poisson lattice Green's function using asymptotic expansion and Fast Fourier Transform

Abstract

We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solid-state physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the Fast Fourier Transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

Figures (10)

• Figure 1: Asymptotic and near field error of the asymptotic form given in \ref{['thm:asym_exp_Katsura']}.
• Figure 1: Error of $G_N$ for various $c$ at $\alpha_1 = 0.75$. We randomly choose 5 points within the square $[0, 10)^2$ and evaluate their $G_N$ approximation using various $N$ at different $c$. We compare the resulting $G_N$ with the solution obtained by evaluating $B_c$ at those points using \ref{['eq:appell_LGF']}. We also show the error bounds given by \ref{['thm:asym_bound']}.
• Figure 1: Error of the trapezoidal rule approximation of $B_c(n,m)$ with various $N_{pts}$, $\alpha_1$, $n$, and $m$. Across all the cases, $c = 0.3$. The error is computed by referencing the analytical expression using \ref{['eq:appell_LGF']}. The error bound is computed using \ref{['eq:adhoc_bound']}.
• Figure 1: Convergence study of solving the Poisson equation using the three-dimensional Poisson LGF with one periodic direction. The ratio $\Delta x_3/\Delta x_2 = 2\pi$ is held constant across all cases. Within each series, the ratio between $\Delta x_1$ and $\Delta x_2$ is fixed. Different series have different ratios of $\Delta x_2$ and $\Delta x_1$. The dashed line indicates the expected second-order convergence rate.
• Figure 1: The return probability, $P_{return}$, at various $n,m$ at different kill probabilities $p_k$.

Theorems & Definitions (25)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Proof 1
• Definition 3.2
• Theorem 3.3
• Proof 2
• Theorem 3.4
• Remark 3.5