Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions

Abstract

We present a hybrid asymptotic/numerical method for the accurate computation of single- and double-layer heat potentials in two dimensions. It has been shown in previous work that simple quadrature schemes suffer from a phenomenon called “geometrically induced stiffness,” meaning that formally high-order accurate methods require excessively small time steps before the rapid convergence rate is observed. This can be overcome by analytic integration in time, requiring the evaluation of a collection of spatial boundary integral operators with non-physical, weakly singular kernels. In our hybrid scheme, we combine a local asymptotic approximation with the evaluation of a few boundary integral operators involving only Gaussian kernels, which are easily accelerated by a new version of the fast Gauss transform. This new scheme is robust, avoids geometrically induced stiffness, and is easy to use in the presence of moving geometries. Its extension to three dimensions is natural and straightforward, and should permit layer heat potentials to become flexible and powerful tools for modeling diffusion processes.

Appendix: A Stirling’s formula, Cramer’s inequality, and Gauss-Legendre quadrature

In the proof of Theorem 2, we make use of Stirling’s formula.

$$ \sqrt{2\pi}\left( \frac{n}{e}\right)^{n}\sqrt{n} \leq n! \leq e\left( \frac{n}{e}\right)^{n}\sqrt{n}. $$

(1)

From this, it is staightforward to derive the following:

Corollary 1

Letn ∈N,we have the following:

$$ \sqrt{(2n)!} \leq C n^{1/4}(2n-1)!! , $$

(2)

where C > 0 is a constant.

We also use Cramer’s inequality [11].

Lemma 1

Leth_sn(t) be the nth order Hermite function, defined by the following:

$$ h_{n}(t)=(-1)^{n} D^{n} e^{-t^{2}}. $$

(3)

Then

$$ |h_{n}(t)|\leq K 2^{n/2}\sqrt{n!}e^{-t^{2}/2}, $$

(4)

where K is some constant with numerical value K ≤ 1.09.

The following lemma is a direct consequence of Leibniz’s product rule for differentiation.

Lemma 2

Let $F(t)=\frac {1}{t}$ and $f(t)=\frac {1}{\sqrt {t}}$ . Then, we have the following:

$$ F^{(2n)}(t)=\sum\limits_{k = 0}^{2n} \binom{2n}{k}|f^{(2n-k)}(t)|\cdot|f^{(k)}(t)|. $$

(5)

Finally, we state the standard error estimate for Gauss-Legendre quadrature [4].

Lemma 3

Letf ∈ C²ⁿ([a,b]) and let {x_s1,⋯ ,x_sn} and {w_s1,⋯ ,w_sn} be the Gauss-Legendre nodes and weights scaled to[a,b].If we denote, the quadrature error byE_sn(f),we have the following:

$$\begin{array}{@{}rcl@{}} E_{n}(f)&=&{{\int}_{a}^{b}} f(x)\text{dx}-\sum\limits_{k = 1}^{n} w_{k} f(x_{k}) \\ &=&\frac{(b-a)^{2n + 1}(n!)^{4}}{(2n + 1)[(2n)!]^{3}}f^{(2n)}(\xi), \end{array} $$

(6)

whereξ ∈ [a,b].