A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media

Abstract

This paper presents a direct solution technique for the scattering of time-harmonic waves from a bounded region of the plane in which the wavenumber varies smoothly in space. The method constructs the interior Dirichlet-to-Neumann (DtN) map for the bounded region via bottom-up recursive merges of (discretization of) certain boundary operators on a quadtree of boxes. These operators take the form of impedance-to-impedance (ItI) maps. Since ItI maps are unitary, this formulation is inherently numerically stable, and is immune to problems of artificial internal resonances. The ItI maps on the smallest (leaf) boxes are built by spectral collocation on tensor-product grids of Chebyshev nodes. At the top level the DtN map is recovered from the ItI map and coupled to a boundary integral formulation of the free space exterior problem, to give a provably second kind equation. Numerical results indicate that the scheme can solve challenging problems 70 wavelengths on a side to 9-digit accuracy with 4 million unknowns, in under 5 min on a desktop workstation. Each additional solve corresponding to a different incident wave (right-hand side) then requires only 0.04 s.

Appendix: Reference solution for plane wave scattering from radial potentials

In this appendix we describe how we generate reference solutions with around 13 digits of accuracy for the scattering problem from smooth radially-symmetric potentials such as

$$\begin{aligned} b(r) = \pm 1.5 e^{-160r^2}, \end{aligned}$$

(7.1)

which are needed in Sect. 5.1. Here $(r,\theta )$ are polar coordinates; in what follows $(x_1,x_2)\in \mathbb {R}^2$ indicate Cartesian coordinates. We choose a solution domain radius $R>0$ such that $b$ is numerically negligible outside the ball $r<R$. A plane wave incident in the positive $x_1$-direction is decomposed into a polar Fourier (“angular momentum”) basis via the Jacobi–Anger expansion [30, 10.12.5],

$$\begin{aligned} e^{i\kappa x_1} = J_0(\kappa r) + 2 \sum _{l=1}^\infty i^l J_l(\kappa r)\cos l\theta . \end{aligned}$$

We write $J_l(z) = (H^{(1)}_l(z) + H^{(2)}_l(z))/2$, and then notice that the effect of the potential $b$ on this field is to modify only the outgoing scattering coefficients. Thus, restricting to a maximum order $L$, the full field becomes

$$\begin{aligned} u(r,\theta ) \approx \frac{1}{2}[H^{(1)}_0(\kappa r) + a_0 H^{(2)}_0(\kappa r)] + \sum _{l=1}^L i^l [H^{(1)}_l(\kappa r) + a_l H^{(2)}_l(\kappa r)] \cos l\theta , \quad r>R. \end{aligned}$$

(7.2)

The coefficients $\{a_l\}$ are known as scattering phases; by flux conservation they lie on the unit circle if $b(r)$ is a real-valued function. Convergence with respect to $L$ is exponential, once $L$ exceeds $\kappa R$. For the case of (7.1) we choose $R=0.5$ and $L=30$.

The phases are found in the following way. For each $l=0,\dots L$ we solve the homogeneous radial ODE,

$$\begin{aligned} u^{''}_{l} + \frac{1}{r} u^{'}_{l} + \left[ -\frac{l^2}{r^2} + \left( 1 - b(r)\right) \kappa ^2\right] u_l = 0, \qquad 0<r<R \end{aligned}$$

with initial conditions that correspond to a regular solution of the form $u_l(r) \sim c r^l$ as $r\rightarrow 0^+$ (we implement the initial condition by restricting the domain to $[r_0,R]$ for some small number $r_0>0$ chosen such that the solution growing with increasing $r$ dominates sufficiently over the decaying one). For the numerical solution we use MATLAB’s ode45 with machine precision requested for absolute and relative tolerances. (We note that the standard transformation $u(r) = r^lU(r)$ which mollifies the behavior at $r=0$ resulted in no improvement in accuracy). After extracting each interior solution’s Robin constant $\beta _l := u'_l(R)/u_l(R)$, and matching value and derivative to (7.2) at $r=R$, we get after simplification,

$$\begin{aligned} a_l = -\frac{\alpha _l^*}{\alpha _l}, \qquad \text{ where } \; \alpha _l = \kappa {H^{(1)}_l}'(\kappa R) - H^{(1)}_l(\kappa R) \beta _l , \end{aligned}$$

which completes the recipe for the phases. The computation time required is a few seconds, due to the large number of steps required by ode45. A simple accuracy test is independence of the phases with respect to variation in $R$. Values of $u(r,\theta )$ for $r\ge R$ may then be found via evaluating the sum in (7.2), and for $r<R$ by summation of the interior solutions $\{u_l(r)\}$.