Extending the unified transform: curvilinear polygons and variable coefficient PDEs

Abstract

We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

“…We have also obtained an approximate analytical representation for the solution for the case that d(s) is even. The exact representation is given by Equation (37), where the terms F j and A j are given in terms of d(s), but the terms B j involve the unknown Neumann boundary value. However, these terms are exponentially small as β → ∞.…”

“…The main achievement of this work is presented in Section 3 and concerns the fourth step: our analysis yields the solution for the case of odd symmetric Dirichlet data in the closed form (34). We study the case of even symmetric data in Section 4, where we derive the expression (37); this expression in addition to known terms also involves an unknown term. In Section 5, Figures 1 and 2 depict the numerical verification of the main result of Section 3; also, Figures 7 and 8 indicate that the unknown term in the expression (37) is exponentially small in the high frequency limit, and hence this result provides an excellent approximation for this physically significant limit.…”

“…The global relation has had important analytical and numerical implications: first, it has led to novel analytical formulations of a variety of important physical problems from water waves [20][21][22][23][24][25][26] to three-dimensional layer scattering [27]. Second, it has led to the development of new techniques for the Laplace, modified Helmholtz, Helmholtz, biharmonic equations, both analytical [28][29][30][31][32][33][34][35] and numerical [36][37][38][39][40][41][42][43][44][45][46][47].…”

The Unified Transform: A Spectral Collocation Method for Acoustic Scattering

“…Whilst formulated for polygonal domains in this paper, this method extends to more arbitrary geometries, 13 including multiple disjoint scatterers with a range of physical boundary conditions. It may also be extended to three-dimensions by allowing for two complex parameters, λ and µ during the selection of an arbitrary solution v. This however leads to much more complicated symmetry transforms and basis selection, thus work is ongoing.…”

“…The global relation (9) may be used directly, however to simplify it in the case of a polygonal domain where ∂D consists of M straight sides, we let q j and q j n denote the Dirichlet and Neumann boundary values a Whilst here we discuss convex polygons, for extensions to non-convex polygons one should consult Colbrook et al 9 The unified transform can also be used for circular domains [10][11][12] and non-polygonal domains with general curved edges. 13 on the j th side which connects corners z j and z j+1 . We then expand q j and q j n in terms of a set of basis functions, S l (t):…”