Related papers: Lattice Green's Functions for High Order Finite Di…
We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast…
This paper develops a finite-difference analogue of the boundary integral/element method for the numerical solution of two-dimensional exterior scattering from scatterers of arbitrary shapes. The discrete fundamental solution, known as the…
We study the face-centered cubic lattice (fcc) in up to six dimensions. In particular, we are concerned with lattice Green's functions (LGF) and return probabilities. Computer algebra techniques, such as the method of creative telescoping,…
We propose a multi-resolution strategy that is compatible with the lattice Green's function (LGF) technique for solving viscous, incompressible flows on unbounded domains. The LGF method exploits the regularity of a finite-volume scheme on…
Computing the atomic geometry of lattice defects--point defects, dislocations, crack tips, surfaces, or boundaries--requires an accurate coupling of the local strain field to the long-range elastic field. Periodic boundary conditions used…
In this study, we address the challenge of obtaining a Green's function operator for linear partial differential equations (PDEs). The Green's function is well-sought after due to its ability to directly map inputs to solutions, bypassing…
We derive formulas for the matrix elements of the lattice Green function for the discrete Poisson equation on an infinite square lattice. The partial difference equation for the matrix elements is solved by reducing it to a series of first…
A computationally efficient method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. The method formally discretizes the incompressible Navier-Stokes equations on an unbounded staggered…
Lattice Green's functions (LGF) and density of states (DOS) for non-interacting models on 3 related lattices are presented. The DOS and LGF at the origin for the kagome and diced lattices are rederived. Furthermore, from the form obtained…
We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2…
We have studied possible applications of a particular pseudo-differential algebra in singular analysis for the construction of fundamental solutions and Green's functions of a certain class of elliptic partial differential operators. The…
A new parallel, computationally efficient immersed boundary method for solving three-dimensional, viscous, incompressible flows on unbounded domains is presented. Immersed surfaces with prescribed motions are generated using the…
A new fast multipole formulation for solving elliptic difference equations on unbounded domains and its parallel implementation are presented. These difference equations can arise directly in the description of physical systems, e.g.…
In this paper a method is presented for evaluating the convolution of the Green's function for the Laplace operator with a specified function $\rho(\vec x)$ at all grid points in a rectangular domain $\Omega \subset {\mathrm R}^{d}$ ($d =…
Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on…
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…
In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets…
We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG)…
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local…
A first order differential equation of Green's Function, at the origin G(0), for the one- dimensional lattice is derived by simple recurrence relation. Green's Function at site (m)is then calculated in terms of G(0). A simple recurrence…