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We explore higher-dimensional generalizations of the Runge-Kutta-Wentzel-Kramers-Brillouin method for integrating coupled systems of first-order ordinary differential equations with highly oscillatory solutions. Such methods could improve…
This contribution is dedicated to the exploration of exponential operator splitting methods for the time integration of evolution equations. It entails the review of previous achievements as well as the depiction of novel results. The…
A new spectral type method for solving the one dimensional quantum-mechanical Lippmann-Schwinger integral equation in configuration space is described. The radial interval is divided into partitions, not necessarily of equal length. Two…
We study the existence of solutions for a class of nonlinear Schr\"odinger equations involving a magnetic field with mixed Dirichlet-Neumann boundary conditions. We use Lyusternik-Shnirelman category and the Morse theory to estimate the…
We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode…
This paper is concerned with boundary integral equation methods for solving the two-dimensional fluid-solid interaction problem. We reduce the problem to three differential systems of boundary integral equations via direct and indirect…
This paper is concerned with solving the Helmholtz exterior Dirichlet and Neumann problems with large wavenumber $k$ and smooth obstacles using the standard second-kind boundary integral equations. We consider Galerkin and collocation…
This paper presents smoothed combined field integral equations for the solution of Dirichlet and Neumann exterior Helmholtz problems. The integral equations introduced in this paper are smooth in the sense that they only involve…
Three different numerical techniques for solving a coupled channel Schroedinger equation are compared. This benchmark equation, which describes the collision between two ultracold atoms, consists of two channels, each containing the same…
We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This…
We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical…
In this paper we generalize and improve a recently developed domain decomposition preconditioner for the iterative solution of discretized Helmholtz equations. We introduce an improved method for transmission at the internal boundaries…
In the case of electromagnetic waves it is necessary to distinguish between inward and outward on-shell integral equations. Both kinds of equation are derived. A correct implementation of the photonic KKR method then requires the inward…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
Partial differential equations (PDE) on manifolds arise in many areas, including mathematics and many applied fields. Among all kinds of PDEs, the Poisson-type equations including the standard Poisson equation and the related eigenproblem…
Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical…
This paper develops the numerical inverse scattering transform (NIST) framework for the coupled modified Korteweg-de Vries (mKdV) equation based on its associated Riemann-Hilbert problem. The coupled system gives rise to a $3\times3$…
This paper is concerned with the cavity scattering problem in an infinite thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. Based on an operator splitting, the scattering problem is…
We develop multilevel methods for interface-driven multiphysics problems that can be coupled across dimensions and where complexity and strength of the interface coupling deteriorates the performance of standard methods. We focus on solvers…
In this paper, we present a systematic procedure to derive discrete analogues of integrable PDEs via Hirota's bilinear method. This approach is mainly based on the compatibility between an integrable system and its B\"acklund…