Related papers: Fast Computation of the Direct Scattering Transfor…
We deal with an initial-boundary value problem for the multidimensional acoustic wave equation, with the variable speed of sound. For a three-level semi-explicit in time higher-order vector compact scheme, we prove stability and derive 4th…
In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection-diffusion equation with variable convection coefficient is proposed. The schemes are fourth order accurate in space and second…
Orthogonal systems in $\mathrm{L}_2(\mathbb{R})$, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation…
This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations [1] with the problem coefficient depending on only one…
We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an $\mathfrak{sl}(3)$ Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic…
We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…
The term "direct scattering study" refers to the calculation and analysis of the discrete eigenvalues of the associated Zakharov-Shabat (ZS) eigenvalue problem. The direct scattering study was applied to time-dependent oscillating solitons…
Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be…
We present two approaches for enhancing the accuracy of second order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth order compact difference scheme and the fourth order scheme based on…
We present quadrature schemes to calculate matrices, where the so-called modified Hilbert transformation is involved. These matrices occur as temporal parts of Galerkin finite element discretizations of parabolic or hyperbolic problems when…
A novel class of high-order linearly implicit energy-preserving integrating factor Runge-Kutta methods are proposed for the nonlinear Schr\"odinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation…
The semiclassical limit of the focusing Nonlinear (cubic) Schr\" odinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse scattering transforms (IST). In this paper, we…
This paper discusses a fast direct solver using boundary integral equations for Helmholtz transmission problems involving multiple inclusions in two dimensions. Efficiently addressing scattering problems in the presence of numerous…
We consider the model of the focusing one-dimensional nonlinear Schr\"odinger equation (fNLSE) in the presence of an unstable constant background, which exhibits coherent solitary wave structures -- breathers. Within the inverse scattering…
The compact finite difference method is a powerful tool for discretizing conservation laws, owing to its inherent flexibility in developing high-resolution and highly stable schemes. In this paper, we propose a framework for the design of…
Many integral equation-based methods are available for problems of time-harmonic electromagnetic scattering from perfect electric conductors. Among the many challenges that arise in such calculations are the avoidance of spurious…
The effort to generate matrix exponentials and associated differentials, required to determine the time evolution of quantum systems, frequently constrains the evaluation of problems in quantum control theory, variational circuit…
We propose a block finite difference, error inhibiting scheme that is fourth-order accurate for short to moderate times and has a six-order convergence rate for long times. This scheme outperforms the standard fourth-order Finite Difference…
We use the well-known observation that the solutions of Jacobi's differential equation can be represented via non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial…
A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the…