Related papers: A fourth-order, multigrid cut-cell method for solv…
We describe a fourth-order accurate finite-difference time-domain scheme for solving dispersive Maxwell's equations with nonlinear multi-level carrier kinetics models. The scheme is based on an efficient single-step three time-level…
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…
This article discusses the search procedure for the Poincar\'e recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system using a previously developed high-precision numerical method. For the resulting…
We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as a closure of sorts of the first order prolongation of a family of second order PDEs. We…
We introduce a three-dimensional (3D) fully tensor train (TT)-assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs) on complex geometries. Our method reformulates IGA discrete operators…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns…
We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems…
In this paper, we consider the Poisson equation on a "long" domain which is the Cartesian product of a one-dimensional long interval with a (d-1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will…
Multiperforated plates exhibit high gradients and a loss of regularity concentrated in a boundary layer for which a direct numerical simulation becomes very expensive. For elliptic equations the solution at some distance of the boundary is…
In this paper, we consider a finite difference grid-based semi-Lagrangian approach in solving the Vlasov-Poisson (VP) system. Many of existing methods are based on dimensional splitting, which decouples the problem into solving linear…
We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are…
Unfitted finite element methods have emerged as a popular alternative to classical finite element methods for the solution of partial differential equations and allow modeling arbitrary geometries without the need for a boundary-conforming…
The $P_1$--nonconforming quadrilateral finite element space with periodic boundary condition is investigated. The dimension and basis for the space are characterized with the concept of minimally essential discrete boundary conditions. We…
In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth…
A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially…
Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large…
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We…
A new method of numerical solution for partial differential equations is proposed. The method is based on a fast matrix multiplication algorithm. Two-dimensional Poison equation is used for comparison of the proposed method with…
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.…