Related papers: Petrov-Galerkin and Spectral Collocation Methods f…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ \gamma $ ($1<\gamma<2$). We establish…
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger operator (FSO). In the problem, in order to get reliable gaps distribution…
Spectral methods for solving partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) often use Fourier or polynomial spectral expansions on either uniform and non-uniform grids. However, while very widely…
A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, to use of the collocation method to approximate the slowly…
In this paper, we focus on designing a well-conditioned Glarkin spectral methods for solving a two-sided fractional diffusion equations with drift, in which the fractional operators are defined neither in Riemann-Liouville nor Caputo sense,…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the…
We consider a sequence of elliptic partial differential equations (PDEs) with different but similar rapidly varying coefficients. Such sequences appear, for example, in splitting schemes for time-dependent problems (with one coefficient per…
In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional…
The elucidation of many physical problems in science and engineering is subject to the accurate numerical modelling of complex wave propagation phenomena. Over the last decades, high-order numerical approximation for partial differential…
In this paper we propose a method to solve the Kadomtsev--Petviashvili equation based on splitting the linear part of the equation from the nonlinear part. The linear part is treated using FFTs, while the nonlinear part is approximated…
In this work we adapt classical residual-based stabilization techniques to the spline collocation setting. Inspired by the Streamline-Upwind-Petrov-Galerkin and Pressure-Stabilizing-Petrov-Galerkin methods, our stabilized collocation…
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of…
We investigate linear dynamical systems consisting of ordinary differential equations with high dimensionality. Model order reduction yields alternative systems of much lower dimensions. However, a reduced system may be unstable, although…
Well-conditioned spectral collocation and spectral methods have recently been proposed to solve differential equations. In this paper, we revisit the well-conditioned spectral collocation methods proposed in [T.~A. Driscoll, {\it J. Comput.…
This paper develops the hybridizable discontinuous Galerkin (HDG) method for the Ostrovsky equation, a nonlinear dispersive wave equation featuring both third-order dispersion and a nonlocal antiderivative term with Coriolis effect. On a…
We propose a compressive spectral collocation method for the numerical approximation of Partial Differential Equations (PDEs). The approach is based on a spectral Sturm-Liouville approximation of the solution and on the collocation of the…
This paper proposes a strong second-order two-step explicit/implicit technique with spectral orthogonal basis Galerkin finite element method for solving a two-dimensional Gray-Scott model subject to appropriate initial and boundary…
We present the Parareal-CG algorithm for time-dependent differential equations in this work. The algorithm is a parallel in time iteration algorithm utilizes Chebyshev-Gauss spectral collocation method for fine propagator F and backward…