Wenjie Liu

Wenjie Liu
Harbin Institute of Technology | HIT · Department of Mathematics

Doctor of Science

About

19
Publications
2,861
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131
Citations
Additional affiliations
November 2016 - November 2017
Nanyang Technological University
Position
  • Reseacher Fellow
September 2012 - July 2016
Harbin Institute of Technology
Position
  • PhD

Publications

Publications (19)
Article
In this paper, we propose a space–time spectral method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions and initial conditions. The proposed method is based on the Legendre–Galerkin spectral method in space and the spectral collocation method (or block spectral collocation method) in...
Article
In this article, we introduce a new space-time spectral collocation method for solving the one-dimensional sine-Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second-order system of ordinary differential eq...
Article
Full-text available
In this paper, we present a high-order accurate method for two-dimensional semilinear parabolic equations. The method is based on a Galerkin-Chebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Op...
Article
Full-text available
In this paper, we construct a set of non-polynomial basis functions from a generalised Birkhoff interpolation problem involving the operator: \({\mathscr {L}}_\lambda ={d^2}/{dx^2}-\lambda ^2 \) with constant \(\lambda .\) With a direct inverting the operator, the basis can be pre-computed in a fast and stable manner. This leads to new collocation...
Article
Full-text available
This paper presents a fully discrete scheme by discretizing the space with the Legendre-Galerkin method and the time with the Crank-Nicolson method to solve the two-dimensional second-order wave equation. Unconditional stability and optimal error estimates in both L2 and H1 norms of the fully discrete Crank-Nicolson Galerkin method are obtained. Nu...
Article
Full-text available
We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and “interpolates” the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and fur...
Preprint
Full-text available
This paper concerns optimal error estimates for Legendre polynomial expansions of singular functions whose regularities are naturally characterised by a certain fractional Sobolev-type space introduced in [34, Math. Comput., 2019]. The regularity is quantified as the Riemann-Liouville (RL) fractional integration (of order $1-s\in (0,1)$) of the hig...
Article
In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoreti...
Article
Full-text available
In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractiona...
Preprint
In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoreti...
Article
Full-text available
We propose a spectral collocation method for the numerical solution of the time‐dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi‐implicit time marching schem...
Article
In this paper, we provide an explicit, stable and fast means to compute the approximate inverse of Hermite/Laguerre collocation differentiation matrices, and also the approximate inverse of the Hermite/Laguerre collocation matrices of a second-order differential operator. The latter offers optimal preconditioners for developing well-conditioned Her...
Article
Full-text available
In this paper, we present a stable and efficient numerical scheme for the linearized Korteweg–de Vries equation on unbounded domain. After employing the Crank–Nicolson method for temporal discretization, the transparent boundary conditions are derived for the time semi-discrete scheme. Then the unconditional stability of the resulting initial bound...
Article
As a generalisation of Gegenbauer polynomials, the generalized Gegenbauer functions of fractional degree (GGF-Fs): ${}^{r\!}G^{(\lambda)}_\nu(x)$ and ${}^{l}G^{(\lambda)}_\nu(x)$ with $\lambda>-1/2$ and real $\nu\ge 0,$ are found indispensable for optimal error estimates of the orthogonal polynomial approximation to functions in fractional Sobolev-...
Article
In this paper, we present a space–time Legendre–Gauss–Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation met...
Article
Full-text available
In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives, and then employ the implicit Runge-Kutt...
Article
In this paper, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can...

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