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Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Spectral and High-Order Methods with Applications

Orthogonal approximations in Sobolev spaces stability and convergence spectral methods and pseudospectral methods spectral methods for multi-dimensional and high order problems mixed spectral methods combined spectral methods spectral methods on the spherical surface.

Spectral Methods and Their Applications

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.

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A Crank-Nicolson adi spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation

Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation

A fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane The medium inside the cavity is assumed to be (vertically) layered By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem A simple finite difference method is then applied to solve the model Helmholtz equation The fast algorithm is designed for solving the resulting discrete system in terms of the discrete Fourier transform in the horizontal direction, a Gaussian elimination in the vertical direction, and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system The existence and uniqueness of the finite difference solution are established for arbitrary wave numbers Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, eg, between 100 and 200 wavelengths, show that the algorithm is extremely efficient The cost for calculating the radar cross section, which is of significant interest in practice, is O(M2) for an $M \times M$ mesh The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity

In this paper, we study the stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the two-dimensional nonstationary Navier-Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank-Nicolson scheme for the linear term and the explicit Adams-Bashforth scheme for the nonlinear term. Moreover, we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent, i.e., stable and convergent when the time step is less than or equal to a constant.

Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) conditions, while all previous works require certain time-step restrictions. Our theoretical results provide a new understanding on commonly used linearized schemes. The proof is based on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Unconditional convergence and optimal error estimates of a galerkin-mixed fem for incompressible miscible flow in porous media ∗

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) condition, while all previous works require certain time-step condition. Our theoretical results provide a new understanding on commonly-used linearized schemes for nonlinear parabolic equations. The proof is based on a splitting of the error function into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Weiwei Sun

Papers

Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation

A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity

Stability and Convergence of the Crank-Nicolson/Adams-Bashforth scheme for the Time-Dependent Navier-Stokes Equations

Unconditional convergence and optimal error estimates of a galerkin-mixed fem for incompressible miscible flow in porous media ∗

Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media