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Numerical Methods For Partial Differential Equations

Soliton Solutions in a Diatomic Lattice System

1. Introduction to Plasma Instabilities.- 1.1 Introduction.- 1.2 Basic Equations.- 1.3 Dispersion Relations.- 1.4 Analysis of Instability.- References for Chapter 1.- 2. Microinstabilities-Instabilities Due to Velocity Space Nonequilibrium.- 2.1 Introduction.- 2.2 Instabilities Due to Two Humped Velocity Distributions..- 2.2a Electrostatic Instabilities.- 2.2b Electromagnetic Instabilities.- 2.2c Effect of Magnetic Mirror.- 2.2d Application to the Field Aligned Current-Anomalous Resistivity.- 2.3 Instabilities Due to Anisotropic Velocity Distributions.- 2.3a Electrostatic Instabilities.- 2.3b Electromagnetic Instabilities.- 2.3c Effect of Nonuniformity on Anisotropic Distributions.- 2.3d Quasi-Linear Diffusion in Velocity Space.- 2.4 Hydromagnetic Instabilities.- 2.4a Introduction.- 2.4b Mirror Instability.- 2.4c.- 2.5 Instabilities in Partially Ionized Plasmas.- 2.5a Introduction.- 2.5b General Properties of Instabilities.- 2.5c Application to Ionospheric Plasma Instabilities.- References for Chapter 2.- 3. Macroinstabilities-Instabilities Due to Coordinate Space Nonequilibrium.- 3.1 Introduction.- 3.2 Drift Wave Instabilities.- 3.2a Low ? Case.- 3.2b Medium ? Case.- 3.2c High ? Case.- 3.2d Notes on the Application of Drift Wave Instabilities.- 3.3 Rayleigh-Taylor Instability.- 3.4 Kelvin-Helmholtz Instability.- 3.4a Hydromagnetic Instability.- 3.4b Electrostatic Instability.- 3.5 Current Pinch Instabilities.- 3.5a Instability of Cylindrical Current with Infinite Conductivity.- 3.5b Instability of a Current with Finite Resistivity: Tearing Mode Instability.- References for Chapter 3.- 4. Nonlinear Effects Associated with Plasma Instabilities.- 4.1a Introduction.- 4.1b Methods of Approach.- 4.2 Nonlinear Effects on Particles.- 4.2a Introduction.- 4.2b Quasilinear Theory.- 4.2c Resonance Broadening.- 4.2d Particle Trapping and Wave Overtaking.- 4.3 Nonlinear Effects on Waves.- 4.3a Introduction.- 4.3b Coherent Three-Wave Interactions.- 4.3c Three Wave Interactions with Random Phase.- 4.3d Nonlinear Two Wave Interactions Due to Scattering by Particles-Nonlinear Landau Damping.- 4.4 Nonlinear Waves and Envelopes.- 4.4a Introduction.- 4.4b Nonlinear Waves-Solitons and Shocks.- 4.4c Nonlinear Envelopes-Envelope Solitons, Envelope Shocks, and Envelope Holes.- 4.4d Modulational Instability.- References for Chapter 4.- General References.- List of Symbols.

Plasma instabilities and nonlinear effects

Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation

A compact finite difference scheme for the fourth‐order time‐fractional integro‐differential equation with a weakly singular kernel

In this paper, a new implicit compact difference scheme is constructed for the fourth-order fractional diffusion-wave system by the method of order reduction. The temporal Caputo fractional derivative is discretized by an L1 scheme. The spatial derivative of order 4 is reduced to one of order 2 by order reduction. Then, the reduced derivative of order 2 is discretized by a difference formula of order 4. Using order reduction, two simple and accurate formulae of discretization for the derivative boundary conditions are obtained. And a new way of proving the stability and convergence of the scheme is presented in this paper. Some numerical results demonstrate the accuracy and efficiency of our new scheme.

A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system

In this paper, a quadratic B-spline finite-element method is proposed for solving the coupled Schrodinger-Boussinesq equations numerically. A semi-discrete finite-element scheme is constructed for this system. The existence and uniqueness of the numerical solutions and the convergence of the discrete scheme are discussed. Numerical results indicate that the proposed method is accurate and efficient.

The quadratic B-spline finite-element method for the coupled Schrodinger-Boussinesq equations

In this article, a decoupled and linearized compact difference scheme is investigated to solve the coupled Schrodinger–Boussinesq equations numerically. We establish the convergence rates f...

Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations

In this paper, we present a conservative fourth-order compact difference scheme for the initial-boundary value problem of the Zakharov equations. Discrete conservation laws, convergence and stabili...

A conservative compact difference scheme for the Zakharov equations in one space dimension

In this paper, we investigate the convergence of explicit finite difference schemes which contain a Richardson scheme and a leap-frog scheme for computing the coupled Gross–Pitaevskii equations in ...

Luming Zhang

Papers

A new implicit compact difference scheme for the fourth-order fractional diffusion-wave system

The quadratic B-spline finite-element method for the coupled Schrodinger-Boussinesq equations

Numerical analysis of a conservative linear compact difference scheme for the coupled Schrödinger–Boussinesq equations

A conservative compact difference scheme for the Zakharov equations in one space dimension

Optimal error estimates of explicit finite difference schemes for the coupled Gross–Pitaevskii equations