Spectral And Implicit Runge-kutta Methods For Two-dimensional Schrodinger Equation

The Schrodinger equation is the fundamental equation of physics for describingquantum mechanical behavior. It is also called the Schrodinger wave equation, and itis a partial differential equation that describes how the wave function of a physicalsystem evolves over time. Also this equation appears in electromagnetic wavepropagations, in underwater acoustics (paraxial approximation of the waveequations) or also in optic (Fresnel equation) and design of certain optoelectronicdevices as it models an electromagnetic wave equation in a two-dimensional weaklyguiding structure. It has also found its application in various quantum dynamicscalculations. For these reasons, the construction of efficient numerical schemes forsolving two-dimensional Schrodinger equation represents an important task.In this paper we combined high order spectrum method and implicitRunge-Kutta methods which based on Gauss-Legendre integral formula to solvetwo-dimensional Schrodinger equation. With spectral method approximating spatialderivatives, we generate linear system of ordinary differential equations via ImplicitRunge-Kutta methods. implicit Runge-Kutta methods has the advantages of bothunconditional stability and high-order accuracy. Spectrum method also has thecharacteristic of high-order accuracy.In this paper, the following work is done.Firstly, we use spectrum method to solve a two-dimensional elliptic partialdifferential equation, then give some numerical examples, which shows spectrummethod has high order accuracy. Secondly, we introduce the general solutionprocedure based on Gauss-Legendre integral formula of implicit Runge-Kuttamethod, and use it to solve a stiff differential equation, which shows implicitRunge-Kutta method has high order accuracy and stability. Thirdly, we combinespectral method and Implicit Runge-Kutta methods to solve the two-dimensional Schrodinger equation, and analyze the error estimates of half-discrete scheme ofspectral method. Finally,we give the numerical simulation of two-dimensionalSchrodinger equation which combines spectrum method and Implicit Runge-Kuttamethod. The numerical results show this method has high accuracy and stability.