High Order Finite Difference Methods For The Photoacoustic Coupled Mode Equations

Fiber Bragg grating, whose core refractive index changes periodically, forms theperiodic distribution of spatial phase in the core and has important applications in thefield of optical fiber communication (fiber lasers, fiber filter) and the field of fiberoptic sensors(displacement, velocity, acceleration, temperature measurement). Photoacoustic coupled equations are to consider the electrostrictive the roledescription followed by the diffusion of light waves and sound waves in fiber Bragggrating and the equations are coupled nonlinear, having a big significance in theresearch on the interaction between light and acoustic. In this paper, some high ordercompact finite difference schemes for coupled mode equations in fiber Bragg gratingand photoacoustic coupled equations are derived and analyzed, and some numericalexperiments are carried on at the same time.The first chapter introduces the physical background of the coupled mode equations infiber Bragg grating and the photoacoustic coupled equations, the physical significanceof each parameter in the equation, the equation satisfied by the conservation laws andthe equations of the status quo, and one of the conservation laws is proved. Finally, itis described that the solution photoacoustic initial boundary value problem of coupledequations, the lemma and a variety of notation and later used are explanated.In the second chapter, a high order compact finite difference scheme for coupledmode equations in fiber Bragg grating is derived and analyzed. Fourier analysis showsthat the linearized scheme is unconditionally stable. Some numerical experiments arepresented to show that the present scheme preserves the conservation law andachieves the expected convergence rate.In the third chapter, using the third order averaging operator, two difference schemesfor solving the photoacoustic coupling equations are established and it is proved thatthe formats are unconditionally stable in the linear case. The numerical test resultsshow that the two kinds of formats with high accuracy and maintain the conservationlaws. Take a different equation parameters, the numerical simulation of the stability ofsoliton solutions, the results meet the theoretical requirements.In the last chapter two difference schemes for the equations are established by usingthe fourth order averaging operator and both of the formats are unconditionally stablein the linear case. In the end of the chapter some numerical experiments are presented.