High Order Finite Difference Approximation For Several Kinds Of Equations With Oscillation Solutions

Many mathematical models arising in science and engineering usually have highly oscillatory solutions,which usually post some challenges in designing high order approximation methods.More specifically,the feature of the solution can not be capatured on coarse discretization meshes,while huge computational costs will be led on very fine meshes.To overcome this difficulty,taking singularly perturbed problem,nonlinear Helmholtz equation and Schr(?)dinger-Possion equation for examples,we research a kind of high order finite difference method in this thesis.As is known to all,when using the finite difference method to approximate the differential equation,the solution is assumed to be smooth enough in some neighborhood of each grid point,and some "high order" terms containing the mesh parameter and high-order derivatives of the solution in the Taylor's expansion will be dropped.But the regularity results suggest that,for the problems mentioned above which have oscillation solutions,the smoothness of the solution heavily depends on some "key parameters" in the equation.Moreover,more smoothness assumptions on the solution,stronger dependence on the "key parameter".For instance,in the singularly perturbed problem,the derivatives of its solution are proportional to the reciprocal of the singularly perturbation parameter which is regarded as the "key parameter".In this case,the "high order" terms neglected in constructing the classical finite difference schemes will be so significant that the classical finite difference method can not work well in approximation.And similar for the other two equations.In this thesis,in order to minimize the effect results from the direct truncation of "higher order" terms,we firstly translate all the high-order derivatives of the solution into the lower ones by using the original equation recursively.Then these modified high-order terms are substituted into the Taylor's expansion and rearranged based on the relationship between the "key parameter" and the mesh parameter.And some elementary functions may be used for simplifying the summation in the Taylor's expansion.Thus,we obtain a new Taylor's expansion whose "high order" terms have no connection with the "key parameter".Finally,the new high order finite difference scheme is constructed by neglecting some "high order" terms in the new Taylor's expansion.The newly proposed finite difference schemes perform very well in solving the equation with highly oscillation solution since the neglected "high order" terms in the new Taylor's expansion don't depend on the "key parameter" in the original equations.According to the above observation,we study the singularly perturbed problem,nonlinear Helmholtz equation and Schr(?)dinger-Possion equations in section 2,section 3and section 4,respectively.For singularly perturbed problem in section 2,the new high order finite difference scheme for 1D equation is constructed firstly,and then the numerical scheme is extended to 2D problem by using the ADI technique.We prove that these new schemes are uniformly convergent with respect to the singularly perturbed parameter.In Section 3,the nonlinear Helmholtz equation is linearized firstly through the error correction iterative method.Then,the high order finite difference schemes for1 D and 2D equations with discontinuous coefficients are proposed for the linearization equation.Actually,the numerical schemes are constructed for a system of equations since the original equation in complex field needs to be divided into real and imaginary parts.More over,we successfully simulate the soliton transmission and the optical bistability by solving the nonlinear Helmholtz equation with the proposed finite difference schemes.The Schr(?)dinger-Possion equation which is used to simulate the resonant tunnel diode(RTD)in section 4 is a coupled nonlinear system,therefore,the Gummel iterative method is applied to decouple this system firstly.Then,we develop the high order finite difference scheme for the Schr(?)dinger equation(with discontinuous coefficient)and Possion equation(with discontinuous source term),respectively.By solving this nonlinear system accurately and efficiently,the phenomenon of electron tunneling in RTD structure is also observed.