A Study Of Hybrid Augmented Compact Finite Volume Method For Nonlinear Singular Differential Equation

Nonlinear singular differential equations are widely used in many fields such as physics and physiology.Since nonlinear singular differential equations contain singular factors,the regularity of the solutions is low,which makes it very difficult to study the theoretical analysis and numerical methods-The theoretical analysis and numerical methods of nonlinear singular differential equations have become one of the hot topics in modern mathematics,which has important scientific significance and practical application.In this thesis,a new hybrid augmented finite volume method is proposed for several kinds of nonlinear singular differential equations.Firstly,for a class of nonlinear singular differential equations,a new augmented compact finite volume method is proposed based on the accurate asymptotic analysis of the singularity properties of solutions.The whole interval is divided into singular interval with singular point and residual regular interval.In the singular interval,we obtain a reconstructed Puiseux series expansion solution at the singular point,which is used to accurately characterize the singularity properties of the solution.A new hybrid asymptotic and augmented compact finite volume scheme is constructed by using an augmented variable related to the singularity.The high precise numerical solution of this class of nonlinear singular differential equation is obtained.The order error estimates in the sense of L2 norm,H1 semi-norm and L? are derived.Numerical experiments are provided to show the effectiveness and accuracy of the hybrid method.Secondly,for Thomas-Fermi equation in semi-unbounded domain with impor-tant physical background,by using the asymptotic series of solution at infinity and the Puiseux series expansion at origin to characterize the singularity,the natural and pre-cise boundary conditions are obtained,and then the original problem is transformed into a well-posed one.Since there is an uncertain augmented variable in the series,the augmented compact finite volume method is constructed,and a high precise numerical algorithm for Thomas-Fermi equation is obtained.The computed results show that the method not only obtains the high precise numerical solution,but also obtains the high precise initial slope.In particular,we find that the initial slope is exactly equal to the augmented variable related to the singularity in the Puiseux series.Initial slope not only has important physical significance,but also its calculation accuracy has become an important symbol to measure the quality of the algorithm.Finally,the optimal control problem governed by nonlinear singular differential equations is considered.By using Lagrange multiplier method,the KKT system cou-pled by state equation,adjoint state equation and variational inequality are obtained for the optimal control problem.In order to overcome the difficulty that state and ad-joint state equations are nonlinear singular differential equations,a hybrid asymptotic and augmented compact finite volume method is constructed,and the high precise numerical solutions of the state,adjoint state and control are obtained.The numerical experiments show that the method is stable and reliable with high accuracy.