Finite Difference Methods For Initial Boundary Value Problem Of Novikov Equation

Many non-linear phenomena in nature can be described by non-linear partial differential equations.The most difficult problem of non-linear problems is that the exact solution of the corresponding problem can not be obtained or the expression of the solution is very complex.With the rapid development of artificial intelligence,big data,cloud computing and Internet of Things,as a new interdisciplinary subject with instrumentality,methodologies and edges,the numerical solution of differential equations has been developed unprecedentedly in the process of scientific computerization.For non-linear partial differential equations,both qualitative and quantitative studies,numerical calculation plays an increasingly important role.The numerical simulation is the main means to investigate the non-linear evolution equation and seek its physical law.In this paper,several conservative difference schemes are designed for the initial-boundary value problem of Novikov equation and generalized Novikov equation respectively.The existence,uniqueness,conservation,boundedness and convergence of solutions are discussed for each difference scheme.Chapter 3 establishes a two-layer nonlinear difference scheme for the initial-boundary value problem of Novikov equation,gives a priori estimate of the solution of Novikov equation by energy method,proves the existence of the solution of the difference scheme by Browder Theorem,calculates the truncation error of the difference scheme,proves the stability and convergence of the solution of the difference scheme,and designs the iterative algorithm of the difference scheme,and gives the numerical simulation results.Chapter 4 establishes two difference schemes for the initial-boundary value problem of the generalized Novikov equation,which satisfy the discrete conservation law,proves the existence of the solutions of the two difference schemes,calculates the truncation errors of the two schemes,analyses and proves the stability and convergence of the solutions of the difference schemes,and compares the advantages and disadvantages of the two difference schemes by designing an iterative algorithm to verify the effectiveness of the two schemes.