Construction And Application Of Higher-order Numerical Schemes For Partial Differential Equations

In solving practical engineering problems,the solution of partial differential equations is often involved.But the analytical solution is hard to come by.Therefore,the numerical simulation of partial differential equations plays an important role in solving practical engineering problems.In many numerical solutions of partial differential equations,compact difference method is a hot research topic in recent years.In this paper,two kinds of partial differential equations are studied by using high-precision ultracompact difference scheme.First of all,this paper briefly states the research background and significance of high order difference scheme.The research status of high-order difference scheme is analyzed and studied.Then the research status and content of the two kinds of partial differential equations are introduced.The structure arrangement of this paper is also given.Secondly,based on the central difference scheme and Taylor expansion,a combinatorial ultra-compact difference scheme is constructed.The truncation error of this scheme is analyzed and compared with other difference schemes.The combined scheme is combined with the time fourth-order Runge-Kutta method to simulate the CahnHilliard equation.The results show that the combined supercompact difference scheme is feasible and can achieve high precision.Finally,a five-point eight-order ultra-compact difference scheme is constructed based on the Taylor expansion.The Fourier analysis method is used to analyze the error of the format,which shows that the format can achieve higher precision.The numerical simulation of the KdV-Burgers equation is carried out using this scheme combined with the time fourth-order Runge-Kutta method,and the results are compared with those of the ordinary eighth-order difference scheme.The feasibility of five-point eight-order ultracompact difference scheme is proved.