The Research Of Reduced-order Extrapolated Formats Of Finite Volume Element And Natural Boundary Element Based On POD For Some Problems

Many practical physical problems can be described by some kind of developmental partial differential equation(groups).However,most of the development partial differential equations(groups)can not find their analytic solutions except for a few of them.The most effective and economical method is to find its numerical solution.The finite volume elemnet method(FVE)and natural boundary element(NBE)method are two common and effective numerical computation methods.For large practical engineering problems,it will produce tens of thousands of unknown quantities when we use the classical numerical method to discrete.In the process of calculation,it will occur floating point overflow and non-convergence after a number of steps of calculation due to the continuous accumulation of truncation error.This paper mainly studies the numerical calculation theory and method of reduced order extrapolation for FVE schemes of hyperbolic equations and a classical NBE for parabolic equation,Sobolev equation,and hyperbolic equation by proper orthogonal decomposition(POD)method.Under the premise that the classical FVE format and NBE format have enough precision,the reduced order extrapolation FVE model and NBE model based on POD can greatly reduce the unknowns and computational load,so as to save computer storage space,improve computational efficiency,and slow down the accumulation of truncation error.In addition,the biggest difference between this paper and the existing literature lies in that we will use error estimation to guide the selection of POD basis,which is an improvement and innovation of existing methods of reduceing order extrapolation based on POD technology.This paper consists of six chapters,mainly including the following four aspects:Part one(Chapter two),the POD reduced order extrapolation method is combined with the FVE method to establish the reduced order extrapolation FVEt scheme for hyperbolic equations.Firstly,we constructed the time semi-discrete scheme of hyperbolic equation and the full-discrete scheme of classical finite volume element and discuss the existence,stability,and convergence of the classical FVE solutions.And then,we proposed a reduced order extrapolation scheme of FVE for hyperbolic partial differential equations based on POD method,discussed the existence,uniqueness stability and convergence of the reduced order extrapolation solution based on POD and selected the number of POD base through error estimation.Finally,numerical examples are given to verify the validity and feasibility of the proposed method.Part two(Chapter three),the reduced order extrapolation(NBE)scheme for parabolic equations based on POD.Firstly,we used the Newmark method to construct the time semi-discrete scheme for parabolic equation,establish a fully discrete scheme by the(NBE)method,discussed the existence,stability,and convergence of the classical NBE solutions.And then,we proposed a reduced order extrapolation scheme of NBE for parabolic partial differential equations based on POD method and discussed the existence,uniqueness stability and convergence of the reduced order extrapolation solution based on POD.Finaly,numerical examples are given to verify the validity and feasibility of the proposed method.Moreover,we analyzed the influence of different number of bases on the accuracy of the reduced extrapolation numerical model and selected the number of POD base through error estimation.Part three(Chapter four),the reduced order extrapolation NBE method for Sobolev equation based on POD was developed.Firstly,the time semi-discrete scheme of Sobolev equation and the full discrete scheme of the classical NBE method are constructed,discuss the existence,stability,and convergence of the classical NBE solutions.Secondly,we proposed a reduced order extrapolation scheme of NBE for Sobolev partial differential equations based on POD method and discussed the existence,uniqueness stability and convergence of the reduced order extrapolation solution based on POD.Finaly,numerical examples are given to verify the validity and feasibility of the proposed method.Part four(Chapter five)is to research of order-reduction of NBE method based on POD for hyperbolic equation.Firstly,the time semi-discrete scheme of hyperbolic equation and the full discrete scheme of the classical NBE method are constructed,discussed the existence,stability,and convergence of the classical NBE solutions.Secondly,we proposed a reduced order extrapolation scheme of NBE for hyperbolic partial differential equations based on POD method and discussed the existence,uniqueness stability and convergence of the reduced order extrapolation solution based on POD.Finaly,numerical examples are given to verify the validity and feasibility of the proposed method.This method improves the precision of time discretization and greatly reduces the number of the degree of freedom and iterative steps in time direction,thus reducing the accumulation of truncation error in actual calculation and improving the calculation accuracy and efficiency.