The Simulation Research For Two Kinds Of Nonlinear Dynamic Equation Based On Corrected Parallel FPM

As a kind of numerical calculation method developed in recent years,the meshless method has been one of the key research contents in the field of computational mechanics.The meshless method is mainly based on the approximate solution of local discrete points so as to obtain the global approximate solution.Since this method does not depend on the background grid at all,it has obvious advantages in solving high dimension higher-order nonlinear dynamic problems,such as high order higher dimensional parabolic partial differential(describe the temperature variation of heat conduction problem),Maxwell's equations(describe the electromagnetic interaction)and nonlinear Schrodinger equation(describe the solitary wave in the physics phenomenon of quantum mechanics problem).In recent years,Finite Pointset Method(FPM)has been widely used in computational physics and related mechanical fields as a particle method.However,at present,there are few reports on numerical simulation of FPM for high-dimensional and high-order nonlinear dynamics problems,mainly because of the shortcomings of existing FPM such as poor stability,low accuracy and low computational efficiencyTherefore,in order to improve the numerical accuracy and stability of FPM in solving parabolic partial differential equations of high dimension and high order,this thesis firstly rewrites the high order derivative to the first derivative based on Taylor expansion and the idea of continuous first derivative,and then develops modified FPM which can be accurately applied to Neumann boundary conditions.The simulation results show that the modified FPM can simulate high order parabolic partial differential equations accurately and reliably.Secondly,for the low computational efficiency of the traditional FPM method for solving three-dimensional high-order parabolic partial differential equations,a 3D modified parallel FPM(I-FPM-3D)method with higher numerical accuracy and stability is proposed combined with MPI parallel technology.Then,the modified FPM method is extended to the solve Maxwell's equations describing the electromagnetic interaction,which provides a basis for solving the nonlinear Schrodinger equation with two components.Finally,the I-FPM-3D method is extended to the higher-order nonlinear Schrodinger equation,and a parallel split-modified high-order parallel FPM(HSS-IPFPM)method is proposed to accurately and efficiently solve the higher-order nonlinear Schrodinger equation The main work of this thesis is as follows(1)In order to improve the accuracy of traditional FPM method in solving parabolic partial differential equations of high dimension and high order,this thesis continuously rewrites the high order derivatives to the first derivative,and each first derivative is solved by using modified FPM method,and a modified FPM method is proposed based on Taylor expansion.(2)In order to improve the computational efficiency of solving high-dimensional high-order parabolic equations,a 3D modified parallel FPM(I-FPM-3D)method is proposed in combination with MPI parallel computing technology.Numerically simulate high dimensional and high order parabolic equations with analytic solutions under different boundary conditions,and analyze the comparison of computational efficiency under different CPU Numbers.The simulation results show that the I-FPM-3D method can solve the parabolic equation with Dirichlet and Neumann boundary in high dimension and high order.I-FPM-3D can stably solve high dimensional and high order parabolic partial differential equations and has second order accuracy and good convergence.I-FPM-3D is accurate in the simulation of high dimensional and high order parabolic equation problems under mixed boundary.(3)Using the I-FPM-3D method proposed above,the two-dimensional Maxwell's equations of electric field/magnetic field are solved,and the numerical results are compared with the finite-difference method.The results show that the proposed I-FPM-3D method can accurately and reliably solve the problem of two-dimensional Maxwell's equations of electric field/magnetic field.(4)Since the nonlinear Schrodinger equation has nonlinear terms and source terms,when the modified parallel FPM is used to solve the nonlinear Schrodinger equation in the complex domain,numerical simulation instability will occur as the calculation time increases.Therefore,a modified parallel FPM(HSS-IPFPM)method is proposed based on high order split format.Then,several kinds of nonlinear Schrodinger equation are simulated and the results are compared with other numerical methods.The numerical results show that the proposed HSS-IPFPM method can efficiently and accurately solve the high-dimensional nonlinear Schrodinger equation with higher accuracy and better stability,and can accurately simulate and predict the nonlinear solitary wave singularity and quantized vortex.