Sixth-order (Quasi) Compact Methods For Several Classes Of Partial Differential Equations

Partial differential equations have extensive applications in science and engineering,and many practical problems can be transformed into solving partial differential equations.High-order compact difference methods,due to their advantages such as a small stencil of grid points,easy treatment of boundaries,and the ability to achieve high resolution,are often used to solve definite problems of partial differential equations.This thesis mainly focuses on designing high-order accuracy(quasi)compact difference schemes and conducting theoretical analyses for the Poisson equations,Helmholtz equations,and time-dependent partial differential equations.The specific research work are included as follows.Firstly,a new high-order compact difference method is constructed for the high-dimensional Poisson equations.Based on the sixth-order compact difference formulas for the second derivatives,sixth-order compact difference schemes for solving the two-dimensional and three-dimensional Poisson equations are established,and the truncation errors of the schemes are analyzed.The truncation error main terms are corrected using the truncation error remainder correction method to improve the accuracy of the sixthorder compact difference schemes without expanding the grids template of unknown.The existence and convergence of schemes are proved using the discrete maximum principle.Numerical experiments are conducted to verify the accuracy and dependability of the present schemes.Then,the sixth-order quasi-compact difference schemes are constructed for the high-dimensional Poisson equations.Since the existing sixth-order compact difference schemes for solving the Poisson equations are established based on the analytical expressions of the derivatives of the source terms are known.To improve the applicability of the sixth-order compact schemes,the interior grid points set is divided into the proper interior grid points set and improper interior grid points set.On these two points sets,using different central difference formulas to approximate the derivatives in the sixth-order compact schemes,the quasi-compact difference schemes for solving the Poisson equations without the derivatives of the source term are obtained.The scheme has a local sixth-order truncation error on the proper interior grid points set and a local fourth-order truncation error on the improper interior grid points set.Moreover,the grid template for unknown is compact,while the non-compact grid template is only for source terms.Theoretically,the global sixth-order convergence of the present scheme is rigorously proved using the discrete maximum principle and the properties of the discrete Green’s function.Numerical experiments are carried out to verify the accuracy and dependability of the present quasi-compact schemes.For the high-dimensional Helmholtz equations,sixth-order difference schemes are established.Based on the six-order compact difference formulas for the second derivatives,sixth-order compact difference schemes for solving the two-dimensional and three-dimensional Helmholtz equations are constructed.Analyzing the truncation errors of the difference schemes and correcting the main error terms,so that the modified difference schemes have better accuracy.Theoretically,the existence and convergence of the new schemes are analyzed.Numerical examples are used to verify the accuracy of the present schemes.Then,sixth-order quasi-compact difference scheme without involving derivative of source terms and parametric functions is constructed for Helmholtz equations with variable coefficients.Using the quasi-compact difference strategy similar to Poisson equations,the compact difference schemes achieve sixth-order convergence when the derivatives of the source term are unknown.This method provides a compact grid template for the unknown term,while non-compact grid templates are only used for the source terms and parameter functions.Theoretically,the global sixth-order convergence of the present schemes under the condition of non-positive wave numbers is proved by the discrete maximum principle and error analysis.Numerical experiments are conducted to demonstrate the accuracy and dependability of the present schemes.Since the proposed sixth-order quasi-compact difference scheme does not include the derivatives of source terms and parametric functions,it can be extended to time-dependent partial differential equations and established.sixth-order quasi-compact difference scheme.For the heat equation and unsteady convection diffusion equation for examples,the time derivative is approximated using the Crank-Nicolson method,and the resulting time semi-discrete equation is transformed into a modified Helmholtz equation.In this case,the analytical expression for the source term of the modified Helmholtz equation is unknown.The sixth-order quasi-compact difference scheme for solving Helmholtz equation is extended to solve time semi-discrete equations,and a quasi-compact difference scheme with second-order precision in time and sixth-order precision in space is obtained.The stability of the scheme is demonstrated using Fourier analysis.Numerically,the Richardson extrapolation algorithm is used to improve the time accuracy to the sixth order.Numerical experiments are done to verify the accuracy and stability of the proposed difference schemes.