Research On Fast And High Order Compact Finite Difference Schemes For Some Partial Differential Equations

As one of the commonly used numerical methods for solving partial differential equations,the finite difference method is widely used in solving various partial differential equations.It is of great research significance to construct a high-precision and efficient difference method.This paper constructs an efficient compact finite difference scheme for several common partial differential equations,which improves the solution accuracy on the one hand and ensures the computational efficiency on the other hand.For Poisson equation and Helmholtz equation,we construct and deduce the eighth-order precision compact finite difference scheme to discretize them in space,and provide the error estimation.For the obtained discrete system,this paper uses the discrete sine transform method to solve it,which avoids the huge computational cost brought by the direct matrix inversion and improves the computational efficiency.Finally,some numerical examples are given to verify the accuracy and computational efficiency of the constructed numerical method,which shows that the algorithm is accurate and effective.For the quasilinear partial differential equation Allen-Cahn equation,the operator splitting technique for time derivative is used to split it into two different sub-equations,namely quasilinear sub-equation and linear sub-equation.For quasi-linear equations,analytical methods are used to solve them exactly;for linear equations,compact finite difference schemes and discrete sine transform methods are combined for numerical solutions.The scheme was then analyzed for stability.Finally,several numerical examples are tested,and the numerical simulation results show that the algorithm is effective.