Alternating Group Difference Method For Partial Differential Equations

This paper focuses on the numerical solutions to some mathematical partial differential equations. This paper studies the alternating group difference scheme for parabolic equations and Burgers equations, solves some partial differential equations by this format, and does theoretical analysis on the alternating group difference scheme of the equations. The analysis results show that the alternating group difference scheme proposed in this paper for several equations is convergent, and this format can be used in the actual calculations. Compared with the traditional Crank-Nicolson format in the numerical examples, the calculation accuracy of the format proposed in this paper is greatly improved.The main results achieved in this paper are as follows:1. The paper constructs the Crank-Nicolson difference scheme and the alternating group difference scheme for parabolic equations. The format is stable and convergent through the theoretical analysis. The accuracy of this format is O(τ2+h4). The numerical solutions is given in numerical experiments, and compared with the numerical solutions in the numerical examples, the calculation accuracy of the alternating group scheme proposed in this paper is greatly improved than the Crank-Nicolson scheme.2. For Burgers equations we often mention in mathematics, this paper constructs the alternating group scheme, which is convergent through the theoretical analysis, then gives the corresponding numerical experiments. In the experiments, we can see that the numerical solutions is a good approximation of the exact solutions.