Finite Difference Method for Solving Second-Order Boundary Value Problems with High-Order Accuracy

Abstract

When researching and solving practical problems in continuous environments, through modeling methods, the vast majority of problems lead to models described by equations containing differential operators. In case the problem model is not complicated, we often obtain simple partial differential equations, then the solution of the problem can be obtained directly through analytical methods. Most complex problems, through the method of approximating differential operators by difference operators, from which differential problems are approximated by corresponding difference schemes and approximate solutions will be obtained. is achieved through solving systems of difference equations based on the tools of electronic computers. Then, building difference schemes to approximate the differential problem with high-order accuracy will play an important role in the accuracy of the obtained approximate solution. In this paper, we propose two difference schemes with high-order accuracy to solve second-order differential problems with Dirichlet and mixed boundary conditions. Theoretical results and experimental calculations have confirmed the accuracy of the proposed schemes.