Numerical Solution Example 1.1.3 from Sun Zhizhong's "Numerical Methods for Partial Differential Equations"

This documentation describes the program implementation for Example 1.1.3 from Sun Zhizhong's textbook "Numerical Methods for Partial Differential Equations." The program employs a compact difference scheme to solve ordinary differential equation boundary value problems. The compact difference scheme represents a sophisticated numerical analysis method for solving differential equations, operating by constructing finite difference approximations of differential operators on discrete grids to approximate the differential operators in partial differential equations. In this specific implementation, the compact difference scheme demonstrates superior efficiency and accuracy in solving boundary value problems through its high-order approximation capability while maintaining a compact computational stencil. The algorithm typically involves constructing a tridiagonal system of equations from the discretized differential operator, which can be efficiently solved using Thomas algorithm (TDMA). Key programming aspects include proper handling of boundary conditions through ghost points or modified finite difference coefficients, and implementing the matrix formulation for the compact scheme that achieves fourth-order accuracy with minimal grid points. This program serves as a valuable resource for academic research in numerical analysis, computational simulations requiring high-precision solutions, and practical engineering applications where efficient and accurate differential equation solutions are critical. The code structure typically includes modules for grid generation, coefficient matrix assembly, boundary condition implementation, and linear system solvers optimized for tridiagonal systems.