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

In Professor Sun Zhizhong's textbook "Numerical Methods for Partial Differential Equations", Example 1.1.3 presents a classical case study using compact difference schemes to solve boundary value problems for ordinary differential equations. Such problems hold significant importance in computational mathematics and engineering applications. This example primarily investigates boundary value problems for second-order linear ordinary differential equations. Compact difference schemes represent high-order precision numerical discretization methods that offer more accurate numerical solutions compared to traditional difference schemes under the same grid size. The core concept involves considering information from multiple nodal points when discretizing equations, thereby constructing difference schemes with higher-order truncation errors. Key implementation challenges typically include: - Special treatment of boundary conditions to maintain scheme accuracy - Determination of difference coefficients through Taylor series expansion - Construction and solution of linear algebraic systems using appropriate solvers The advantage of compact difference schemes lies in their ability to maintain computational accuracy while reducing the number of grid points, thereby decreasing computational overhead. This scheme is particularly suitable for problems requiring high-precision results, such as certain precision calculation problems encountered in engineering applications. A typical program implementation for this example generally involves several critical steps: first establishing a discrete grid through mesh generation functions, then constructing algebraic equations according to the compact difference scheme formulation, and finally obtaining numerical solutions using appropriate linear algebra solvers like LU decomposition or iterative methods. The entire process demonstrates the transformation techniques from continuous problems to discrete problems in numerical computation, where careful handling of boundary conditions and coefficient matrices ensures solution stability and accuracy.