ABSTRACT
High-order finite difference schemes can be classified into two main categories: explicit schemes and Pade-type or compact schemes. Explicit schemes compute the numerical derivatives directly at each grid by using large stencils, while compact schemes obtain all the numerical derivatives along a grid line using smaller stencils and solving a linear system of equations. This chapter introduces the basic idea for developing compact difference schemes, and presents a detailed construction of the scheme and MATLAB exemplary code. It extends the discussion to high-dimensional problems and introduces some other ways of constructing compact difference schemes. The chapter discusses a popular way to construct high-order compact difference schemes. The basic idea is to apply central differences to the governing PDE and then repeatedly replace those higher-order derivatives in the truncation error by low-order derivatives using the PDE.
