The Compatibility, Convergence, and Stability of Difference Schemes

In this chapter, we mainly discuss the topic of the compatibility, convergence, and stability of difference schemes for linear initial value problems. Firstly, some theoretical bases of functional analysis such as the definition and properties of linear space, linear operator, and norm are introduced. In the stability analysis part, we focus on the Von-Neumann stability analysis and introduce other methods, including maximum modulus method, discrete perturbation method, energy method, and Hirt heuristic method, to analyse the numerical stability. Besides, a simple method using difference operator transform to calculate the transition factor is proposed, and an adequate discussion about various forms of stability conditions and the Lax equivalence theorem is given.