This study describes the Lax-Richtmyer Theorem and its application to the
study of convergence of the finite difference method, for which 4 different models
based on linear and nonlinear, ordinary and partial differential equations are used.
After carrying out a bibliographic study, it was confirmed that there is great
diversity in the way the method mentioned is presented. Moreover, it should be
noted that this numerical method continues to be the basis of more advanced
methods and that it is widely used in applied sciences currently. Therefore, this
work stems from the need to unify and improve the presentation of this topic.
Specifically, the convergence of the numerical scheme based on the finite difference
method is demonstrated for two ordinary differential equations and for two partial
differential equations. In each case, one equation is linear while the other is
nonlinear. Finally, definitions, structure, notation, analysis, unified proofs and
adaptation of the writing of the Lax-Richtmayer Theorem are provided. For future
projections, the computational implementation of the cases studied is considered.