The synergistic relationship between applied mathematics and computational science has catalyzed transformative advances in our ability to model, simulate, and understand complex physical systems. This review examines the foundational principles and contemporary developments in numerical methods that underpin modern scientific computing, with particular emphasis on the triumvirate of stability, convergence, and computational efficiency. The exposition begins with an overview of classical discretization frameworks—finite difference, finite element, and spectral methods—elucidating their theoretical foundations through consistency analysis, variational principles, and approximation theory. The discussion progresses to advanced computational techniques including mesh-free methods, multigrid solvers, and adaptive strategies that address the limitations of traditional approaches when confronted with complex geometries or multiscale phenomena. Central to the review is a rigorous treatment of stability and convergence theory, encompassing the Lax equivalence theorem for linear problems, energy methods for parabolic and hyperbolic systems, and nonlinear stability concepts including entropy conditions and monotonicity preservation. Error estimation frameworks—both a priori and posteriori—are examined in the context of adaptive refinement and solution verification. Applications in structural mechanics, fluid dynamics, and large-scale linear algebra illustrate the practical realization of these theoretical constructs. The review concludes by identifying persistent challenges and emerging directions, including multiscale coupling, uncertainty quantification, and structure-preserving algorithms that define the frontier of numerical innovation in applied mathematics.