Finite Difference Methods for Compressible Two-Fluid Dynamics provides the essentials of high-order numerical methods for compressible single-fluid and two-fluid transport phenomena. This book can serve as a first course on the numerical methods for transport phenomena or fluid dynamics for students in mechanical, aerospace, and chemical engineering, applied mathematics, and physics at the senior level of an undergraduate or graduate degree. It also provides foundations and algorithmic details for implementing the most recent numerical schemes for compressible flows and extending them to include other physics, such as elasticity, reaction, and magnetohydrodynamics. The book's presented schemes enable computations for broad applications, including shock-induced interfacial instability and turbulence, shock-bubble interactions, and detonation, to name a few. For a broad reach and impact, the numerical schemes satisfy the simultaneous requirements of simplicity, extendibility, and efficiency on serial and parallel computers. The physics of the compressible single- and two-fluid system also guide the design and analysis of the numerical methods. The enabled direct numerical simulations also help obtain accurate data for tuning the emerging physics-based neuromorphic algorithms.