Numerical Methods for Chemical Engineers: Homotopy and Bifurcation

In chemical engineering, non-linear systems often present multiple steady states, sudden behavior shifts, and equations that standard solvers fail to resolve. Understanding how to track these solutions and predict system stability is critical for designing safe and efficient chemical reactors and separation processes. This text-based course guides you through the foundational mathematics and numerical strategies needed to solve highly non-linear engineering equations. You will transition from basic algebraic solvers to advanced tracking techniques, learning how to map out system behavior when parameters change. This course also introduces modern computational practices, such as writing clean, structured code with type hints to ensure your numerical scripts are robust and maintainable. What you'll learn: - Understand the foundational theory of homotopy continuation and how it guarantees convergence for difficult non-linear equations - Analyze bifurcation points to predict stability limits and multiple steady states in chemical reactors - Apply numerical arc-length continuation techniques to trace complex solution curves - Configure robust solver algorithms that handle limit points and hysteresis loops without failing - Implement clean numerical scripts using modern coding standards and structured error handling Starting with essential definitions of non-linearity and stability, this course walks you through the mathematical formulation of homotopy, step-by-step curve-tracking algorithms, and practical engineering applications. Each concept is reinforced with clear written explanations and structured code snippets. This course is designed for engineering students, researchers, and practicing chemical engineers who have a basic understanding of calculus and linear algebra. No advanced numerical analysis background is required. Begin mastering non-linear engineering systems today with this comprehensive written guide.