This module covers convection-diffusion equations and conservation laws, exploring their applications in engineering. Students will analyze how these equations model real-world phenomena, enhancing their analytical capabilities.
This module introduces students to the fundamentals of difference methods used for solving ordinary differential equations (ODEs). Emphasis is placed on understanding how to apply these methods effectively in engineering scenarios.
This module covers the crucial aspects of finite differences, focusing on their accuracy, stability, and convergence properties. Students will learn to evaluate and apply these concepts in various engineering problems.
In this module, the one-way wave equation is analyzed alongside the CFL (Courant-Friedrichs-Lewy) condition and von Neumann stability analysis. Students will develop a solid understanding of wave propagation and its numerical representation.
This module compares various numerical methods for solving the wave equation, highlighting their strengths and weaknesses. Students will engage in practical applications, enhancing their problem-solving skills in dynamic scenarios.
Focusing on the second-order wave equation, including the leapfrog method, this module provides students with insights into effective numerical techniques for wave equations, emphasizing their application in real-world engineering problems.
This module investigates wave profiles and the heat equation, including point sources. Students will learn how to model these phenomena and apply numerical methods to analyze heat transfer in various contexts.
Students will learn finite difference methods specifically for the heat equation in this module. Emphasis will be placed on stability and convergence, allowing for accurate modeling of heat transfer phenomena.
This module covers convection-diffusion equations and conservation laws, exploring their applications in engineering. Students will analyze how these equations model real-world phenomena, enhancing their analytical capabilities.
In this module, students will analyze conservation laws, focusing on their behavior under various conditions, including shock waves. The practical implications of these laws in engineering applications will be emphasized.
This module examines shocks and fans resulting from point sources, providing insights into their numerical treatment. Students will engage with practical examples to solidify their understanding of these phenomena.
The level set method is introduced in this module, allowing students to understand its application in tracking interfaces and shapes. Practical applications in engineering and computational science will be emphasized.
This module focuses on using matrices in difference equations across one, two, and three dimensions. Students will understand how to apply matrix techniques in numerical methods to enhance their analytical skills.
This module covers elimination techniques with reordering for sparse matrices. Students will learn how to efficiently solve systems of equations involving sparse data, crucial for large-scale engineering problems.
The financial mathematics aspect of this module focuses on the Black-Scholes equation. Students will learn to model financial processes using mathematical methods, enhancing their skills in quantitative finance.
This module introduces iterative methods and preconditioners, which are vital for solving large systems of equations efficiently. Students will engage with practical examples to deepen their understanding of these techniques.
General methods for sparse systems are examined in this module, focusing on techniques that improve computational efficiency. Students will learn to apply these methods to real-world engineering problems.
This module covers multigrid methods, a powerful technique for solving differential equations. Students will explore the theoretical foundations and practical applications of multigrid methods in engineering contexts.
This module continues the exploration of Krylov methods and their integration with multigrid techniques. Students will understand the synergy between these methods in enhancing computational efficiency.
The conjugate gradient method is introduced in this module, focusing on its application for solving large systems of linear equations. Students will learn about its advantages and effective strategies for implementation.
This module provides insights into fast Poisson solvers, essential for various engineering applications. Students will explore different algorithms and their efficiency in solving Poisson's equation numerically.
This module focuses on optimization with constraints, teaching students how to formulate and solve constrained optimization problems. Practical applications in engineering contexts will be emphasized throughout the module.
Weighted least squares methods are covered in this module, emphasizing their importance in fitting models to data. Students will learn to apply these methods in engineering and scientific applications.
This module introduces the calculus of variations and the weak form, providing students with the theoretical framework necessary for understanding variational problems in engineering applications.
Error estimates and projections are the focus of this module, where students will learn to assess the accuracy of numerical methods and how to project solutions effectively in engineering contexts.
This module covers saddle points and the inf-sup condition, essential concepts in optimization and numerical analysis. Students will learn about their implications in engineering applications and problem-solving.
This module introduces concepts related to two squares and the equality constraint Bu = d. Students will learn to apply these concepts in optimization and modeling scenarios.
Regularization by penalty term is discussed in this module, where students will understand how to apply regularization techniques to improve solution stability in the presence of noisy data.
This module covers linear programming and duality, equipping students with the skills needed to solve optimization problems using linear programming techniques and understand the duality principle.
This final module introduces the duality puzzle, inverse problems, and integral equations, providing students with a comprehensive understanding of these advanced mathematical concepts and their applications in engineering.