Finite Difference

Finite difference methods are fundamental numerical techniques for approximating solutions to differential equations, primarily by discretizing the equations onto a grid and approximating derivatives using differences between neighboring grid points. Current research focuses on enhancing accuracy and efficiency, particularly for high-dimensional problems and those with complex features like boundary layers or discontinuities, often incorporating neural networks (e.g., Physics-Informed Neural Networks) or novel algorithms like complex-step finite differences to improve derivative estimation and reduce computational cost. These advancements are impacting diverse fields, from fluid dynamics simulations and image processing to the development of more efficient and robust solvers for partial differential equations in scientific computing and engineering applications.