Fluid simulation plays a vital role in computer graphics, allowing for the realistic representation of natural phenomena such as water, smoke, and fire in applications like films, video games, and virtual environments. The scale of fluid simulations can vary greatly, making the selection of an appropriate discretization method critical to achieving both accuracy and efficiency. While Eulerian methods are valued for their computational efficiency, they often suffer from numerical dissipation, which can result in the loss of fine details in fluid motion. Hybrid Lagrangian/Eulerian methods, on the other hand, better capture these intricate details but are prone to issues like volume loss and uneven particle distribution. Additionally, designing a robust framework for accurately and efficiently handling solid-fluid interactions remains a complex challenge.In this dissertation, we focus on the errors and limitations introduced by different discretization schemes and propose a set of novel algorithms to address long-standing challenges, such as numerical dissipation, volume loss, and the accurate two-way coupling between solids and fluids. Through extensive experiments and comparisons, we demonstrate that our framework not only improves accuracy but also extends the capabilities of current simulation methods to capture a broader range of phenomena. By overcoming key limitations in existing discretization techniques, this dissertation lays the groundwork for future advancements in fluid simulation and offers more reliable and realistic solutions for both academic research and industry applications