Pengembangan Algoritma Numerik yang Efisien untuk Menyelesaikan Persamaan Diferensial Parsial dalam Simulasi Matematika Modern

Abstract

Partial Differential Equations (PDEs) are fundamental mathematical tools for modeling physical and engineering phenomena such as fluid dynamics, heat transfer, and electromagnetism. Due to their structural complexity and boundary conditions, analytical solutions to PDEs are often infeasible, making numerical approaches essential. This study aims to develop efficient, stable, and adaptive numerical algorithms to solve PDEs in the context of modern mathematical simulations. Using a systematic literature review method based on the PRISMA protocol, this research analyzes ten recent scientific articles from reputable international journals. Data were analyzed through content and thematic analysis to identify patterns and trends in numerical algorithm development. The findings indicate that recent innovations include polynomial matrix collocation methods, Haar wavelets, numerical neural networks, mesh-free methods, and hybrid approaches such as Physics-Informed Neural Networks (PINNs). Furthermore, the integration of techniques such as adaptive mesh refinement (AMR), domain decomposition methods, and GPU acceleration shows strong potential in improving computational efficiency and numerical stability. This study concludes that the advancement of sophisticated numerical algorithms is a critical element in enabling high-precision simulations across various applied fields such as engineering, physics, and computer science. These findings provide theoretical contributions in the expansion of numerical theory and practical value in supporting the design of complex systems in the digital era.