Numerical Methods for Solving Problems of Mathematical Physics using Modern Libraries for High-Performance Computing

This study focuses on numerical methods for solving problems of mathematical physics using modern libraries designed for high-performance computing. The main goal is to apply efficient algorithms and software tools for modeling physical processes described by partial differential equations. Classical numerical methods, such as finite difference, finite element, and finite volume methods, are analyzed, along with their implementation using NumPy, SciPy, CuPy, Numba, Dask, PETSc, and Trilinos libraries. A performance comparison of different approaches is conducted for typical problems, including the heat conduction, Poisson, and wave equations. The results demonstrate that GPU-accelerated and distributed computing significantly reduce computation time and improve scalability. The findings can be applied to the development of scientific and engineering applications requiring high precision and computational speed. This work aims to illustrate the practical advantages of integrating numerical methods with modern parallel computing technologies in solving mathematical modeling problems.