Computational Approaches to Nonlinear Partial Differential Equations
DOI:
https://doi.org/10.63075/zcxz3k50Keywords:
nonlinear partial differential equations, numerical methods, finite element method, spectral methods, physics-informed neural networks, computational modeling.Abstract
Background: Nonlinear partial differential equations (PDEs) play a pivotal role in the study of complex systems in physics, biology and engineering but the solutions of these equations are analytically intractable because of their non linearity and dynamic aspects. Computational methods are the vital requirements of solution approximation as they differ in accuracy, stability, and efficiency.
Aim: The study was motivated by the following goal: It is the purpose of this work to assess and compare various computational approaches to the solution of nonlinear PDEs ranging across classical numerical methods and recent data-driven approaches to rate their respective efficacies and practical efficiency.
Method: Three nonlinear PDEs that serve as benchmarks (Burgers equation, nonlinear Schrodinger equation and Korteweg de Vries equation) were solved based on four computational methods: Finite Difference Method (FDM), Finite Element Method (FEM), Spectral Method and Physics-Informed Neural Networks (PINNs). Simulations were made keeping common discretization parameters and results were compared using root-mean-square error (RMSE), convergence rate, CPU time, and numerical stability criteria.
Results: The highest accuracy was obtained by the spectral methods and the lowest RMSE in all test case scenarios. FEM illustrated good tradeoff between precision and stability more so with adaptive grid. FDM although computationally efficient had limitations in stability when it was combined with higher CFL conditions. PINNs did not show any mesh dependency and provided an accurate result but needed much more time to train and much more computational resources.
Conclusion: There is no single method, which will be better than others across the board. High-accuracy problems are the closest fit to spectral and FEM methods whereas PINN is flexible with regard to complex or data-intensive tasks. Hybrid approaches can offer a favorable line of inquiry in future.
