Обзор современных численных методов для уравнений в частных производных с акцентом на физически-информированные и операторные подходы
Expanded Comparative Analysis of Modern Numerical Methods for PDEs
Method | Accuracy (Order of Convergence) | Computational Cost (Per Iteration) | Memory Footprint | Stability | Scalability (High-D) | Implementation Complexity | Ideal Applications |
Spectral | Exponential (for smooth solutions) | High (Setup), Low (Solution) | High | High | Low | High | Smooth solutions, simple domains, high- fidelity benchmarks |
Discontinuous Galerkin (DG) | High-Order Algebraic O(hp+1) | Moderate/High | Moderate | High | Moderate | High | Hyperbolic problems, complex geometries, local conservation |
Meshfree (MPM/SPH) | Algebraic O(hp) | High (Particle- Grid Transfer) | High | Moderate (Requires Stabilization) | Moderate | Moderate | Complex/moving boundaries, large deformation, FSI |
Adaptive (AMR) | Algebraic O(hp) | Low (Effective Cost) | Moderate | Moderate | Very High | High | Problems with localized features (shocks, boundary layers) , |
AI-based (PINNs) | Moderate/High (Algebraic) | Very High (Training), Low (Inference) | Moderate | High | Moderate | High | High- dimensional, inverse problems, data- rich scenarios , |
Operator Learning (DeepONet/FNO) | High (Algebraic) | Very High (Training), Very Low (Inference) | Moderate | High | Moderate | Moderate | Real-time prediction, surrogate modeling, learning solution operators , |