Manuscript Topics

Partial differential equations (PDEs) play a central role in modeling a wide range of natural and engineered systems, including materials science, fluid dynamics, finance, biology, and data-driven sciences, etc. Analytical solutions to PDEs are often limited to special cases, and thus numerical approaches have become indispensable tools for exploring their complex behaviors. Recent advances in high-performance computing, scientific machine learning, and multi-scale modeling have significantly expanded the possibilities of PDE simulations. New numerical approaches are being developed to ensure stability, accuracy, and efficiency for problems on complex geometries, nonlinear systems, and high-dimensional domains. In addition, structure-preserving algorithms, adaptive methods, and stochastic approaches provide deeper insights into PDE-driven dynamics across scientific disciplines.

This Special Issue aims to bring together original research and review papers on novel numerical methods, algorithmic developments, and computational simulations for PDEs.

The scope includes, but is not limited to:
● Numerical analysis and convergence theory for PDE solvers.
● Numerical approaches for PDEs on complex geometry domains.
● Scientific machine learning approaches for PDEs.
● Finite difference, and spectral methods.
● Operator splitting and time-integration techniques.
● Adaptive methods for PDEs.
● Applications in biology, physics, finance, and engineering.

By fostering interdisciplinary dialogue, this Special Issue seeks to advance both theoretical foundations and computational practices, highlighting innovative strategies that push the boundaries of PDE research and applications.

Keywords: Numerical PDEs; Mathematical modeling; Numerical methods; Scientific computing; Finite difference method; Spectral method; High-order method; Stable method; Operator splitting method; Adaptive method; Monte Carlo method.

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