The field of partial differential equations (PDEs) is witnessing significant developments in numerical methods, driven by the need for efficient and accurate solutions to complex problems. A key direction is the development of stable and high-order methods for solving PDEs, including the use of gnomonic cubed-sphere grids and Hermite-type discretizations. Another area of focus is the improvement of existing methods, such as the Anderson acceleration of the Picard iteration for Navier-Stokes equations, and the development of new methods like the two-level sketching Alternating Anderson acceleration. The use of machine learning techniques, such as generative models and neural networks, is also becoming increasingly popular for solving PDEs and discovering new equations. Furthermore, researchers are exploring the application of reduced order models and adaptive methods to mitigate the Kolmogorov barrier in multiscale kinetic transport equations. Noteworthy papers in this area include the work on 'Generative Discovery of Partial Differential Equations by Learning from Math Handbooks', which introduces a knowledge-guided approach for discovering PDEs, and 'CodePDE: An Inference Framework for LLM-driven PDE Solver Generation', which presents a framework for generating PDE solvers using large language models. Overall, these advancements have the potential to significantly impact various fields, including fluid dynamics, physics, and engineering.
Advances in Numerical Methods for Partial Differential Equations
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Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere grids
On the Stability Barrier of Hermite Type Discretizations of Advection Equations
Generative Discovery of Partial Differential Equations by Learning from Math Handbooks
Unconditionally local bounds preserving numerical scheme based on inverse Lax-Wendroff procedure for advection on networks
On the choice of optimization norm for Anderson acceleration of the Picard iteration for Navier-Stokes equations
Adaptive and hybrid reduced order models to mitigate Kolmogorov barrier in a multiscale kinetic transport equation
The Lax--Wendroff theorem for Patankar-type methods applied to hyperbolic conservation laws
Numerical Analysis of Stabilization for Random Hyperbolic Systems of Balance Laws
Two-Level Sketching Alternating Anderson acceleration for Complex Physics Applications
CodePDE: An Inference Framework for LLM-driven PDE Solver Generation
On a Modified Random Genetic Drift Model: Derivation and a Structure-Preserving Operator-Splitting Discretization
Approximation of viscous transport and conservative equations with one sided Lipschitz velocity fields
Single-shot prediction of parametric partial differential equations