Recent decades have witnessed remarkable progress in the mathematical foundations of numerical methods for partial differential equations. Yet the scientific and engineering problems we now confront—long-time integration, multiscale and multiphysics coupling, high dimensionality, uncertainty quantification, and extreme computational scale—have outgrown the classical theoretical and algorithmic frameworks developed 20–30 years ago.

This weeklong workshop, Advances and Challenges in Computational Partial Differential Equations, will bring together leading experts and emerging researchers to rethink and further advance the design, analysis, and implementation of modern partial differential equation solvers in regimes that remain challenging for existing methods. Building on ICERM’s tradition of fostering deep interaction between mathematics and computation, the program will emphasize the interplay between:

      •     High-order discretizations (e.g., discontinuous Galerkin and (weighted) essentially non-oscillatory schemes, spectral and Arbitrary Lagrangian-Eulerian formulations),

      •     Structure-preserving /invariant domain methods as unifying design principles,

      •     Robust treatment of boundaries and interfaces (e.g., inverse Lax–Wendroff techniques, interface-preserving remapping),

      •     Stable and efficient time integration for stiff nonlinear systems (e.g., implicit-explicit, strong stability preserving, multistage and multiderivative methods),

      •     Complexity-reduction strategies, including reduced-order modeling and emerging low-rank tensor network approaches for high-dimensional partial differential equations,

      •     Uncertainty quantification, stochastic modeling, and optimal control in nonlinear and multiscale settings.

While many of the foundational numerical ideas are well established, their modern deployment in high-dimensional, data-rich, and multiphysics environments raises fundamentally new mathematical questions including maintaining nonlinear internal stability in under-resolved simulations, entropy convergence in long-time simulations, the coupling of adaptive meshes and reduced models, and computational scalability on next-generation architectures. The workshop will highlight both the theoretical advances and computational innovations addressing these challenges, and discuss emergent challenges in the field.