The field of numerical methods for partial differential equations is rapidly advancing, with a focus on developing innovative and efficient methods for solving complex problems. Recent research has centered around improving the accuracy and stability of existing methods, such as the proximal Galerkin method and discontinuous Galerkin methods, as well as exploring new approaches like entropy stable nodal discontinuous Galerkin methods and super-time-stepping methods. Additionally, there is a growing interest in applying machine learning techniques, such as denoisers and automatic differentiation, to improve the performance of numerical simulations. Notable papers in this area include: The paper on entropy stable nodal discontinuous Galerkin methods via quadratic knapsack limiting, which presents a novel approach to enforcing cell entropy inequalities. The work on a conservative and positivity-preserving discontinuous Galerkin method for the population balance equation, which develops a new algorithm for simulating particle growth and aggregation.
Advances in Numerical Methods for Partial Differential Equations
Sources
A priori error analysis of the proximal Galerkin method
Entropy Stable Nodal Discontinuous Galerkin Methods via Quadratic Knapsack Limiting
Spectral Analysis of Node- and Cell-Centered Higher-Order Compact Schemes for Fully Discrete One and Two-Dimensional Convection-Dispersion Equation
Convergence analysis of Anderson acceleration for nonlinear equations with H\"older continuous derivatives
Superconvergence points of Hermite spectral interpolation
MAP Estimation with Denoisers: Convergence Rates and Guarantees
A Conservative and Positivity-Preserving Discontinuous Galerkin Method for the Population Balance Equation
Explicit Monotone Stable Super-Time-Stepping Methods for Finite Time Singularities
Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation
On MAP estimates and source conditions for drift identification in SDEs