The recent developments in the research area of numerical methods and computational techniques have shown a significant shift towards enhancing efficiency, accuracy, and robustness in solving complex partial differential equations (PDEs). A notable trend is the integration of optimization-based limiters and adaptive-rank algorithms to address the challenges posed by non-linearities and heterogeneous coefficients in PDEs. These advancements are particularly evident in the fields of Fokker-Planck equations, advection-diffusion equations, and structural health monitoring, where multi-input multi-output frameworks are being refined for better damage detection. Additionally, there is a growing emphasis on developing implicit-explicit time discretization schemes and weak Galerkin finite element methods to handle non-convex polyhedral meshes and semilinear wave equations with nonautonomous dampings. The field is also witnessing innovations in preconditioning techniques for matrix computations and the application of hybrid adaptive methods for large-scale non-linear boundary conditions. These developments collectively aim to push the boundaries of computational efficiency and accuracy, making it possible to tackle more complex and large-scale problems in various scientific and engineering domains.
Noteworthy papers include one that introduces an optimization-based positivity-preserving limiter for semi-implicit discontinuous Galerkin schemes, significantly enhancing the efficiency of enforcing positivity in high-order accurate solutions. Another paper presents a novel staggered discontinuous Galerkin method on polytopal meshes, which improves stability and accuracy by leveraging a primal-dual grid framework. Furthermore, a study on the hybrid adaptive dual reciprocity method demonstrates its effectiveness in solving large-scale non-linear boundary conditions, offering a robust solution for complex problems.
