The field of numerical methods is experiencing significant growth, with a focus on developing innovative techniques for solving complex problems. A common theme among recent research areas is the pursuit of high-order methods, stability, and computational efficiency.
In the context of hyperbolic conservation laws, notable developments include the use of active flux methods and collocation methods for well-balanced schemes. The introduction of a novel active flux method with overlapping finite-volume meshes, as seen in A New Semi-Discrete Finite-Volume Active Flux Method for Hyperbolic Conservation Laws, has shown promise in capturing discontinuities and preserving physical properties. Additionally, the application of collocation RK methods for solving local non-linear problems, as presented in Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws, has demonstrated improved accuracy and stability.
The development of more efficient and accurate techniques for solving coupled problems is also a major focus in the field of numerical methods for complex systems. Researchers are exploring the use of adaptive sampling procedures, hyperreduction, and novel finite element methods to improve the performance of reduced-order models and computational simulations. A high-order combined interpolation/finite element technique for evolutionary coupled groundwater-surface water problems has demonstrated unconditionally stable and temporally second-order accurate results. Furthermore, the polyhedral scaled boundary finite element method for three-dimensional seepage analysis has achieved higher accuracy and faster convergence compared to conventional FEM.
In the area of linear algebra and matrix factorization, innovations in techniques such as canonical decomposition, singular value decomposition, and perturbation analysis have led to significant advances. The development of new optimization algorithms and frameworks, such as variable projection and adaptive mixed precision, has expanded the range of problems that can be effectively solved. Notable papers in this area include those on the calculation of canonical decomposition via local discrepancies and constructive solutions to the common invariant cone problem.
The field of numerical methods for complex materials and systems is moving towards the development of more efficient and accurate techniques for handling heterogeneous materials, fluid-structure interaction problems, and complex geometries. Partial integration-based regularization and fictitious domain methods are being explored to mitigate the challenges associated with dense system matrices and singular integral kernels. The Phantom Domain Finite Element Method has offered significant advantages in handling complex inclusion geometries and improving computational efficiency.
Overall, these advancements demonstrate the ongoing efforts to improve the efficiency and accuracy of numerical methods for complex problems. The development of fast and reliable algorithms for solving linear response calculations and the real Monge-Ampere equation has shown promising results. The introduction of enrichment functions in immersed finite element methods has improved the accuracy and stability of interface problems. As research in these areas continues to evolve, we can expect to see significant impacts on various fields, including aerodynamics, hydrodynamics, geotechnical engineering, and materials science.
