The field of numerical methods for complex systems is rapidly evolving, with a focus on developing innovative techniques for solving nonlinear problems, improving computational efficiency, and enhancing accuracy. A significant trend is the development of high-order methods, such as the third-order finite volume semi-implicit method for the Shallow Water-Exner model, which demonstrates improved stability and accuracy. Another area of research is the application of Variational Multiscale methods to nonlinear problems, including the Navier-Stokes equations, which shows promise in preserving high-order accuracy and desirable conservation properties. Additionally, there is a growing interest in developing preconditioners for linear poroelasticity and elasticity, with a focus on creating parameter-robust preconditioners that can efficiently handle large-scale saddle-point systems. Noteworthy papers in this area include the introduction of a quasi-Grassmannian gradient flow model for eigenvalue problems, which ensures asymptotic orthogonality and exponential convergence, and the development of a generalized framework for phase field-based modeling of coupled problems, which demonstrates good agreement with experimental data and existing numerical solutions. Overall, these advancements have the potential to significantly impact various fields, including engineering, geophysics, and biology, by providing more accurate and efficient numerical simulations of complex systems.
Advancements in Numerical Methods for Complex Systems
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Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier--Stokes Equations
Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources
Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES
Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials
Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem
A generalised framework for phase field-based modelling of coupled problems: application to thermo-mechanical fracture, hydraulic fracture, hydrogen embrittlement and corrosion
A Hereditary Integral, Transient Network Approach to Modeling Permanent Set and Viscoelastic Response in Polymers
Robust space-time multiscale upscaling via multicontinuum homogenization for evolving perforated media