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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Preconditioning and Linearly Implicit Time Integration for the Serre-Green-Naghdi Equations
Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier--Stokes Equations
From eigenvector nonlinearities to eigenvalue nonlinearities
A third-order finite volume semi-implicit method for the Shallow Water-Exner model
Quasiseparable LU decay bounds for inverses of banded matrices
Fast Converging Single Trace Quasi-local PMCHWT Equation for the Modelling of Composite Systems
Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources
Error analysis of BDF schemes for the evolutionary incompressible Navier--Stokes equations
Structure-preserving scheme for 1D KWC system
Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES
Spectral approximation to fractional integral operator
A High-Order Compact Hermite Difference Method for Double-Diffusive Convection
Sharp numerical approximation of the Hardy constant
A Generalized Framework for Higher-Order Localized Orthogonal Decomposition Methods
Anisotropic approximation on space-time domains
A Spline-Based Stress Function Approach for the Principle of Minimum Complementary Energy
Hybrid high-order approximations of div-curl systems on domains with general topology
Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials
A Hybrid High-Order Method for the Gross--Pitaevskii Eigenvalue Problem
A quasi-Grassmannian gradient flow model for eigenvalue problems
Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem
Solver Performance of Accelerated MoM for Connected Arrays
A Taylor-Hood finite element method for the surface Stokes problem without penalization
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
Multicontinuum Homogenization for Poroelasticity Model
An energy-stable parametric finite element method for the Willmore flow in three dimensions
Entropy-stable in- and outflow boundary conditions for the compressible Navier-Stokes equations
Robust space-time multiscale upscaling via multicontinuum homogenization for evolving perforated media
Runge--Kutta generalized Convolution Quadrature for sectorial problems
A discontinuous in time Streamline Diffusion Virtual Element Method for Darcy-transport problem
Efficient parameter-robust preconditioners for linear poroelasticity and elasticity in the primal formulation
Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes