The field of numerical methods for complex systems is rapidly evolving, with a focus on developing efficient and accurate algorithms for solving nonlinear equations and simulating complex phenomena. Recent developments have centered around the creation of novel finite element methods, such as the weighted implicit-explicit discontinuous Galerkin method and the cut finite element exterior calculus, which offer improved stability and accuracy for a wide range of applications. Additionally, researchers have made significant progress in the development of adaptive time-stepping strategies, such as the Energy-Variation Moving Average technique, which enable more efficient and robust simulations of complex systems. Noteworthy papers in this area include the development of novel superconvergence and ultraconvergence structures for the finite volume element method, which achieve higher-order accuracy and improved stability. Another significant contribution is the construction of basis functions for the geometry conforming immersed finite element method, which enables the simulation of complex interface problems with high accuracy. Overall, these advances have the potential to significantly impact a wide range of fields, from materials science to fluid dynamics.
Advances in Numerical Methods for Complex Systems
Sources
Weighted implicit-explicit discontinuous Galerkin methods for two-dimensional Ginzburg-Landau equations on general meshes
Novel superconvergence and ultraconvergence structures for the finite volume element method
Multilevel correction type of adaptive finite element method for Hartree-Fock equation
An efficient iteration method to reconstruct the drift term from the final measurement
An adaptive time-stepping strategy for the modified phase field crystal model with a strong nonlinear vacancy potential
Convergence Analysis of Galerkin Approximations for the Lindblad Master Equation
Construction of Basis Functions for the Geometry Conforming Immersed Finite Element Method
A priori error estimates for stable generalized finite element discretization of parabolic interface optimal control problems
Optimal $L^2$ error estimation for the unfitted interface finite element method based on the non-symmetric Nitsche's methods
On the maximum bound principle and energy dissipation of exponential time differencing methods for the chiral liquid crystal blue phases
On Convergence of the Secant Method
A decoupled Crank-Nicolson leap-frog scheme for the unsteady bioconvection flows problem with concentration dependent viscosity
Superconvergent and Divergence-Free Finite Element Methods for Stokes Equation
High-order mass- and energy-conserving methods for the nonlinear Schr\"odinger equation and its hyperbolization
Ghost stabilisation for cut finite element exterior calculus
Efficient and Flexible Multirate Temporal Adaptivity