The field of numerical methods for complex systems is witnessing significant advancements, driven by the need for more accurate and efficient simulations. A key direction is the development of novel finite element methods, such as the enriched Galerkin method and the Zipped Finite Element Method, which offer improved stability and convergence properties. Another area of focus is the creation of adaptive methods, including p-adaptive high-order mesh-free frameworks, which can effectively capture complex geometries and highly non-linear regions. Additionally, researchers are exploring new techniques for solving specific problems, such as the Ginzburg-Landau equation and the Boltzmann-BGK equation, using innovative methods like the fully implicit Crank-Nicolson discontinuous Galerkin method and a modified BGK collision operator. Noteworthy papers include the proposal of a stabilized coupling of conforming and mixed finite element spaces for wave propagation in thermo-poroelasticity, and the development of a provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations.
Advancements in Numerical Methods for Complex Systems
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Coupling of conforming and mixed finite element methods for a model of wave propagation in thermo-poroelasticity in the frequency domain
Space-time adaptive methods for parabolic evolution equations
Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids
Benchmarking stabilized and self-stabilized p-virtual element methods with variable coefficients
The Ginzburg-Landau equations: Vortex states and numerical multiscale approximations
Provably fully discrete energy-stable and asymptotic-preserving scheme for barotropic Euler equations
Multiscale Methods for wave propagation in materials with sign-changing coefficients
High order tracer variance stable transport with low order energy conserving dynamics for the thermal shallow water equations
A finite element method for a non-Newtonian dilute polymer fluid
Enriched Galerkin Method for Navier-Stokes Equations
A Modified BGK Collision Operator for Exact Conservation in Numerical Solutions of Boltzmann-BGK
A new analytical technique of the fully implicit Crank-Nicolson discontinuous Galerkin method for the Ginzburg-Landau Model
A p-adaptive high-order mesh-free framework for fluid simulations in complex geometries
The Zipped Finite Element Method: High-order Shape Functions for Polygons