Advances in Numerical Methods for Hyperbolic PDEs and Electromagnetic Analysis

The field of numerical methods for hyperbolic PDEs and electromagnetic analysis is experiencing significant developments, with a focus on improving the accuracy and efficiency of simulations. Researchers are exploring new approaches to preserve stationarity and divergence in multi-dimensional problems, such as the use of global flux formulations and weighted essentially non-oscillatory (WENO) schemes. Additionally, there is a growing interest in applying machine learning techniques, like Reservoir Computing, to predict complex dynamics and improve long-term forecasting abilities. The development of hybrid methods, combining different numerical techniques, is also gaining traction, enabling the analysis of complex, multi-material environments and heterogeneous media. Noteworthy papers include:

  • A paper on a genuinely multi-dimensional stationarity preserving global flux Finite Volume formulation, which significantly outperforms existing methods.
  • A study on model-free forecasting of rogue waves using Reservoir Computing, demonstrating remarkable agreement between predicted and actual dynamics.
  • A work on a hybrid DEC-SIE framework for potential-based electromagnetic analysis, offering a unified and efficient approach for solving electromagnetic scattering and radiation problems.

Sources

Genuinely multi-dimensional stationarity preserving global flux Finite Volume formulation for nonlinear hyperbolic PDEs

Model-free Forecasting of Rogue Waves using Reservoir Computing

An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems

CAD-Integrated Electrostatic Boundary Element Simulations with Non-Conforming Higher-Order Meshes

High order global flux schemes for general steady state preservation of shallow water moment equations with non-conservative products

A hyperboloidal method for numerical simulations of multidimensional nonlinear wave equations

A Hybrid DEC-SIE Framework for Potential-Based Electromagnetic Analysis of Heterogeneous Media

A second-order and unconditionally stable time filtered scheme for the Cahn-Hilliard-Navier-Stokes system

A modified Crank-Nicolson scheme for the Vlasov-Poisson system with a strong external magnetic field

High order uniform in time schemes for weakly nonlinear Schr\"odinger equation and wave turbulence

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