Numerical Methods for Complex Systems

The field of numerical methods for complex systems is witnessing significant advancements, driven by the need for efficient and accurate solutions to multidisciplinary problems. A key direction in this area is the development of robust and unified approaches for handling heterogeneous porous media, contaminant transport, and parabolic-type equations. Researchers are exploring innovative finite element methods, block-centered schemes, and one-step schemes to address the challenges inherent in these systems. Noteworthy papers in this area include: A paper introducing a novel weak Galerkin finite element method for the Brinkman equations, which offers a unified approach for capturing both Stokes- and Darcy-dominated regimes. Another paper presents a scalable ADER-DG transport method with a polynomial order-independent CFL limit, achieving a maximum stable time step governed by an element-width based CFL condition.

Sources

Weak Galerkin Methods for the Brinkman Equations

A Family of Block-Centered Schemes for Contaminant Transport Equations with Adsorption via Integral Method with Variational Limit

A new class of one-step A-stable and L-stable schemes of high-order accuracy for parabolic type equations

Scalable ADER-DG Transport Method with Polynomial Order Independent CFL Limit

A structure-preserving finite element framework for the Vlasov-Maxwell system

A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

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