The field of numerical methods for complex systems is rapidly advancing, with a focus on developing innovative techniques for solving challenging problems. One of the key directions is the development of high-order methods, which offer superior accuracy and resolution at a relatively modest computational cost. These methods are being applied to a range of problems, including compressible flow simulations, hyperbolic systems, and multiscale models. Another important area of research is the development of efficient and robust methods for solving time-fractional differential equations, which are crucial for modeling complex phenomena in various fields. Additionally, there is a growing interest in the development of stochastic approaches for solving space-time fractional diffusion models, which are widely used to describe anomalous diffusion dynamics. Noteworthy papers in this area include: A Nodal Discontinuous Galerkin Method with Low-Rank Velocity Space Representation for the Multi-Scale BGK Model, which establishes a foundation for extending modern low-rank techniques to solve the Boltzmann equation in realistic settings. The paper High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation presents a class of nonuniform time-stepping, high-order linear stabilized schemes that can preserve both the discrete energy stability and maximum-bound principle for the time-fractional Allen-Cahn equation.
Advances in Numerical Methods for Complex Systems
Sources
A Nodal Discontinuous Galerkin Method with Low-Rank Velocity Space Representation for the Multi-Scale BGK Model
A Relaxed Step-Ratio Constraint for Time-Fractional Cahn--Hilliard Equations: Analysis and Computation
Parallel Nodal Interior-Penalty Discontinuous Galerkin Methods for the Subsonic Compressible Navier-Stokes Equations: Applications to Vortical Flows and VIV Problems
Performance evaluation of high-order compact and second-order gas-kinetic schemes in compressible flow simulations
High-order nonuniform time-stepping and MBP-preserving linear schemes for the time-fractional Allen-Cahn equation