Report on Current Developments in Numerical Methods and Computational Mathematics

General Direction of the Field

The recent advancements in numerical methods and computational mathematics have shown a strong emphasis on developing innovative techniques to address complex, nonlinear, and nonlocal problems. The field is moving towards higher-order methods that not only improve accuracy but also ensure stability and robustness, particularly in the context of evolving geometries, nonlinear differential equations, and nonlocal models. There is a growing interest in preserving physical properties such as energy dissipation and entropy in numerical schemes, which is crucial for the reliability of simulations in various applications, from fluid dynamics to biological transport networks.

One of the key trends is the integration of advanced discretization techniques with rigorous mathematical analysis to guarantee the convergence and stability of numerical methods. This includes the development of finite element methods (FEMs) and finite volume methods (FVMs) that can handle irregular meshes, curved boundaries, and high-dimensional problems. The use of virtual element methods (VEMs) and hybrid approaches is also gaining traction, offering flexibility in mesh construction and improved accuracy for problems with complex geometries.

Another significant development is the exploration of novel numerical schemes for specific types of differential equations, such as the Cahn-Hilliard equation, the Ginzburg-Landau model, and the Navier-Stokes equations. These schemes often incorporate innovative strategies to preserve key properties of the underlying physical systems, such as positivity, entropy dissipation, and steady-state solutions. The incorporation of auxiliary variables and the use of entropy-dissipative schemes are notable examples of these advancements.

Noteworthy Innovations

1. Numerical Method for Abstract Cauchy Problem with Nonlinear Nonlocal Condition:

   • The proposed method reduces the problem to an abstract Hammerstein equation, leveraging Sinc-based numerical evaluation of operator exponentials.

2. Maximal Regularity of Evolving FEMs for Parabolic Equations on an Evolving Surface:

   • The method preserves maximal $L^p$-regularity at the discrete level, extending results from stationary to evolving surfaces.

3. New Mixed Finite Element Method for the Cahn-Hilliard Equation:

   • The method provides well-posedness and error estimates for the linearized fully discrete scheme, validated through numerical experiments.

4. Original Energy Dissipation Preserving Corrections for Integrating Factor Runge-Kutta Methods:

   • The new methods enforce the preservation of steady-state solutions and original energy dissipation, validated through extensive numerical experiments.

5. Time Splitting and Error Estimates for Nonlinear Schrödinger Equations with a Potential:

   • The spectral localization approach yields low regularity error estimates for the time discretization, supported by discrete Strichartz estimates.

6. Influence of Gauges in the Numerical Simulation of the Time-Dependent Ginzburg-Landau Model:

   • The study reveals a tipping point value for the gauge parameter, influencing the convergence behavior and suggesting strategies to avoid numerical artefacts.

7. SAV-based Entropy-Dissipative Schemes for a Class of Kinetic Equations:

   • The schemes ensure robustness by preserving positivity and entropy dissipation, validated for the Boltzmann and Landau equations.

8. Fully-Discretely Nonlinearly-Stable Flux Reconstruction Methods for Compressible Flows:

   • The FD-NSFR scheme ensures entropy stability in both space and time, allowing larger time step sizes while maintaining robustness.

These innovations represent significant strides in the development of numerical methods that are both theoretically sound and practically effective, advancing the field towards more accurate, stable, and efficient computational solutions for complex mathematical problems.