Anisotropic Slope Limiting Techniques for Ocean Flow Simulation on Unstructured Meshes

• Applicants: Professor Dr. Vadym Aizinger; Dr. Sergey Danilov; Professor Dr. Dmitri Kuzmin
• Subject Area: Oceanography
• Term: from 2013 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 246518690

• Final Report Year: 2018

Final Report Abstract

The research conducted in this project under the auspices of DFG has significantly advanced the state of the art in the field of physics-compatible high-resolution finite element methods for numerical simulation of ocean flows. The new limiting approaches are readily applicable to other systems of conservation laws, such as the Euler equations of gas dynamics and the equations of magnetohydrodynamics (MHD). In the context of multiresolution ocean modeling, the use of anisotropic limiting is required to reduce spurious mixing for salinity and temperature and enhance the fidelity of overflow simulations. Overflows play a fundamental role in the deep water formation in the ocean. Simulating them in the presence of a complex topography represents a particularly challenging application for high-resolution finite element schemes. Other highlights of the developed methodology (objective directional vector limiters, product rule limiters for velocity fields, extensions to high-order finite elements, accuracy-preserving smoothness indicators, failsafe mechanisms for a posteriori control of derived quantities) also lead to significant improvements in terms of robustness, accuracy, and efficiency. Further research endeavors will focus on theoretical studies of proposed schemes and their use in other applications. The development and analysis of improved limiting techniques in this project was particularly aimed at improving the fidelity, accuracy, and efficiency of unstructured mesh multiresolution models of coastal and global ocean circulation. Any significant improvement in the accuracy of a numerical advection scheme and reduction in the rate of numerical mixing is of vital importance for maintaining the model’s ability to preserve watermasses. The methodology developed in this project will ultimately improve the predictive power of coupled climate simulations suffering from the biases inherent in current ocean simulators. While extensive knowledge has been accumulated in the field of structured mesh methods for ocean modeling, the experience with multiresolution models that support unstructured meshes still lags far behind. The FESOM package is the first high-performance finite element code that is as efficient as structured grid codes for ocean flows. The effort invested in the development of anisotropic limiting techniques for the underlying FEM-FCT schemes will make it even more competitive. The findings of this project will also be of profound importance for further development of UTBEST3D.

Publications

• Scale separation in fast hierarchical solvers for discontinuous Galerkin methods. Appl. Math. & Computation 266 (2015) 838–849
  V. Aizinger, D. Kuzmin and L. Korous
  
• Embedded discontinuous Galerkin transport schemes with localised limiters. J. Comput. Phys. 311 (2016) 363–373
  C.J. Cotter and D. Kuzmin
  
• Anisotropic slope limiting for discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 84 (2017) 543-565
  V. Aizinger, A. Kosik, D. Kuzmin and B. Reuter
  
• Bathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization. Ergebnisberichte Angew. Math. 586, TU Dortmund University, 2018
  H. Hajduk, D. Kuzmin and V. Aizinger
  
• Frame-invariant directional vector limiters for discontinuous Galerkin methods. Ergebnisberichte Angew. Math. 585, TU Dortmund University, 2018
  H. Hajduk, D. Kuzmin and V. Aizinger
  
• Sequential limiting in continuous and discontinuous Galerkin methods for the Euler equations. J. Comput. Phys. 356 (2018) 372-390
  V. Dobrev, Tz. Kolev, D. Kuzmin, R. Rieben and V. Tomov