Prof. D. S. Balsara of University of Notre Dame, IN, United States

Adaptive mesh refinement (AMR) is the art of solving PDEs on a mesh hierarchy with increasing mesh refinement at each level of the hierarchy. Accurate treatment on AMR hierarchies requires accurate prolongation of the solution from a coarse mesh to a newly-defined finer mesh. For scalar variables, suitably high order finite volume WENO methods can carry out such a prolongation. However, classes of PDEs, like computational electrodynamics (CED) and magnetohydrodynamics (MHD), require that vector fields preserve a divergence constraint. The primal variables in such schemes consist of normal components of the vector field that are collocated at the faces of the mesh. As a result, the reconstruction and prolongation strategies for divergence constraint-preserving vector fields are necessarily more intricate. In this talk, we present a fourth order divergence constraint-preserving prolongation strategy that is analytically exact. Extension to higher orders using analytically exact methods is very challenging. To overcome that challenge, a novel WENO-like reconstruction strategy is invented that matches the moments of the vector field in the faces where the vector field components are collocated. This approach is almost divergence constraint-preserving; so we call it WENO-ADP. To make it exactly divergence constraint-preserving, a touch-up procedure is developed that is based on a constrained least squares (CLSQ) based method for restoring the divergence constraint up to machine accuracy. With the touch-up, it is called WENO-ADPT. It is shown that refinement ratios of two and higher can be accommodated. An item of broader interest in this work is that we have also been able to invent very efficient finite volume WENO methods where the coefficients are very easily obtained and the multidimensional smoothness indicators can be expressed as perfect squares. We demonstrate that the divergence constraint-preserving strategy works at several high orders for divergence-free vector fields as well as vector fields where the divergence of the vector field has to match a charge density and its higher moments. We also show that our methods overcome the late time instability that has been known to plague adaptive computations in Computational Electrodynamics.

Biography:

Dinshaw S. Balsara received the Ph.D. degree in computational physics and astrophysics from the University of Illinois at Urbana-Champaign, Champaign, IL, USA, in 1990. He is currently a Professor with the Department of Physics and the Department of Applied and Computational Mathematics and Statistics. He has developed computational algorithms and applications in the areas of interstellar medium, turbulence, star formation, planet formation, the physics of accretion disks, compact objects, and relativistic astrophysics. Many of the algorithms developed by him for higher order methods have seen extensive use and have been copiously cited. Dr. Balsara was the recipient of the 2014 Department of Energy Award of Excellence for significant contributions to the Stockpile Stewardship Program and the 2017 Global Initiative on Academic Networks Award from the Government of India. He serves the community as an Associate Editor of Journal of Computational Physics and Computational Astrophysics and Cosmology.