In the past two decades, embedded boundary (EB) methods have emerged as an attractive alternative to unstructured, body-conforming grid methods for simulations of flows involving complex geometries. These methods employ reconstruction of the flow field at the near boundary cells up to a desired accuracy. Stability of the reconstruction is essential for reliable numerical simulations. However, presence of the embedded boundaries typically render the discrete system non-normal and the usual eigenvalue analysis may not provide sufficient conditions for stability.
This talk presents non-normal linear stability of embedded boundary (EB) methods employing pseudospectra and resolvent norms. Stability of the linear wave equation is analyzed in a domain with an embedded boundary in one and two dimensions. In particular, the stability is characterized in terms of the normalized distance of the EB to the nearest ghost node (α). An important objective is that the CFL condition remains unaffected by the EB which is taken to be that of the underlying grid (as if the EB is absent or aligned with the regular grid). Various spatial and temporal discretization methods are considered including both central and upwind-biased schemes. Stability is guaranteed when α ≤ αmax where αmax ranges between 0.5 and 0.77 depending on the discretization scheme. Sharper limits on the sufficient conditions for stability are obtained based on the pseudospectral radius (the Kreiss constant) than the restrictive limits based on the usual singular value decomposition analysis. A resulting sufficient condition for stability is that the ghost node must lie within 0.5Δx distance from the EB. A simple and robust reclassification scheme is presented for the ghost cells (dubbed "hybrid ghost cells'') to ensure Lax stability of the discrete system. This has been tested successfully for both low and high order discretization schemes with transient growth of at most order one.
Dr. Narsimha R. Rapaka joined KAUST in Dec. 2015 as a postdoctoral fellow at Fluid and Plasma Simulation (FPS) laboratory lead by Prof. Ravi Samtaney (PSE Division). At FPS lab, his research is focused on simulations of fluid flow in complex geometry using embedded boundary method and non-modal stability analysis. He received his Ph.D. from University of California San Diego, M.Tech from Indian Institute of Technology Kanpur and B. Tech from Jawaharlal Nehru Technological University Hyderabad, all in Mechanical Engineering. His Ph.D. thesis work was focused on simulations of stratified turbulent flows in complex geometry using immersed boundary method. Before his Ph.D., he worked at TATA Motors, Pune as a Development Manager and at IIT Kanpur as Sr. Project Associate.