Internal tide generation at Luzon Strait simulated using Immersed Boundary Method
Embedded Boundary Methods
Fluid flow problems involving complex geometries are ubiquitous in many engineering and environmental applications. Body fitted meshes are widely used in engineering applications, however, they suffer from time consuming mesh generation, poor mesh quality, and increased memory requirements. In the present work, a Cartesian mesh is used to alleviate these issues and take advantage of efficient solution techniques. Since the Cartesian mesh is usually not aligned with the embedded boundaries, effects of complex geometry are handled through an embedded boundary (EB) method coupled with adaptively refined mesh (AMR) techniques. An important step in this approach is to reconstruct the flow at cells near EB and satisfy prescribed boundary conditions. Closed form analytical expressions are developed for EB reconstruction along with efficient, non-iterative solution methodology which is particularly desirable for moving boundary simulations. Further, non-normal stability of EB method is analysed through pseudospectra and resolvent norms (Rapaka & Samtaney, JCP 2018). Based on sufficient conditions for stability, hybrid ghost cells (HGC) are designed which offer superior stability properties compared with conventional ghost cells. The HGC are demonstrated to improve the overall stability of EB method in both incompressible and compressible Navier-Stokes solvers.
Pseudospectra and Non-normal Stability Analysis
For numerical solution of initial boundary value problems, (Lax-) stability of the underlying discretization is essential for convergence. Presence of physical boundaries (including EB) or upwind biased discretizations typically render a discrete system non-normal, i.e., the eigenvectors of the system form a non-orthogonal basis. Non-normal systems typically exhibit transient growth and the eigenvalues provide only the asymptotic growth rates of the system. Thus, normal mode/Fourier stability analysis provides merely a necessary condition of (asymptotic) stability for such systems. Although singular value decomposition (SVD) provides sufficient conditions for stability, these conditions are far from sharp and practically not much useful in many cases. Better, though not sharp, stability estimates can be obtained through numerical range. According to state of the art theories in the applied mathematics literature, pseudospectra provides by far the sharpest estimates of sufficient conditions for Lax-stability.
The theory of pseudospectra is employed for the first time to study non-normal stability of EB methods in the context of linear scalar advection and advection-diffusion equations (Rapaka & Samtaney, JCP 2018). Various low and high order spatio-temporal finite difference discretization schemes are considered including central and upwind biased, explicit and compact schemes. Method of lines approach is employed to study the semi and the full discretizations. Stability estimates for full discretizations obtained through pseudospectra are compared with those through eigenvalue analysis, numerical range and SVD. Uniform upper bounds on transient growth are obtained in terms of the Kreiss constant. The study shows that the sharpest stability estimates are obtained through the pseudospectra. Further, SVD fails to provide sufficient conditions of stability for some of the discretization schemes.
Fast Multipole Method + Boundary Element Method
A novel EB Poisson solver is developed for complex domains that outperforms exiting methods and alleviates the need for multi-level representation of complex boundaries present in the existing multigrid methods. The solver integrates geometric multi-grid method with boundary element method (BEM) and is integrated into an incompressible Navier-Stokes solver (Rapaka & Samtaney, JCP 2020). Conventional implementation of BEM is known to have O(N2) operational count for a discrete system with N degrees of freedom. A fast multipole method (FMM) is used to bring down the computational complexity to O(N).
High Resolution Numerical Schemes
Higher resolution numerical schemes for finite difference discretization are developed through multi-parameter optimization. Near spectral resolution is achieved for non-dimensional wave numbers up to 2.95 (maximum possible limit is 3.14, see the figure below). These schemes have O(N) computational complexity for a system with N degrees of freedom and performance evaluation of these schemes for both linear and non-linear partial differential equations is a subject of future work.
