High Order Multi-Scale Numerical Methods for All-Mach Number Flows

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The primary goal of this project is the design of highly accurate and multi-scale numerical methods for all-speed flow simulations. To the best of our knowledge, highly accurate multi-scale solvers for all-Mach number flows are still underdeveloped. The proposed work is expected to establish close connections on existing state-of-art computational tools for compressible flow (with shock capturing techniques) and incompressible flows (with projection and divergence cleaning techniques). Successful methodology developments for all-speed flow simulations will have a broad impact for a wide range of computational fluid dynamics (CFD)-related applications. The proposed research work in algorithm design and analysis will promote/benefit from the demands from CFD applications fields. Further impact comes from the multidisciplinary nature of the proposed research, publications, participation and organization of mini-symposium sections in conferences, as well as the training of students.High order multi-scale solvers for all-Mach number flows are proposed. The PI seeks to develop a uniform framework to build up high order solvers that could effectively capture shocks without numerical oscillations for the compressible Euler system, and would successfully approximate the incompressible solutions of the system with proper divergence cleaning steps for low Mach flows without the need in resolving acoustic waves. In particular, the main focus would be finite difference schemes with weighted essentially non-oscillatory (WENO) reconstructions coupled with proper implicit-explicit (IMEX) Runge Kutta (RK) treatments with the following properties: (1) the schemes can robustly capture shock fronts in the compressible regime when the Mach number is of order 1; (2) the schemes automatically become high order, stable and consistent solvers for the incompressible Euler system when the Mach number approaches 0; (3) the schemes are high order accurate in both space and in time both when the acoustic waves are well-resolved and are under-resolved. Along this direction, the PI develops a thorough plan in methodology development, stability and asymptotic preserving analysis, as well as extensive benchmarked tests for all-Mach number flows. Further extensions to the Navier-Stokes system with additional consideration of viscous terms and special focus on boundary conditions will be explored.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.