Qualitative Analysis, Structure-Preserving Discretizations and Scalable Solvers for Multiphysics Problems

Project: Research project

Project Details


Objectives of the project. This proposal is devoted to the development of advanced numerical simulation methods for multiphysics problems. Simulation for these problems typically involve solving coupled systems of partial differential equations (PDEs). Accurate simulations for these coupled PDEs are extremely challenging in a number of fundamental ways and there is a great demand of robust, efficient, and practical numerical algorithms to address these challenges. In this project, we propose to investigate an integrated study of qualitative analysis, robust discretization and scalable solvers for multiphysics problems with applications to important DOE-mission science problems. The proposed project is aimed to develop a theoretical foundation of integrated study of these problems. More specifically, we will focus our research on two science applications: magnetohydrodynamics (MHD) models with application to fusion energy and fluid-structure interaction (FSI) model with application to hydropower. These two applications represent two different categories of multiphysics problems: everywhere coupling (MHD) between Navier-Stokes and Maxwell equations and subdomain coupling (FSI) between Navier-Stokes equations and elasticity equations. However, these two multiphysics models share similar mathematical properties on the abstract level, which motivates us to study them in a unified fashion. They are both highly nonlinear, strongly coupled, and multiscale for both time and space and they are computationally challenging to simulate. We aim to develop an integrated framework for the design, analysis, and application of numerical simulations for these multiphysics problems.Description of the project. We propose to carry out theoretical investigations on fundamental issues such as existence, well-posedness, and stability of both the continuous and the discretized multiphysics systems, design robust discretizations which preserve fundamental physics laws (e.g. conservation of mass, volume, energy and Gauss law) and properties of physical quantities (e.g. positive of the density, incompressibility), and scalable and robust solvers which are based on multilevel techniques and robust with respect to physical and discretization parameters. More specifically, we propose to investigate the following topics:(1) Perform qualitative analysis on both continuous and discrete levels, investigate the solvability and stability for the linearized operators obtained by both simple fixed point iteration and Newtown method.(2) Develop new structure preserving finite element discretizations preserve certain crucial physical laws and properties in both MHD and FSI problems. Carry out error analysis between the finite element solutions and the exact solutions of the original MHD models and FSI models.(3) Design robust and scalable preconditioners for the linearized discretized systems and analyze the convergence rate for the corresponding preconditioned Krylov methods by using properties of the structure preserving finite element schemes and analytical techniques.Besides the theoretical studies and algorithmic developments, we plan to apply proposed discretizations and solvers for certain benchmark problems of DOE's interests and provide prototype simulations to demonstrate the potential of our proposed numerical techniques, especially on the robustness and scalability.Impact of the project. We expect that our proposed research on the integrated study will lead to significant advances in the numerical simulations for MHD and FSI. Through qualitative analysis of the continuous and discrete problems, we will develop more robust discretizations that preservesimportant physical properties in MHD and FSI models, e.g. incompressibility and Gauss's law. Efficient and salable preconditioners will be developed to solve the discretized large-scale systems. As a result, (nearly) optimal and high scalable MHD and FSI solvers will be developed and tested. Such new solvers will make it possible to use highly-adapted meshed and thus allow for resolutions that are currently out of reach.The results of this project will have an impact on computational and applied mathematics, especially the theoretical analysis for PDEs, development of robust and structure preserving discretizations, and the design of efficient and scalable solvers, as well as on the target applications in MHD and FSI. More generally, the theories, algorithms, and frameworks developed in this proposal will contribute to the advancement of numerical simulations for multiphysics applications and provide tools for exploration of multiphysics problems in other fields.
Effective start/end date9/1/158/31/18


  • Advanced Scientific Computing Research


Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.