This project develops and analyzes mesh and method-adaptive high-order numerical methods for solving partial differential equations. That is, high order methods such as spectral methods, WENO schemes, radial basis functions and discrete variable expansions will be incorporated with various mesh sizes to create a highly accurate, flexible and robust method that varies by domain depending on the smoothness of the solution in that region. High order adaptive viscosity discontinuous Galerkin methods will also be considered to stabilize the nonlinear problem and enhance accuracy. Such techniques can be considered as optimal, since they combine the best mesh with the highest possible order accuracy throughout the computational domain. In one important application, an efficient three dimensional parallel multi-domain mesh-adaptive spectral penalty code will be created for the simulation of the reactive flow in a scramjet engine's cavity flame holder, which is critical for designing a supersonic engine. This work will use an already successful two dimensional static multi-domain spectral penalty reactive code. One and two dimensional mesh-adaptive hybrid methods will be simultaneously developed.
These robust, accurate and adaptive techniques have a broad range of applications, including combustion, wave propagation in heterogeneous media and supersonic shock waves. Such problems demand highly accurate and efficient numerical algorithms in order to capture small-scale features of the solution. The proposed techniques in this project are well suited to perform these types of simulations. In addition to the supersonic scramjet engine example, one interesting application is the modeling of water propagation in a fuel cell unit. Understanding the small-scale phenomenon is critical to enhance the fuel efficiency of a fuel cell. In this case, the adaptive high order numerical simulations are crucial since the laboratory experiments are costly, difficult to measure and even fail to capture such small-scale phenomenon. Finally, this project advances state of the art computational methods and will provide valuable insight into the investigation of various challenging physical and engineering problems.
