UC Santa Cruz

The recent advent of high-performance computing hardware enables large-scale, multi-physics simulation that provides accurate physical pictures in various fields of study. In order to utilize the high-performance computing system more efficiently, the high-order numerical approximations have become one of the central themes in computational fluid dynamics (CFD) due to their potential in achieving highly accurate predictions in a limited memory capacity.The single-stage or single-step high-order temporal discretizations have shown great promise in delivering high-order temporal accuracy in fast performance. Fundamentally, the single-stage time integrators are based on a Taylor series in the time domain. Although its high performance, the single-stage time integrators are less attractive and less flexible compared to the multi-stage methods due to the complexities in calculating the coefficient of time-Taylor expansion, which usually demands the flux Jacobians and Hessians. This dissertation develops a new single-stage high-order temporal integrator under finite difference discretization. The proposed high-order temporal method is based on the Lax-Wendroff type time discretization, with an algorithmic extension that provides the system independence property. The new approach, called system-free (SF) method, furnishes ease of implementation as well as portability and flexibility of the single-stage time integration method while maintaining the accuracy and stability of the numerical solution.