The ability to discretize and solve time-dependent Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) remains of great importance to a variety of physical and engineering applications. Recent progress in supercomputing or high-performance computing has opened new opportunities for numerical simulation of the partial differential equations (PDEs) that appear in many transient physical phenomena, including the equations governing fluid flow. In addition, accurate and stable space-time discretization of the partial differential equations governing the dynamic behavior of complex physical phenomena, such as fluid flow, is still an outstanding challenge. Even though significant attention has been paid to high and low-order spatial schemes over the last several years, temporal schemes still rely on relatively inefficient approaches.
Furthermore, academia and industry mostly rely on implicit time marching methods. These implicit schemes require significant memory once combined with high-order spatial discretizations. However, since the advent of high-performance general-purpose computing on GPUs (GPGPU), renewed interest has been focused on explicit methods. These explicit schemes are particularly appealing due to their low memory consumption and simplicity of implementation.
This study proposes low and high-order optimal Runge-Kutta schemes for FR/DG high-order spatial discretizations with multi-dimensional element types. These optimal stability polynomials improve the stability of the numerical solution and speed up the simulation for high-order element types once compared to classical Runge-Kutta methods. We then develop third-order accurate Paired Explicit Runge-Kutta (P-ERK) schemes for locally stiff systems of equations. These third-order P-ERK schemes allow Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria, while seamlessly pairing at their interface. We then generate families of schemes optimized for the high-order flux reconstruction spatial discretization. Finally, We propose optimal explicit schemes for Ansys Fluent finite volume density-based solver, and we investigate the effect of updating and freezing reconstruction gradient in intermediate Runge-Kutta schemes. Moreover, we explore the impact of optimal schemes combined with the updated gradients in scale-resolving simulations with Fluent's finite volume solver. We then show that even though freezing the reconstruction gradients in intermediate Runge-Kutta stages can reduce computational cost per time step, it significantly increases the error and hampers stability by limiting the time-step size.