Performance analysis of relaxation Runge-Kutta methods

Recently, global and local relaxation Runge-Kutta methods have been developed for guaranteeing the conservation, dissipation, or other solution properties for general convex functionals whose dynamics are crucial for an ordinary differential equation solution. These novel time integration procedures have an application in a wide range of problems that require dynamics-consistent and stable numerical methods. The application of a relaxation scheme involves solving scalar nonlinear algebraic equations to find the relaxation parameter. Even though root-finding may seem to be a problem technically straightforward and computationally insignificant, we address the problem at scale as we solve full-scale industrial problems on a CPU-powered supercomputer and show its cost to be considerable. In particular, we apply the relaxation schemes in the context of the compressible Navier-Stokes equations and use them to enforce the correct entropy evolution. We use seven different algorithms to solve for the global and local relaxation parameters and analyze their strong scalability. As a result of this analysis, within the global relaxation scheme, we recommend using Brent's method for problems with a low polynomial degree and of small sizes for the global relaxation scheme, while secant proves to be the best choice for higher polynomial degree solutions and large problem sizes. For the local relaxation scheme, we recommend secant. Further, we compare the schemes' performance using their most efficient implementations, where we look at their effect on the timestep size, overhead, and weak scalability. We show the global relaxation scheme to be always more expensive than the local approach-typically 1.1-1.5 times the cost. At the same time, we highlight scenarios where the global relaxation scheme might underperform due to its increased communication requirements. Finally, we present an analysis that sets expectations on the computational overhead anticipated based on the system properties.