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