Numerical analysis of an H(div)-conforming divergence-free DG method with a second-order explicit Runge-Kutta scheme for incompressible flows

Abstract: Recently, H(div)-conforming DG type methods coupled with Runge-Kutta (RK) time stepping have been widely employed for simulating high Reynolds number flows, with the convective terms treated explicitly. Although the analysis techniques of RKDG methods were well developed, the extension to incompressible flows is highly nontrivial due to the exactly divergence-free constraint, where the key lies in analyzing the convective terms. We neglect viscosity effects, and conduct an error analysis for an H(div)-conforming divergence-free DG method combined with a second-order explicit RK scheme, for the incompressible Euler equations. We derive an a priori error estimate of $O(h^{k+1 / 2}+\tau^2)$ under a restrictive CFL condition $\tau \lesssim h^{4 / 3}$ for polynomials of degree $k \geq 1$, where $h$ and $\tau$ are the mesh size and time step size, respectively, assuming that the exact solution is smooth. For the case of linear polynomials, we investigate whether existing analytical techniques can relax the restrictive CFL condition to a standard CFL condition $\tau \lesssim h$. It is demonstrated that the exactly divergence-free constraint prevents the application of these techniques. We conjecture that the error estimates for linear polynomials cannot be derived under a standard CFL condition. Finally, we mention that based on our analytical framework, our analytical results will be readily extended to the Navier-Stokes equations at high mesh Reynolds number, with the viscous and convective terms treated explicitly. Numerical experiments are conducted, supporting our analytical results and the conjecture for linear polynomials.