On semi-implicit schemes for the incompressible Euler equations via the vanishing viscosity limit

TL;DR

This work addresses numerically approximating the incompressible Euler equations by leveraging the vanishing viscosity limit from the Navier–Stokes system. It introduces a semi-implicit Fourier spectral scheme that is unconditionally energy-stable and provides uniform $H^s$ bounds, along with an iterative full discretization method; a novel integration-by-parts technique lowers the $L^2$-error regularity from $H^4$ to $H^3$, enabling first-order convergence in $L^2$ and in $H^s$ under suitable regularity. The authors prove error bounds $\\sup_n \|u^{n+1}-v(t_{n+1})\\|_{L^2} \le C_1(\\nu+\\tau+N^{-s+1})$ and, for $u_0\in H^{s+3}$, $\\sup_n \|u^{n+1}-v(t_{n+1})\\|_{H^s} \le C_2(\\tau+N^{-2}+N^2\\nu)$, and show the full discretization converges under $\\tau\\le C^*\\max\\{\\nu,N^{-1}\\}$. Numerical experiments on benchmark flows confirm the predicted error behavior for low and high regularity and demonstrate the method’s ability to capture Euler dynamics in the inviscid limit. Overall, the paper provides a principled, spectrally accurate framework for Euler simulations via the inviscid limit with rigorous stability and convergence analysis.

Abstract

A new type of systematic approach to study the incompressible Euler equations numerically via the vanishing viscosity limit is proposed in this work. We show the new strategy is unconditionally stable that the $L^2$-energy dissipates and $H^s$-norm is uniformly bounded in time without any restriction on the time step. Moreover, first-order convergence of the proposed method is established including both low regularity and high regularity error estimates. The proposed method is extended to full discretization with a newly developed iterative Fourier spectral scheme. Another main contributions of this work is to propose a new integration by parts technique to lower the regularity requirement from $H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best knowledge, this is one of the very first work to study incompressible Euler equations by designing stable numerical schemes via the inviscid limit with rigorous analysis. Furthermore, we will present both low and high regularity errors from numerical experiments and demonstrate the dynamics in several benchmark examples.

Figures (3)

• Figure 1: Dynamics of 2D incompressible Euler equations (NS equations) by scheme \ref{['5.1']} where $\nu = 0.0001$, $\tau= 0.01,~N_x=N_y = 128$ and the initial data $u_0$ is given in \ref{['6.1']}. $m=2$ in the first line, $m=8$ in the second line and $m=20$ in the third line.
• Figure 2: Dynamics of 2D incompressible Euler (NS) equations by scheme \ref{['5.1']} where $\tau= 0.001,~N_x=N_y = 128$ and the initial data $u_0$ is a double shear flow given in \ref{['6.2']}. We vary the choice $\nu = 0,0.001,0.1$. $\nu=0$ in the first line, $\nu=0.001$ in the second line and $\nu=0.1$ in the third line.
• Figure 3: Dynamics of 2D incompresible Euler (NS) equations by scheme \ref{['5.1']} where $\tau= 0.001,~N_x=N_y = 128$ and the initial data $u_0$ is composed of two Guassian vortices given in \ref{['6.3']}.

Theorems & Definitions (14)

• Theorem 1.1: Unconditional stability
• Remark 1.1
• Corollary 1.2: $H^1$-energy stability
• Theorem 1.3: Error estimates
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1: Commutator estimate
• proof
• Lemma 2.2: Inviscid limit