Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

Abstract

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

Figures (12)

• Figure 1: A fluid domain $\Omega^f$ embedded into a torus $\mathbb{T}^d$.
• Figure 2: Dual mesh $D_\sigma = D_{\sigma,K} \cup D_{\sigma,L}$.
• Figure 3: Experiment 1: Numerical solutions $\varrho_h$ (left) and $\bm{u}_h$ (right) obtained with $h = 0.2 \cdot 2^{-4}$ for different $\epsilon = 4^{-m-2}, m = 1, \dots, 4$ from top to bottom.
• Figure 4: Experiment 1: The errors $E_\varrho^{\epsilon}, E_{\bm{u}}^{\epsilon}, E_{\nabla_x \bm{u}}^{\epsilon}, R_E^{\epsilon}$ with respect to $h$ for different but fixed $\epsilon$. The black and red solid lines without any marker denote the reference slope of $h$ and $h^2$, respectively.
• Figure 5: Experiment 1: Errors $E_\varrho, E_{\bm{u}}, E_{\nabla_x \bm{u}}$ and relative energy $R_E$ with respect to the pairs $(h,\epsilon(h)) = \left(0.2\cdot 2^{-m}, 2^{-(m+14)/2} \right)$ (left), $\ \left(0.2\cdot 2^{-m}, 4^{-(m+2)} \right)$ (middle) and $\left(0.2\cdot 2^{-m}, 16^{-m} \right)$ (right), $m = 0,1,2,3$. The black solid and red dashed lines without any marker denote the reference slope of $h$ and $h^2$, respectively.

Theorems & Definitions (46)

• Definition 2.1: DW solution of the penalized problem
• Definition 2.2: DW solution of the Dirichlet problem
• Remark 2.3
• Definition 3.1: Finite volume method
• Remark 3.2
• Lemma 4.1: Properties FeLMMiSh
• Lemma 4.2: Energy stability
• proof
• Lemma 4.3: Uniform bounds
• Lemma 4.4: Consistency formulation