Error Estimates of the Godunov Method for the Multidimensional Compressible Euler System

Abstract

We derive a priori error estimates of the Godunov method for the multidimensional compressible Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the $$L^2$$ L 2 -norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the $$L^1$$ L 1 -norm, that is to say, a convergence rate of 1/4 for the $$L^2$$ L 2 -error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the $$L^1$$ L 1 -norm, consequently, the numerical solution converges in the $$L^2$$ L 2 -norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis.

Compressible Navier–Stokes Equations with Potential Temperature Transport: Stability of the Strong Solution and Numerical Error Estimates

“…Our numerical experiments indicate that theoretical results obtained in Sect. 4.5 might be suboptimal, such a behaviour was observed in the literature also for other numerical methods and models, see, e.g., [9,13,16]. Figure 1 illustrates time evolution of the solution computed at different times on a mesh with h = 1/128 and for γ = 1.4.…”

“…To this end, we assume that the strong solution exists and apply a relative energy inequality and a consistency formulation for the numerical method. Such an approach has already been applied successfully to the compressible Navier-Stokes equations, see Kwon and Novotný [13], and to the compressible Euler system, see [16]. However, in those works, the approximation of a sufficiently smooth domain Ω ⊂ R d by a sequence of polygonal approximations Ω h ⊂ R d , h ↓ 0, was not considered.…”

On the convergence of a finite volume method for the Navier–Stokes–Fourier system

“…dxdt − e h S,1 (ϕ) − e h S,2 (ϕ) (A.21) Proof of Lemma A.7. The proof follows the lines of [19,Section 4], [7, Section 4], [30,Section 4] and also [18]. Here, we only concentrate on (A.17), (A.18), (A.21) and (A.22), which are new estimates compared to the above mentioned literature.…”

“…Hence, analogously to [7, Appendix C] and [30,Appendix D.2], applying the interpolation estimates, boundedness assumption (A.11) and a priori uniform bounds (A.12), we have…”

Convergence of first‐order finite volume method based on exact Riemann solver for the complete compressible Euler equations

“…In this case that the strong solution exists, convergence rates can be studied. Typically a relative energy or relative entropy is used to study error estimates, see References [6,29,31].…”