New Invariant Domain Preserving Finite Volume Schemes for Compressible Flows

Abstract

We present new invariant domain preserving finite volume schemes for the compressible Euler and Navier–Stokes–Fourier systems. The schemes are entropy stable and preserve positivity of density and internal energy. More importantly, their convergence towards a strong solution of the limit system has been proved rigorously in [9, 11]. We will demonstrate their accuracy and robustness on a series of numerical experiments.

1 Introduction

Numerical simulations of compressible flows find their applications in many everyday problems, ranging from engineering, oceanography, meteorology to hemodynamics. Over the years a large variety of powerful numerical schemes has been developed. Let us point out a few well-established and practical schemes, e.g., [1, 5, 6, 12, 16, 18, 19, 23, 25]. Despite of their practical success the rigorous numerical analysis, in particular, in multiple space dimensions, is still open in general.

In [13, 14] the concept of invariant domain preserving schemes for hyperbolic conservation laws has been introduced. These methods satisfy some important structure preserving properties, such as positivity of some quantities, entropy production or the minimum entropy principle. In our recent works [9,10,11] we have proposed new finite volume schemes for the compressible Euler equations of gas dynamics, compressible Navier–Stokes and Navier–Stokes–Fourier equations, respectively. Our new finite volume methods belong to the class of the invariant domain preserving schemes. Their properties further allowed us to study the convergence of the schemes rigorously. More precisely, we proved a nonlinear variant of the Lax equivalence theorem: a consistent numerical scheme is convergent if and only if it is stable.

Of course, the compressible Euler and Navier–Stokes–Fourier equations are truly nonlinear, thus we have to overcome difficulties arising due to nonlinear terms. To this goal, we apply a concept of dissipative measure–valued solutions developed in [2, 3, 8] for the above mentioned systems, respectively. Indeed, the Young measures which are the space-time parametrized probability measures replace the linearity setting. They allow us to pass to the limit in nonlinear terms and show the convergence of our finite volume schemes. A limit is in general only a measure, more precisely a dissipative measure–valued solution. We refer a reader to [2, 3, 8] and [9,10,11] for more details on its definition.

A crucial ingredient of our convergence analysis is the fact that we have the weak-strong uniqueness principle for all systems mentioned above. More precisely, if the strong solution exists our dissipative measure–valued solution coincides with the former on its lifespan. Consequently, we get the strong convergence of our numerical solutions to the strong solution in appropriate Lebesgue spaces. The main aim of this paper is to illustrate experimentally the behavior of our new invariant domain preserving finite volume schemes for compressible fluids, namely for the Euler and the Navier–Stokes–Fourier systems, cf. [9, 11].

The gas dynamics of inviscid compressible flows is governed by the Euler equations

$$\begin{aligned} \partial _{t} \varrho + {\text {div}}_{x}{\boldsymbol{m}}= & {} 0, \nonumber \\ \partial _t {\boldsymbol{m}}+ {\text {div}}_{x}({\boldsymbol{m}}\otimes {\boldsymbol{u}}) + \nabla _xp= & {} 0, \nonumber \\ \partial _t E + {\text {div}}_{x}( (E + p) {\boldsymbol{u}})= & {} 0, \end{aligned}$$

(1)

where $\varrho , p, {\boldsymbol{u}}, {\boldsymbol{m}}= \varrho {\boldsymbol{u}}$, and E represent the density, pressure, velocity, momentum and the total energy of a fluid, respectively. Taking into account the viscous and heat conducting effects yields the Navier–Stokes–Fourier equations

$$\begin{aligned} \partial _t \varrho + {\text {div}}_{x}{\boldsymbol{m}}= & {} 0, \nonumber \\ \partial _t {\boldsymbol{m}}+ {\text {div}}_{x}({\boldsymbol{m}}\otimes {\boldsymbol{u}}) + \nabla _xp= & {} {\text {div}}_{x}\mathbb {S}(\mathbf{D} ({\boldsymbol{u}})) , \nonumber \\ \partial _t (\varrho e) + {\text {div}}_{x}( \varrho e{\boldsymbol{u}}) - {\text {div}}_{x}(\kappa \nabla _x\vartheta )= & {} \mathbb {S}(\mathbf{D} ({\boldsymbol{u}})) : \nabla _x{\boldsymbol{u}}-p {\text {div}}_{x}{\boldsymbol{u}}, \end{aligned}$$

(2)

where the viscous stress tensor $\mathbb {S}$ reads

$$ \mathbb {S}(\mathbf{D} ({\boldsymbol{u}})) = 2\mu \mathbf{D} ({\boldsymbol{u}}) + \lambda {\text {div}}_{x}{\boldsymbol{u}}\mathbb {I}, \quad \mathbf{D} ({\boldsymbol{u}})=\frac{\nabla _x{\boldsymbol{u}}+ \nabla _x{\boldsymbol{u}}^T}{2}. $$

The systems Eq. 1 and Eq. 2 are closed by the standard pressure law for a perfect gas $p = p(\varrho , \vartheta ) = \varrho \vartheta ,$ $\vartheta$ is the temperature. Denoting further e the specific internal energy, s the physical entropy, $\gamma >1$ the adiabatic coefficient and $c_v=\frac{1}{\gamma -1}$ the specific heat at constant volume we have

$$ e(\varrho , \vartheta ) = c_v \vartheta ,\ s(\varrho , \vartheta ) = \log \left( \frac{\vartheta ^{c_v}}{\varrho } \right) = \frac{1}{\gamma - 1} \log \left( \frac{p}{\varrho ^\gamma } \right) . $$

The total energy $E = \frac{1}{2}\frac{{\boldsymbol{m}}^2}{ \varrho } + \varrho e$ consists of the kinetic energy and the internal energy.

Both systems Eq. 1 and Eq. 2 are solved in the time-space cylinder $(0,T)\times {\boldsymbol{\Omega }},$ ${\boldsymbol{\Omega }} \subset R^d,$ $d=2,3.$ We assume that these systems are accompanied with appropriate boundary conditions: either the periodic boundary conditions when the domain ${\boldsymbol{\Omega }}$ is identified with a flat torus, or the no-flux boundary conditions,

$$\begin{aligned} {\boldsymbol{u}}|_{\partial \Omega } \cdot \boldsymbol{n}=0, \ \nabla _x\vartheta \cdot \boldsymbol{n} = 0 \end{aligned}$$

in the case of the Euler equations, see Eq. 1, or the no–slip boundary conditions,

$$\begin{aligned} {\boldsymbol{u}}|_{\partial \Omega }=0 \end{aligned}$$

for the Navier–Stokes–Fourier system, see Eq. 2. To close the formulation of the problem we impose the initial conditions

$$\begin{aligned} \mathbf{U} (0) =\mathbf{U} _0 , \text { with } \varrho _0>0\text { and } { E_0-\frac{1}{2} \frac{|{\boldsymbol{m}}_0|^2}{{\varrho _0}}>0, } \end{aligned}$$

(3)

where $\mathbf{U} =(\varrho ,{\boldsymbol{m}},E)$ or $\mathbf{U} =(\varrho ,{\boldsymbol{m}},\vartheta )$ for the Euler and the Navier–Stokes–Fourier equations, respectively.

2 Finite Volume Schemes

We start by introducing the mesh, space discretization and suitable discrete spaces.

2.1 Mesh and Space Discretization

Primary Grid. We suppose the physical space to be a polyhedral domain ${\boldsymbol{\Omega }} \subset R^d$, $d=2,3$, that is decomposed into compact elements,

$$ \overline{\Omega } = \bigcup _{K \in \mathcal {T}_{h}} K. $$

The elements K are sharing either a common face, edge, or vortex.

They can be chosen to be triangular, rectangular, or any combination of them. The primary mesh $\mathcal {T}_{h}$ is assumed to satisfy the standard regularity assumptions, cf. [4, 7]. The set of all faces is denoted by $\Sigma ,$ while the set of faces on the boundary is denoted by $\Sigma _{ext},$ and the set of interior faces by $\Sigma _{int}=\Sigma \backslash \Sigma _{ext}$. Note that there is no boundary if the flow is periodic:

$$\begin{aligned} \Sigma _{ext}= \emptyset \ \text{ and } \ \Sigma _{int}=\Sigma . \end{aligned}$$

Each face is associated with an outer normal vector $\boldsymbol{n}$. Let |K|, $|\sigma |$ be the Lebesgue measure of an element K and a face $\sigma$, respectively. We shall suppose

$$ |K| \approx h^d, \ |\sigma | \approx h^{d-1} \ \text{ for } \text{ any }\ K \in \mathcal {T}_{h}, \ \sigma \in \Sigma . $$

The parameter $h\in (0,1)$ is the maximum element size, i.e., the size of the mesh $\mathcal {T}_{h}.$

For the discretization of the Navier–Stokes–Fourier system, see Eq. 2, we additionally require the primary grid $\mathcal {T}_{h}$ to satisfy the following property: there is a family of control points $P_h=\{ {x}_K\,|\, {x}_K \in K, K \in \mathcal {T}_{h}\}$, such that the segment $[{x}_K,{x}_L]$ for any adjacent elements K and L is perpendicular to their common face $\sigma =K\cap L$. We denote by $d_\sigma =({x}_K,{x}_L)$ the Euclidean distance between the points ${x}_K$ and ${x}_L$ in $R^d$. This requirement is naturally satisfied by any rectangular mesh with $P_h$ being the set of gravity centers of all elements. For a triangular mesh, we can use the well-centered mesh [24], where $P_h$ is the set of circumcenters of all elements.

Dual Grid. For the theoretical analysis of our finite volume scheme for the Navier–Stokes–Fourier system it is convenient to introduce a dual mesh $\mathcal {D}_h$. A dual cell $D_\sigma$ associated to a face $\sigma = K\cap L$ is defined as $D_\sigma = D_{\sigma ,K} \cup D_{\sigma ,L}$, where $D_{\sigma ,K}$ ($D_{\sigma ,L}$) is a triangle (tetrahedron) built by ${x}_K$ and the common vertices of K and L, see Fig. 1 for a two-dimensional example.

Fig. 1

Dual grid

Discrete Function Spaces. We denote by $Q_h$ and $W_h$ the set of piecewise constant functions on the primary grid $\mathcal {T}_{h}$ and the dual grid $\mathcal {D}_h$, respectively. Moreover, ${\boldsymbol{v}}_{h}\in Q_h$ (resp. ${\boldsymbol{v}}_{h}\in W_h$) means that each component of ${\boldsymbol{v}}_{h}$ belongs to $Q_h$ (resp. $W_h$). Further, for a piecewise continuous function v, whenever $x \in \sigma \in \Sigma _{int}$, we define

$$\begin{aligned}v^{\text {out}}(x)&= \lim _{\delta \rightarrow 0+} v(x + \delta {\boldsymbol{n}}),&v^{\text {in}}(x)&= \lim _{\delta \rightarrow 0+} v(x - \delta {\boldsymbol{n}}),\\ \overline{v}(x)&= \frac{v^{\text {in}}(x) + v^{\text {out}}(x) }{2},&\llbracket {v}\rrbracket&= v^{\text {out}}(x) - v^{\text {in}}(x). \end{aligned}$$

Upwind Flux. Given a velocity ${\boldsymbol{u}}_h\in Q_h$ and a function $r_h \in Q_h,$ we define for each face $\sigma \in \Sigma _{int}$ the upwind flux

$$\begin{aligned}Up [r_h, {\boldsymbol{u}}_h]&=r_h^ \mathrm{up}{\boldsymbol{u}}_h\cdot {\boldsymbol{n}} =r_h^{\text {in}} [\overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}}]^+ + r_{h}^{\text {out}} [\overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}}]^- \\&= \overline{r_h} \ \overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}} - \frac{1}{2} |\overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}}| \llbracket {r_h}\rrbracket , \end{aligned}$$

where

$$\begin{aligned}{}[f]^{\pm } = \frac{f\pm |f|}{2} \quad \text{ and } \quad r^ \mathrm{up}= {\left\{ \begin{array}{ll} r^{\text {in}} &{} \text{ if } \ \overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}} \ge 0, \\ r^{\text {out}} &{} \text{ if } \ \overline{{\boldsymbol{u}}_h} \cdot {\boldsymbol{n}} < 0. \end{array}\right. } \end{aligned}$$

Furthermore, we define the numerical flux function

$$\begin{aligned}F_h(r_h,{\boldsymbol{u}}_h) ={Up}[r_h, {\boldsymbol{u}}_h] - h^\beta \llbracket { r_h }\rrbracket , \ 0< \beta <1.\end{aligned}$$

Discrete Operators. For any $r_h, {\boldsymbol{v}}_{h}\in Q_h$ and ${\boldsymbol{q}}_h\in W_h$ we define the following discrete gradient and Laplace operators

$$\begin{aligned}\begin{aligned}&\nabla _{\mathcal {D}}: Q_h\rightarrow W_h&\\&\nabla _{\mathcal {D}}r_h = \sum _{\sigma \in \Sigma } (\nabla _{\mathcal {D}}r_h)_\sigma 1_{D_{\sigma }},&(\nabla _{\mathcal {D}}r_h)_\sigma&= \frac{1}{d_\sigma } \llbracket {r_h}\rrbracket {\boldsymbol{n}}, \\&\nabla _h: Q_h\rightarrow Q_h&\\&\nabla _hr_h = \sum _{K \in \mathcal {T}_{h}} (\nabla _hr_h)_K 1_K,&(\nabla _hr_h)_K&= \sum _{\sigma \in \partial K} \frac{|\sigma |}{|K|} \overline{r_h} {\boldsymbol{n}}, \\&\Delta _h: Q_h\rightarrow Q_h&\\&\Delta _h r_h = \sum _{K\in \mathcal {T}_{h}} (\Delta _h r_h)_K 1_K ,&(\Delta _h r_h)_K&= \sum _{\sigma \in \partial K} \frac{|\sigma |}{|K|} \frac{\llbracket {r_h}\rrbracket }{d_\sigma } , \end{aligned}\end{aligned}$$

and discrete divergence operators

$$\begin{aligned} \begin{aligned}&\mathrm{div}_{\mathcal {T}}: W_h\rightarrow Q_h\\&\mathrm{div}_{\mathcal {T}}{\boldsymbol{q}}_h= \sum _{K \in \mathcal {T}_{h}} (\mathrm{div}_{\mathcal {T}}{\boldsymbol{q}}_h)_K 1_K, \quad (\mathrm{div}_{\mathcal {T}}{\boldsymbol{q}}_h)_K= \sum _{\sigma \in \partial K} \frac{|\sigma |}{|K|} {\boldsymbol{q}}_h\cdot {\boldsymbol{n}}, \\&\mathrm{div}_h: Q_h\rightarrow Q_h\\&\mathrm{div}_h{\boldsymbol{v}}_{h}= \sum _{K \in \mathcal {T}_{h}} (\mathrm{div}_h{\boldsymbol{v}}_{h})_K 1_K, \quad (\mathrm{div}_h{\boldsymbol{v}}_{h})_K = \sum _{\sigma \in \partial K} \frac{|\sigma |}{|K|} \overline{{\boldsymbol{v}}_{h}} \cdot {\boldsymbol{n}}, \\&\mathrm{div}_h^{\mathrm{up}}: Q_h\rightarrow Q_h\\&\mathrm{div}_h^{\mathrm{up}}(r_h {\boldsymbol{v}}_{h}) = \sum _{K \in \mathcal {T}_{h}}1_K \mathrm{div}_h^{\mathrm{up}}(r_h {\boldsymbol{v}}_{h})_K, \quad \mathrm{div}_h^{\mathrm{up}}(r_h {\boldsymbol{v}}_{h})_K = \sum _{\sigma \in \partial K}\frac{|\sigma |}{|K|} F_h(r_h,{\boldsymbol{v}}_{h}). \end{aligned} \end{aligned}$$

Further, the discrete symmetric gradient operator is given by

$$ D_h({\boldsymbol{v}}_{h}) = \frac{1}{2} (\nabla _h{\boldsymbol{v}}_{h}+ \nabla _h{\boldsymbol{v}}_{h}^T), \quad {\boldsymbol{v}}_{h}\in Q_h. $$

Note that the operators $\nabla _{\mathcal {D}}$ and $\Delta _h$ can be extended to vector-valued functions componentwisely. Let $\boldsymbol{v}_h =(v_{1,h},\ldots , v_{d,h}) \in Q_h.$ Then we have

$$\begin{aligned} \begin{aligned} \nabla _{\mathcal {D}}\boldsymbol{v}_h = \left( \nabla _{\mathcal {D}}v_{1,h} , \ldots , \nabla _{\mathcal {D}}v_{d,h}\right) , \quad \Delta _h \boldsymbol{v}_h = \left( \Delta _h v_{1,h} , \ldots , \Delta _h v_{d,h}\right) , \end{aligned} \end{aligned}$$

and

$$\begin{aligned}(\nabla _{\mathcal {D}}\boldsymbol{v}_h)_\sigma = \frac{1}{d_\sigma } \llbracket {\boldsymbol{v}_h}\rrbracket \otimes \boldsymbol{n} , \quad (\Delta _h \boldsymbol{v}_h)_K = \sum _{\sigma \in \partial K}\frac{|\sigma |}{|K|}\frac{\llbracket {\boldsymbol{v}_h}\rrbracket }{d_\sigma }.\end{aligned}$$

2.2 Numerical Scheme for the Euler System

We recall a semi-discrete finite volume scheme for the Euler system Eq. 1,

$$\begin{aligned} D_t \varrho _{h}+ \mathrm{div}_h^{\mathrm{up}}(\varrho _{h}{\boldsymbol{u}}_h)&=0,\\ D_t {\boldsymbol{m}}_h+ \mathrm{div}_h^{\mathrm{up}}F_h({\boldsymbol{m}}_h,{\boldsymbol{u}}_h) +\nabla _hp_h&= h^{\alpha - 1} \Delta _h {\boldsymbol{u}}_h,\\ D_t E_h + \mathrm{div}_h^{\mathrm{up}}F_h[E_h , {\boldsymbol{u}}_h] + {\boldsymbol{u}}_h\cdot \nabla _hp_h + p_h \mathrm{div}_h{\boldsymbol{u}}_h&=\frac{h^{\alpha - 1}}{2} \Delta _h ({\boldsymbol{u}}_h^2), \end{aligned}$$

where ${\boldsymbol{u}}_h=\frac{{\boldsymbol{m}}_h}{\varrho _{h}},$ $p_h=(\gamma -1)\left( E_h - \frac{1}{2} \frac{|{\boldsymbol{m}}_h|^2}{\varrho _{h}} \right)$ and $D_t$ stands for the time derivative. The scheme was firstly introduced and studied in its weak form in our recent work [9]. Hereafter we will refer to it as the FLM method.

Definition 1

(FLM method) Given the initial values $(\varrho _{0,h},{\boldsymbol{m}}_{0,h},E_{0,h}) \in Q_h \times Q_h \times Q_h,$ we seek a piecewise constant approximation $(\varrho _{h},{\boldsymbol{m}}_h,E_h) \in Q_h \times Q_h \times Q_h$ which solves at any time $t\in (0,T]$ the following equations:

$$\begin{aligned} \int _{\Omega } D_t \varrho _{h}\phi _h \,\mathrm{d} {x} - \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } F_h(\varrho _{h},{\boldsymbol{u}}_h) \llbracket {\phi _h}\rrbracket \mathrm{d}S_x = 0, \ \forall \ \phi _h \in Q_h, \end{aligned}$$

(5a)

$$\begin{aligned} \int _{\Omega } D_t {\boldsymbol{m}}_h\cdot \boldsymbol{\phi }_h \,\mathrm{d} {x} - \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } \mathbf{F}_h({\boldsymbol{m}}_h,{\boldsymbol{u}}_h) \cdot \llbracket {{\boldsymbol{\phi }_h}}\rrbracket \mathrm{d}S_x- \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } \overline{p_h} {\boldsymbol{n}}\cdot \llbracket {{\boldsymbol{\phi }_h} }\rrbracket \mathrm{d}S_x \\ = - h^{\alpha -1} \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } \llbracket { {\boldsymbol{u}}_h}\rrbracket \cdot \llbracket {{\boldsymbol{\phi }_h} }\rrbracket \mathrm{d}S_x , \ \forall \ \boldsymbol{\phi }_h \in Q_h, \end{aligned}$$

(5b)

$$\begin{aligned} \int _{\Omega } D_t E_h \phi _h \,\mathrm{d} {x} - \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } F_h(E_h,{\boldsymbol{u}}_h) \llbracket {\phi _h}\rrbracket \mathrm{d}S_x - \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } \overline{ p_h } \llbracket {\phi _h {\boldsymbol{u}}_h}\rrbracket \cdot {\boldsymbol{n}} \mathrm{d}S_x \\ + \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } \overline{ p_h \phi _h} \llbracket {{\boldsymbol{u}}_h}\rrbracket \cdot {\boldsymbol{n}} \mathrm{d}S_x = - \frac{h^{\alpha - 1}}{2} \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } \llbracket { {\boldsymbol{u}}_h^2}\rrbracket \llbracket {\phi _h}\rrbracket \mathrm{d}S_x,\ \forall \ \phi _h \in Q_{h}. \end{aligned}$$

(5c)

The initial values can be obtained by a standard projection onto the space $Q_h,$

$$\begin{aligned} \Pi _h[r]_{|_K}=\frac{1}{|K|}\int _K r\,\mathrm{d} {x}\ \text { for any } K\in \mathcal {T}_{h}, \end{aligned}$$

i.e. $(\varrho _{0,h},{\boldsymbol{m}}_{0,h},E_{0,h}) =(\Pi _h[\varrho _0], \Pi _h[{\boldsymbol{m}}_0],\Pi _h[E_0]).$

Remark 1

The FLM method in Eq. 5 can be also rewritten in the following per-cell flux formulation

$$\begin{aligned}D_t \varrho _K + \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} F_h(\varrho _{h},{\boldsymbol{u}}_h)&= 0 , \\ D_t {\boldsymbol{m}}_K + \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \left( \mathbf{F}_h({\boldsymbol{m}}_h,{\boldsymbol{u}}_h) + \overline{p_h } {\boldsymbol{n}}\right)&= h^{\alpha - 1} \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \llbracket { {\boldsymbol{u}}_h}\rrbracket , \\ D_t E_K + \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \bigg ( F_h(E_h , {\boldsymbol{u}}_h) + (\overline{p_h } {\boldsymbol{u}}_h+p_h \overline{{\boldsymbol{u}}_h})\cdot {\boldsymbol{n}}\bigg )&= \frac{h^{\alpha - 1}}{2} \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \llbracket {{\boldsymbol{u}}_h^2}\rrbracket ,\end{aligned}$$

for any $K\in \mathcal {T}_{h}$ .

Remark 2

Our finite volume method is based on an upwinding, which naturally yields a numerical diffusion. In addition we include a numerical diffusion of the form $h^{\beta + 1} \Delta _h r_h$, where $r_h = \varrho _h, {\boldsymbol{m}}_h, E_h.$ Altogether this diffusion is of the form $(\frac{1}{2} | {\boldsymbol{u}}_h \cdot {\boldsymbol{n}}| + h^{\beta } ) h \Delta _h r_h$. Note that we have an additional numerical diffusion term $h^\alpha \Delta _h {\boldsymbol{u}}_h$ and $\frac{1}{2} h^\alpha \Delta _h {\boldsymbol{u}}_h^2$ in the momentum and energy equation, respectively. In the case the sound speed is larger than $h^\beta$, the numerical diffusion w.r.t. $\Delta _h r_h$ is smaller than that of standard numerical fluxes based on the Riemann problem solution.

It is truth, that we do not take a special care for the approximation of the contacts. On the other hand, a better resolution can be achieved by introducing a linear reconstruction and limiting to obtained second-order extension.

2.2.1 Properties of the FLM Method

For the rigorous convergence analysis of scheme in Eq. 5 a few important properties are inevitable.

• Existence of numerical solution

  The discrete problem Eq. 5 admits a solution $(\varrho _{h}(t), {\boldsymbol{m}}_h(t),E_h(t)) \in Q_h \times Q_h \times Q_h,$ for any $t\ge 0$. As shown in [9], the result follows from the standard theory of ODEs and sufficiently strong a priori bounds.

• Conservation of discrete mass and energy

  In a straightforward way it can be shown that

  $$\begin{aligned}&\int _{\Omega } \varrho _{h}(t, \cdot ) \,\mathrm{d} {x} = \int _{\Omega } \varrho _{0,h} \,\mathrm{d} {x} = \tilde{M}_0> 0,\\&\int _{\Omega } E_h (t, \cdot ) \,\mathrm{d} {x} = \int _{\Omega } E_{0,h} \,\mathrm{d} {x} = \tilde{E}_0 > 0, \ t \ge 0. \end{aligned}$$

• Positivity of the discrete density, pressure and temperature

  For any fixed h,  the approximate density, pressure and consequently also temperature remain strictly positive on any finite time interval. We refer the reader to [9, Sections 4.3, 4.4] for more details.

• Discrete entropy inequality

  The discrete (renormalized) entropy inequality in the sense of Tadmor is satisfied, cf. [20, 21]. More precisely, it holds that

  $$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t} \int _{\mathcal {T}_{h}} \varrho _h \chi (s_h) \Phi _h \,\mathrm{d} {x} \ge \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } Up[\varrho _h \chi (s_h) , {\boldsymbol{u}}_h] [[ \Phi _h]] \mathrm{d}S_x +\\&+ \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } \mu _h \left( \overline{\nabla _{\varrho } (\varrho _h \chi (s_h))}[[\varrho _h ]]+\overline{\nabla _p (\varrho _h \chi (s_h))}[[p_h ]]\right) [[ \Phi _h]] \mathrm{d}S_x, \end{aligned}$$

  where $\chi$ is a non-decreasing, concave, twice continuously differentiable function on R that is bounded from above. For the derivation and proof see [9, Section 3.2].

• Minimum entropy principle

  The discrete physical entropy $\displaystyle s_h= \log \left( \vartheta _{h}^{c_v}/\varrho _{h}\right)$ attains its minimum at the initial time, cf. [13, 22], i.e.,

  $$\begin{aligned} s_h(t) \ge s_0, \ t \ge 0, \ \text{ where } \ -\infty< s_0 < \min s_h(0). \end{aligned}$$

  The entropy is either constant or produced over time, thus the second law of thermodynamics holds. See [9, Section 4.2] for more details.

Clearly, the FLM method belongs to the class of invariant domain preserving schemes introduced in [13, 14]. Based on the above properties the following convergence result for the FLM method was proved in [9].

Theorem 1

(Convergence of the FLM method) Let the initial data $(\varrho _{0,h},{\boldsymbol{m}}_{0,h},E_{0,h})$ satisfy

$$ \varrho _{0,h} \ge \underline{\varrho }> 0,\ E_{0,h} - \frac{1}{2} \frac{ |{\boldsymbol{m}}_{0,h}|^2 }{\varrho _{0,h}} > 0. $$

Let $( \varrho _{h}, {\boldsymbol{m}}_h, E_h ) \in Q_h \times Q_h \times Q_h$ be the solution of the scheme Eq. 5 such that

$$\begin{aligned} 0< \beta< 1,\ 0< \alpha < \frac{4}{3}, \end{aligned}$$

and

$$\begin{aligned} 0 < \underline{\varrho } \le \varrho _{h}(t), \ \vartheta _{h}(t) \le \overline{\vartheta } \ \text{ for } \text{ all }\ t \in [0,T] \ \text{ uniformly } \text{ for }\ h \rightarrow 0. \end{aligned}$$

Then the family of approximate solutions $\{ \varrho _{h}, {\boldsymbol{m}}_h, E_h \}_{h > 0}$ generates a dissipative measure–valued (DMV) solution of the complete Euler system Eq. 1 in the sense of [2].

Let us point out that a DMV solution of the Euler system is a time-space parametrized probability measure, i.e. the Young measure. The expected values of density and entropy with respect to this Young measure satisfy the corresponding weak formulation of mass conservation and entropy inequality, respectively. The weak formulation for the expected value of the momentum allows a concentration defect that is controlled by the dissipation in the energy balance. The energy conservation is relaxed and the expected value of the energy dissipates in time, see [2] and [9].

Furthermore, evoking the DMV–strong uniqueness result proved in [2, Theorem 3.3] we obtain the following strong convergence result.

Theorem 2

(Strong convergence of the FLM method) In addition to the hypotheses of Theorem 1, suppose that the complete Euler system Eq. 1 admits a Lipschitz–continuous solution $(\varrho , {\boldsymbol{m}}, E)$ defined on [0, T].

Then

$$ \varrho _{h}\rightarrow \varrho ,\ {\boldsymbol{m}}_h \rightarrow {\boldsymbol{m}}, \ E_h \rightarrow E \ \text{(strongly) } \text{ in }\ L^1((0,T) \times \Omega ). $$

In Sect. 3 we will illustrate numerical behavior of the FLM method on a series of well-known benchmarks. In what follows we recall the extension of the FLM method to the finite volume method for the Navier–Stokes–Fourier system introduced in [11]. It turned out that for the convergence analysis of the latter system it is more convenient to work with the temperature formulation instead of the internal energy in the last equation of Eq. 2.

2.3 Numerical Scheme for the Navier–Stokes–Fourier System

Having introduced the notation in Sect. 2.1, we now present a semi-discrete finite volume approximation of the Navier–Stokes–Fourier (NSF) system Eq. 2,

$$\begin{aligned} D_t \varrho _{h}+ \mathrm{div}_h^{\mathrm{up}}(\varrho _{h}{\boldsymbol{u}}_h)&=0, \\ D_t (\varrho _{h}{\boldsymbol{u}}_h) + \mathrm{div}_h^{\mathrm{up}}(\varrho _{h}{\boldsymbol{u}}_h, {\boldsymbol{u}}_h) +\nabla _hp_h&=2 \mu \mathrm{div}_hD_h({\boldsymbol{u}}_h) + \lambda \nabla _h\mathrm{div}_h{\boldsymbol{u}}_h,\\ c_v D_t (\varrho _{h}\vartheta _{h}) + c_v \mathrm{div}_h^{\mathrm{up}}(\varrho _{h}\vartheta _{h}, {\boldsymbol{u}}_h) -\kappa \Delta _h \vartheta _{h}&\\& =2\mu \left| \mathbf{D} _h({\boldsymbol{u}}_h)\right| ^2 +\lambda \left| \mathrm{div}_h{\boldsymbol{u}}_h\right| ^2 - p_h \mathrm{div}_h{\boldsymbol{u}}_h. \qquad \end{aligned}$$

Note that a fully discrete (implicit in time) version of this scheme was analyzed in our work [11].

Definition 2

(Finite volume (FV) method for NSF) Given the initial values $(\varrho _{0,h},{\boldsymbol{u}}_{0,h},\vartheta _{0,h}) \in Q_h \times Q_h \times Q_h,$ we seek a piecewise constant approximation $(\varrho _{h},{\boldsymbol{u}}_h,\vartheta _{h}) \in Q_h \times Q_h \times Q_h$ which solves at any time $t\in (0,T]$ the following equations:

$$\begin{aligned} \int _{\Omega } D_t \varrho _{h}\; \phi _h \,\mathrm{d} {x} - \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } F_h(\varrho _{h},{\boldsymbol{u}}_h) \llbracket {\phi _h}\rrbracket \mathrm{d}S_x = 0, \ \forall \phi _h \in Q_h, \end{aligned}$$

(6a)

$$\begin{aligned} \int _{\Omega } D_t(\varrho _{h}{\boldsymbol{u}}_h) \cdot \boldsymbol{\phi }_h \,\mathrm{d} {x} - \sum _{ \sigma \in \Sigma _{int}} \int _{\sigma } \mathbf{F}_h(\varrho _{h}{\boldsymbol{u}}_h,{\boldsymbol{u}}_h) \cdot \llbracket {\boldsymbol{\phi }_h}\rrbracket \mathrm{d}S_x -\int _{\Omega } p_h \mathrm{div}_h\boldsymbol{\phi }_h \,\mathrm{d} {x} \\ = - 2\mu \int _{\Omega } D_h( {\boldsymbol{u}}_h) : \nabla _h\boldsymbol{\phi }_h \,\mathrm{d} {x} - \lambda \int _{\Omega } \mathrm{div}_h{\boldsymbol{u}}_h\mathrm{div}_h\boldsymbol{\phi }_h \,\mathrm{d} {x} , \ \forall \boldsymbol{\phi }_h \in Q_h, \end{aligned}$$

(6b)

$$\begin{aligned} c_v \int _{\Omega } D_t (\varrho _{h}\vartheta _{h}) \; \phi _h \,\mathrm{d} {x} - c_v \sum _{ \sigma \in \Sigma _{int} } \int _{\sigma } F_h(\varrho _{h}\vartheta _{h},{\boldsymbol{u}}_h) \llbracket {\phi _h}\rrbracket \mathrm{d}S_x - \kappa \int _{\Omega } \Delta _h \vartheta _{h}\; \phi _h \,\mathrm{d} {x} \\= \int _{\Omega } \big (2\mu \left| \mathbf{D} _h({\boldsymbol{u}}_h)\right| ^2+\lambda \left| \mathrm{div}_h{\boldsymbol{u}}_h\right| ^2 - p_h \mathrm{div}_h{\boldsymbol{u}}_h\big ) \phi _h \,\mathrm{d} {x} , \ \forall \phi _h \in Q_{h}. \end{aligned}$$

(6c)

Remark 3

Let us point out that the $h^{\alpha -1}$-terms in Eq. 5b and Eq. 5c yield an additional diffusion and make the FLM method a particular vanishing viscosity approximation of the Euler system. Since the physical viscosity is naturally included in the Navier–Stokes–Fourier system, we do not need to include the additional diffusion in Eq. 6.

Remark 4

The numerical scheme in Eq. 6 can be also rewritten in the usual finite volume formulation for any $K\in \mathcal {T}_{h},$

$$\begin{aligned}D_t \varrho _K&+ \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} F_h(\varrho _{h},{\boldsymbol{u}}_h) = 0 ,\\ D_t (\varrho {\boldsymbol{u}})_K&+ \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \left( \mathbf{F}_h(\varrho _{h}{\boldsymbol{u}}_h,{\boldsymbol{u}}_h) + \overline{p_h } {\boldsymbol{n}}\right) \\&={ \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \left( 2\mu \overline{ D_h({\boldsymbol{u}}_h)}\cdot {\boldsymbol{n}}+ \lambda \overline{ \mathrm{div}_h{\boldsymbol{u}}_h} {\boldsymbol{n}}\right) },\\ c_v D_t (\varrho \vartheta )_K&+ \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \left( c_v F_h(\varrho _{h}\vartheta _{h}, {\boldsymbol{u}}_h) - \kappa \frac{ \llbracket {\vartheta _{h}}\rrbracket }{d_\sigma }\right) \\&= \sum _{ \sigma \in \partial K } \frac{|\sigma |}{|K|} \left( 2\mu \left| \mathbf{D} _h({\boldsymbol{u}}_h)\right| ^2_K +\lambda \left| \mathrm{div}_h{\boldsymbol{u}}_h\right| ^2_K - p_K (\mathrm{div}_h{\boldsymbol{u}}_h)_K\right) .\end{aligned}$$

2.3.1 Properties of the FV Method for NSF

Analogously as in the inviscid case for the convergence analysis it is fundamental that our numerical scheme fulfills some invariant domain preserving properties. In [11] we have proved the following:

• Conservation of discrete mass

  One can easily show that

  $$\begin{aligned} \int _{\Omega } \varrho _{h}(t, \cdot ) \,\mathrm{d} {x} = \int _{\Omega } \varrho _{0,h} \,\mathrm{d} {x} = \tilde{M}_0 > 0,\, \ t \ge 0. \end{aligned}$$

• Non-negativity of the discrete density

  The approximate density remains non-negative on any finite time interval.

• Discrete total energy dissipation

  Let $(\varrho _{h},{\boldsymbol{u}}_h,\vartheta _{h}) \in Q_h\times Q_h \times Q_h$ be a solution to Eq. 6. Then

  $$ E_h(t) \le E_0, \ t \ge 0, $$

  where

  $$ E_h (t) = \int _{\Omega } \left( \frac{1}{2} \varrho _{h}(t) |{\boldsymbol{u}}_h(t)|^2 +c_v \varrho _{h}(t) \vartheta _{h}(t) \right) \,\mathrm{d} {x}. $$

  See [11, Theorem 3.1] for the proof.

• Discrete entropy inequality

  The scheme in Eq. 6 is entropy stable. It holds that

  $$\begin{aligned} \int _{\Omega } D_t \left( \varrho _{h}s_h \right) \,\mathrm{d} {x} \ge&- \int _{\Omega } \kappa \nabla _{\mathcal {D}}\vartheta _{h}\cdot \nabla _{\mathcal {D}}\!\!\left( \frac{1}{\vartheta _{h}}\right) \,\mathrm{d} {x} \\&+ \int _{\Omega } \frac{1}{\vartheta _{h}}\left( 2\mu |\mathbf{D} ({\boldsymbol{u}}_h)|^2 + \lambda |\mathrm{div}_h{\boldsymbol{u}}_h|^2 \right) \,\mathrm{d} {x}, \end{aligned}$$

  see [11, Lemma 3.4].

Remark 5

Note that the above properties shown in [11] for a fully discrete implicit in time version of scheme Eq. 6 can be proven in a straightforward manner for the semi-discrete scheme presented here.

The structure preserving properties listed above, together with the assumptions on uniform boundedness of the discrete density and temperature, are sufficient to derive suitable a priori estimates and consistency formulation of scheme Eq. 6 which are inevitable for the convergence of its solutions. We now recall the convergence results proved in [11].

Theorem 3

(Convergence of the FV method for NSF) Let the initial data satisfy the assumptions

$$\begin{aligned} 0< \underline{\varrho } \le \varrho _{0,h} \le \overline{\varrho }, \ 0< \underline{\vartheta } \le \vartheta _{0,h} \le \overline{\vartheta }, \ \Vert {\boldsymbol{u}}_{0,h}\Vert _{L^2} \le \overline{u}, \end{aligned}$$

for some positive constants $\underline{\varrho },$ $\overline{\varrho },$ $\underline{\vartheta },$ $\overline{\vartheta },$ $\overline{u}.$ Let $(\varrho _h, \vartheta _h, {\boldsymbol{u}}_h) \in Q_h \times Q_h \times Q_h$ be the solution of the finite volume scheme Eq. 6, satisfying the assumptions

$$\begin{aligned} 0< \underline{\varrho } \le \varrho _{h}(t) \le \overline{\varrho }, 0< \underline{\vartheta } \le \vartheta _{h}(t) \le \overline{\vartheta } \text{ uniformly } \text{ for } h \rightarrow 0, \text{ and } \text{ all } t \in (0,T). \end{aligned}$$

Then the family $\{\varrho _h,\vartheta _h,{\boldsymbol{u}}_h,\mathbf{D} _h({\boldsymbol{u}}_h),\nabla _{\mathcal {D}}\vartheta _h \}_{h>0}$ generates a DMV solution of the Navier–Stokes–Fourier system Eq. 2 in the sense of [3].

Analogously as for the inviscid flows a DMV solution is the Young measure. Expected values of density, momentum, energy and entropy satisfy appropriate generalized formulation of Eq. 2. Further, applying the DMV–strong uniqueness principle established in [3, Theorem 6.1] and [11, Theorem 5.5] we have the following strong convergence result.

Theorem 4

(Strong convergence of the FV method for NSF) In addition to the hypotheses of Theorem 3 assume that $\{ \mathcal {V}_{t,x} \}_{(t,x) \in (0,T)\times \Omega }$ is a DMV solution of the Navier–Stokes–Fourier system Eq. 2 in the sense of [3] such that

$$\begin{aligned} \mathcal {V}_{t,x} \left\{ 0 < \underline{\varrho } \le \varrho \le \overline{\varrho }, \ \vartheta \le \overline{\vartheta }, \ |{\boldsymbol{u}}| \le \overline{u} \right\} = 1 \ \text{ for } \text{ a.a. } \ (t,x) \in (0,T)\times \Omega \end{aligned}$$

(7)

for some constants $\underline{\varrho }$, $\overline{\varrho }$, $\overline{\vartheta }$, and $\overline{u}$. Let, moreover,

$$ \mathcal {V}_{0,x} = \delta _{\varrho _0(x), \vartheta _0(x), {\boldsymbol{u}}_0(x)} \ \ \ \text{ for } \text{ a.a. }\ x \in \Omega , $$

where $(\varrho _0, \vartheta _0, {\boldsymbol{u}}_0)$ belongs to the regularity class

$$\begin{aligned} \varrho _0, \vartheta _0 \in W^{3,2}(\Omega ), \ \varrho _0, \ \vartheta _0 > 0 \ \ \text{ in } \ \ \Omega ,\ {\boldsymbol{u}}_0 \in W^{3,2}_0 (\Omega ; R^3). \end{aligned}$$

(8)

Finally, suppose that the Navier–Stokes–Fourier system Eq. 2 is endowed with the initial data $(\varrho _0,\vartheta _0, \boldsymbol{u}_0)$ satisfying Eq. 8. Let $(\varrho _h, \vartheta _h, {\boldsymbol{u}}_h)$ be the solution of the finite volume scheme Eq. 6, and in addition,

$$\begin{aligned} |{\boldsymbol{u}}_h (t)| \le \overline{u} \text{ uniformly } \text{ for } \ h \rightarrow 0 \ \text{ and } \text{ all } \ t \in (0,T). \end{aligned}$$

Then

$$\begin{aligned}&\varrho _{h}\rightarrow \varrho \text{(strongly) } \text{ in } L^{p}\left( (0,T)\times \Omega \right) ,\\&\vartheta _{h}\rightarrow \vartheta \text{(strongly) } \text{ in } L^{p}\left( (0,T)\times \Omega \right) ,\\&\boldsymbol{u}_h \rightarrow \boldsymbol{u} \text{(strongly) } \text{ in } L^p\left( (0,T)\times \Omega ; R^d \right) ,\ p \in [1,\infty ), \end{aligned}$$

where $(\varrho , \vartheta ,\boldsymbol{u})$ is a strong (classical) solution of the Navier–Stokes–Fourier system.

3 Numerical Experiments

In this section we demonstrate the performance of both finite volume methods, the FLM method, see Eq. 5, for the Euler equations, and the finite volume method, see Eq. 6, for the Navier–Stokes–Fourier equations.

For time discretization we use the forward finite differences which yield the explicit finite volume scheme for the Euler system. Diffusive fluxes in the Navier–Stokes–Fourier equations are approximated by the backward finite differences and thus implicitly in time. For stability reasons, we set the time step as $\Delta t = \min \{\Delta t_a,\Delta t_b\}$ in each sub-iteration. The first term arises from the CFL stability condition: $\Delta t_a =\text {CFL}\, h / \max \{|{{\boldsymbol{u}}}| + c \}$, $c=\sqrt{ \vartheta }$. In our numerical simulations we set $\text {CFL}=0.5$ if not explicitly claimed otherwise. The second term is due to the parabolic regularization: $\Delta t_b= h^{1-\beta }/ (2d).$

3.1 Numerical Experiments for the FLM Method

3.1.1 Experimental Order of Convergence (EOC)

We aim to validate the theoretical result on the convergence of $\varrho ,{\boldsymbol{m}},E$ presented in Theorem 2 by computing the corresponding norms of numerical errors

$$\begin{aligned} \left\Vert e_f\right\Vert =\frac{\left\Vert f-f_{ref} \right\Vert _{L^1_tL^1_x} }{\left\Vert f_{ref}\right\Vert _{L^1_tL^1_x}}, \quad f\in \{\varrho ,{\boldsymbol{m}},E\}, \end{aligned}$$

where $L^1_tL^1_x$ is a shortening for $L^1(0,T; L^1(\Omega )).$ Analogous notation is used for other Bochner spaces below. Additionally, we also provide the numerical errors of the velocity ${\boldsymbol{u}}$ in $L^2_tL^2_x-$norm and pressure p in $L^\infty _t L^1_x-$norm. The reference solution is the exact solution to Eq. 1

$$\begin{aligned} \begin{aligned} \varrho _{ref}&=2+\cos (2\pi x),\quad {\boldsymbol{u}}_{ref}= \frac{\sin (\pi t)}{2+\cos (2\pi x)}\left( \begin{array}{c} 1 \\ -1 \end{array}\right) , \\ \quad p_{ref}&= (2+\cos (2\pi x))(2+\sin (2\pi x)), \qquad x \in [0,1]. \end{aligned} \end{aligned}$$

(9)

Setting $\gamma =1.4, \;\alpha =1.3, \;\beta =0.2$ and $\text {CFL}=0.6$, we observe the first order convergence rate for the FLM method, see Table 1.

Table 1 Relative errors and EOC for the FLM method at time $t=0.1$

3.1.2 1D Benchmark Problems

We test one-dimensional Riemann problems studied in [15, 23] with the initial data

$$\begin{aligned} (\varrho ,u,p)= {\left\{ \begin{array}{ll} (\varrho _L,u_L,p_L) &{} \text{ if } 0 \le x<x_m,\\ (\varrho _R,u_R,p_R) &{} \text{ if } x_m \le x \le 1, \end{array}\right. } \end{aligned}$$

with the corresponding values presented in Table 2.

Table 2 Initial data of 1D tests

Fig. 2

1D tests: from top to bottom are Tests 1 to 4, from left to right numerical solutions of $\varrho$, ${\boldsymbol{u}}$, p. The solid blue lines represent solutions obtained by the exact Riemann solver. The dotted red lines and the dashed black lines are solutions obtained by the FLM scheme with $\alpha =1.5$ and $\alpha =3,$ respectively

Test 1 has a weak solution consisting of two rarefaction waves and it is typically used for checking the positivity of density; Test 2 is designed for strong shock; Test 3 and 4 are designed to capture stationary contact waves. We set $\gamma =1.4$, $\beta =0.2$ and aim to show the numerical performance of the scheme Eq. 5 on the domain $\Omega =[0,1]$ with mesh size $h=1/400$. First, we present in Fig. 2 the results of numerical simulations for different choices of $\alpha ,$ that is the parameter appearing in the artificial diffusion terms in Eq. 5b and Eq. 5c. Secondly, we show in Fig. 3 the comparison of the numerical solutions obtained by the FLM method with that of the HLL finite volume method [23].

Fig. 3

1D tests: from top to bottom are Tests 1 to 4, from left to right numerical solutions of $\varrho$, ${\boldsymbol{u}}$, p. The solid blue lines, the dotted red lines and the dashed black lines are solutions obtained by the exact Riemann, HLL, and FLM ($\alpha =1.5$) solvers, respectively

3.1.3 2D Benchmark Problems

Now we test the two-dimensional Riemann problems studied in [15,16,17] with $\Omega =[-1,1]^2$. Boundary values are obtained by extrapolation of conservative variables $(\varrho , {\boldsymbol{m}}, E).$

Test 1: circular two-dimensional Sod problem with the initial data

$$\begin{aligned} (\varrho , u_1, u_2,p)={\left\{ \begin{array}{ll} (1.0,0,0,1.0), &{}|x|<0.4, \\ (1.0 ,0,0, 0.1), &{} \text {else}. \end{array}\right. } \end{aligned}$$

Figure 4 displays the contour lines of the numerical solution of density, velocity components, and pressure at time $t=0.2$ which are in a very good agreement with the results presented in literature, cf., e.g., [23].

Fig. 4

Test 1: Sod problem solution on rectangular mesh $h_x=h_y=0.05$ with $\alpha =1.5$, $\beta =0.2$ at time $t=0.2$

Test 2: two-dimensional benchmark Riemann problem consisting of two moving shocks and two standing slip lines. The initial values are set as

$$\begin{aligned} (\varrho , u_1, u_2,p)={\left\{ \begin{array}{ll} (0.5313,0, 0.7276,0.4), &{} x>0, y>0,\\ (1.0,0.7276, 0,1.0), &{} x<0, y>0, \\ (0.8,0,0,1.0), &{}x<0, y<0, \\ (1.0 ,0,0.7276, 1.0), &{} x>0, y<0. \end{array}\right. } \end{aligned}$$

Figure 5 shows the numerical solution for density and pressure for different CFL numbers. Numerical solutions obtained by the FLM method are in good agreement with the results presented in literature, see, e.g., [16].

Fig. 5

Test 2: solution of $\varrho$ (upper row) and p (lower row) on rectangular mesh $h_x=h_y=0.05$ with $\alpha =1.5$, $\beta =0.5$ at time $t=0.52$

Test 3: two-dimensional Riemann problem with the initial condition

$$\begin{aligned} (\varrho , u_1, u_2,p)={\left\{ \begin{array}{ll} (1.1,0, 0,1.1), &{} x>0, y>0,\\ (0.5065,0, 0.8939,0.35), &{} x<0, y>0, \\ (1.1,0.8939,0.8939,1.1), &{}x<0, y<0, \\ (0.5065 ,0,0.8939, 0.35), &{} x>0, y<0. \end{array}\right. } \end{aligned}$$

In this configuration there are two forward moving shocks and two backward moving shocks. Figure 6 depicts the contour lines of the numerical solution of density, velocity components, and pressure at time $t=0.25$. We can again confirm that the numerical solution is in good agreement with the results presented in the literature, cf., e.g., [16].

Fig. 6

Test 3: solution of $\varrho ,$ $u_1,$ $u_2,$ and p on rectangular mesh $h_x=h_y=0.05$ at time $t=0.25$

3.2 Numerical Experiments for the FV Method for NSF

3.2.1 Experimental Order of Convergence (EOC)

Our aim in this section is to validate theoretical results on the convergence of $\varrho ,{\boldsymbol{u}},\vartheta$ presented in Theorem 4 by computing the numerical errors

$$\begin{aligned} \Vert e_f\Vert =\frac{\Vert f-f_{ref}\Vert _{L^q_tL^q_x}}{\Vert f_{ref}\Vert _{L^q_tL^q_x}}, \ \ f \in \{\varrho ,{\boldsymbol{u}},\vartheta \}, \; q=1,2. \end{aligned}$$

Here the reference solution is the same as in Eq. 9. Thus, we have a manufactured exact solution with a suitable external force in the momentum and energy equation. Setting $\mu =\lambda =\kappa =1,$ $\beta =0.2$ and $\text {CFL}=0.6$ we observe the first order convergence rate for the scheme Eq. 6, see Table 3. We can observe the first order convergence on rectangular as well as triangular mesh.

Table 3 Relative errors and EOC for the FV method for NSF at time $t=0.1$

3.2.2 2D Benchmark Problems

Test 4: Circular shock problem.

We again test the two-dimensional Sod problem using the same initial data as in the first experiment of Sect. 3.1.3 with $\mu =\lambda = \kappa =0.001$ and $\text {CFL}=\beta =0.6$. The contour lines of the numerical solutions are shown in Fig. 7. Small viscosity effects can be noticed but overall the numerical solutions for inviscid and viscous case are similar as expected.

Test 5: Gresho Vortex problem with the initial data [15]

$$\begin{aligned} (u,p)(r) ={\left\{ \begin{array}{ll} (5r , \; 5+ 12.5r^2) &{}r<0.2, \\ (2-5r ,\; 9-4 \ln 0.2 + 12.5r^2 - 20r + 4\ln r) &{} 0.2\le r<0.4 ,\\ (0 , \; 3 + 4\ln 2) &{} r>0.4. \end{array}\right. } \end{aligned}$$

Figure 8 displays the contour lines of the numerical solutions obtained by the scheme Eq. 6 with the parameters $\mu =\lambda = \kappa =0.01,$ and $\text {CFL}=\beta =0.6$ at time $t=0.2$.

Fig. 7

Test 4: Circular shock solution on rectangular mesh $h_x=h_y=0.05$ at time $t=0.2$

Fig. 8

Test 5: Gresho vortex solution on rectangular mesh $h_x=h_y=0.05$ at time $t=0.2$

Conclusion

We have presented behavior and performance of two new convergent finite volume methods for compressible fluids, both inviscid and viscous. These new finite volume methods satisfy some important invariant domain preserving properties, such as the minimum entropy principle, mass and energy conservation, positivity preservation, total energy dissipation and entropy production. These are crucial for showing the stability and consistency of the schemes. In the framework of a nonlinear version of the Lax equivalence theorem, see [9, 11], these properties directly imply the strong convergence of numerical solutions to a strong solution on its lifespan. Our numerical experiments presented in Sect. 3 confirm these theoretical convergence results.