A Second-Order Finite-Volume Scheme for Euler Equations: Kinetic Energy Preserving and Staggering Effects

Abstract

A finite-volume method for Euler equations is presented. Interfaces fluxes are reconstructed with a linear interpolation, which leads to a second order approximation of the spatial derivatives. Since shocked flows have been considered as test cases, a Rusanov-like artificial dissipation is used in order to prevent spurious oscillations. The conservative form of a scheme does not ensure the correct balance of quantities like kinetic energy and inner energy because they are embedded into the total energy, which is instead conserved. We present how to define the fluxes of a conservative scheme taking also into account the kinetic energy balance. Moreover, two grid arrangements are used. One with all conserved variables collocated at the center of the volume, the other one with kinetic quantities collocated at the edges. We discuss the effect of the kinetic energy preservation constraint and of the kinetic variables staggering analyzing two shock tube problems: the modified Sod’s problem and the Shu–Osher’s problem.

Appendices

Appendix 1: Control of Spurious Oscillations via Artificial Dissipation

A centered flux computation gives a second order approximation of derivatives, then in presence of discontinuities it is well known that spurious oscillations are bound to appear. In order to reduce this numerical effect, we introduce artificial diffusion terms which smear the solution. The idea is to solve an approximated Riemann problem, for the sake of clarity we consider a one dimensional scalar conservation law

$$\begin{aligned} \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} = 0 \end{aligned}$$

(68)

where $u \in {\mathbb {R}}$ is the conserved variable and $f:{\mathbb {R}} \rightarrow {\mathbb {R}}$ is the flux.

In presence of a discontinuity we know that this singularity propagates following the eigenvalues of the jacobian matrix $\partial f / \partial u$, that in this case is only one. If a is an approximation of the jacobian $\partial f / \partial u$ at the discontinuity abscissa $x_{d}$ and $u_{L}$ and $u_{R}$ are respectively the left and right value of the discontinuity, then

$$\begin{aligned} u(x_{d}, t) = {\left\{ \begin{array}{ll} u_{L}, &{} \text {if a > 0} \\ u_{R}, &{} \text {if a < 0} \end{array}\right. } \end{aligned}$$

(69)

is the solution of the approximate Riemann problem. When the problem is not scalar and there are more than one eigenvalues, things get more complicated and several jacobian matrix approximations were proposed. However, if we consider a finite volume discretization, the flux computation is always written as follows:

$$\begin{aligned} f_{j+\frac{1}{2}} = \frac{f_{j + 1} + f_{j}}{2} - d_{j+\frac{1}{2}}, \end{aligned}$$

(70)

where $d_{j + 1 / 2} = |a_{j + 1 / 2} |(u_{j+1} - u_{j}) / 2$ is the so-called dissipative term and it is characterized by the approximation of the jacobian, see [13]. These schemes belong to the class of linear upstream schemes which contain a large amount of dissipation [12]. In order to reduce this effect, we want to build a control function that enables a scheme with artificial dissipation to reach the second order of accuracy.

When the system of conservation laws is the Euler equations system, the jacobian matrix has three distinct eigenvalues: $u - a$, u and $u + a$; where a is the speed of sound. As we said there are several approximations of $\partial \mathbf {F} / \partial \mathbf {U}$ where $\mathbf {U} = \begin{bmatrix} \rho , m, \rho E\end{bmatrix}$ and $\mathbf {F} = \begin{bmatrix} m, um + p, \rho u H \end{bmatrix}$. In this appendix we approximate the propagation velocity $a_{j + 1 / 2}$, which is actually a family of characteristics, with the maximum absolute value of the eigenvalues of the jacobian $\partial \mathbf {F} / \partial \mathbf {U}$ over the cells j and $j+1$, that is

$$\begin{aligned} a_{j+\frac{1}{2}} = \max \left\{ \rho \left( \frac{\partial \mathbf {F}_{j}}{\partial \mathbf {U}}\right) , \rho \left( \frac{\partial \mathbf {F}_{j+1}}{\partial \mathbf {U}} \right) \right\} , \end{aligned}$$

where $\rho \left( \partial \mathbf {F} / \partial \mathbf {U}\right)$ is the spectral radius of the jacobian, in this case it reduces to $\max \{ |u\pm a |, |u |\} = |u | + a$. The numerical propagation velocity is therefore given by

$$\begin{aligned} a_{j+\frac{1}{2}} = \max \left\{ |u_{j} | + a_{j}, |u_{j+1} |+ a_{j+1}\right\} \end{aligned}$$

(71)

and the flux with this kind of dissipation is referred to as Rusanov flux [26]. Instead of using the simple average of the fluxes $F_{j}$ and $F_{j+1}$ as centered high order flux, we can use whatever high order flux $F^{HI}$. The collocated TVD flux has therefore the following form:

$$\begin{aligned} F_{j+\frac{1}{2}}^{TVD} = F^{HI}_{j+\frac{1}{2}} - \frac{1}{2}a_{j+\frac{1}{2}} \phi _{j+\frac{1}{2}}(U_{j+1} - U_{j}), \end{aligned}$$

(72)

where $\phi$ is a dissipation multiplier or oscillations control function, which is inherited by TVD methods for linear problems.

Remark 3

The control function $s_{j + 1 / 2}$ needs two quantities: $c = a_{j + 1 / 2}\varDelta t / h$ and r; the former is the local Courant number and the latter is the ratio of the slopes of the quantity, whose we want to check the oscillations, for instance the density $\rho$.

When staggered variables are considered, we can not define the flux and consequently the jacobian in a single point of the discretization. A different approach from the approximation of the propagation speed will be used in this situation. We introduce a staggered version of the upwind scheme, as in [29], thereby taking advantage of the dissipative term that will appear. The momentum flux is

$$\begin{aligned} \begin{aligned} f_{j}^{m}&= u_{j} m_{j}\\&= u_{j}\frac{m_{j+\frac{1}{2}} + m_{j-\frac{1}{2}}}{2} - \frac{ |u_{j} |}{2}(m_{j+\frac{1}{2}} - m_{j-\frac{1}{2}}) + p_{j}, \end{aligned} \end{aligned}$$

(73)

which yields the following discretization of the momentum equation

$$\begin{aligned} m_{j+\frac{1}{2}}^{n+1} = m_{j+\frac{1}{2}}^{n} - \frac{\varDelta t}{h} \left[ f_{j+1}^{m} - f_{j}^{m}\right] . \end{aligned}$$

(74)

This value, $m_{j + 1 / 2}^{n+1}$, enables us to define the staggered velocity:

$$\begin{aligned} u_{j+\frac{1}{2}} = \frac{m_{j+\frac{1}{2}}^{n+1}}{\overline{\rho }_{j+\frac{1}{2}}} = \frac{2m_{j+\frac{1}{2}}^{n+1}}{\rho _{j+1} + \rho _{j}}; \end{aligned}$$

which is used within the mass and energy discretization.

The density and energy flux become, respectively,

$$\begin{aligned} f_{j+\frac{1}{2}}^{\rho }= & {} u_{j+\frac{1}{2}} \frac{\rho _{j+1} + \rho _{j}}{2} - \frac{ |u_{j+\frac{1}{2}} |}{2}(\rho _{j+1} - \rho _{j}). \end{aligned}$$

(75)

$$\begin{aligned} f_{j+\frac{1}{2}}^{\rho E}= & {} u_{j+\frac{1}{2}} \frac{\rho H_{j} + \rho H_{j+1}}{2} - \frac{ |u_{j+\frac{1}{2}} |}{2}(\rho H_{j+1} - \rho H_{j}). \end{aligned}$$

(76)

where $\rho H$ is the enthalpy. Replacing the centered terms of the fluxes with whatever high order reconstruction you want and multiplying each dissipative term by the dissipation control function, we have:

$$\begin{aligned} f_{j+\frac{1}{2}}^{\rho }&= m_{j+\frac{1}{2}} - \frac{1}{2}\phi _{j+\frac{1}{2}} |u_{j+\frac{1}{2}} |(\rho _{j+1} - \rho _{j}) \end{aligned}$$

(77a)

$$\begin{aligned} f_{j}^{m}&= \overline{u}_{j}\overline{m}_{j} + p_{j} - \frac{1}{2}\phi _{j} |u_{j} |(m_{j+\frac{1}{2}} - m_{j-\frac{1}{2}}) \end{aligned}$$

(77b)

$$\begin{aligned} f_{j+\frac{1}{2}}^{\rho E}&= \frac{1}{2} \left. \overline{\overline{m} \widetilde{uu}}\right| _{j+\frac{1}{2}} + \frac{\gamma }{\gamma - 1}u_{j+\frac{1}{2}}\overline{p}_{j+\frac{1}{2}} - \frac{1}{2} \phi _{j+\frac{1}{2}} |u_{j+\frac{1}{2}} |(\rho H_{j+1} - \rho H_{j}); \end{aligned}$$

(77c)

keeping in mind that we have to solve the momentum equation first and then the mass and energy one.

Remark 4

Within (77) we see $\phi _{j + 1 / 2}$ and $\phi _{j}$; the former is computed in the same way as in the collocated case, the latter checks the momentum oscillations, instead of density, because at the interface where it is applied we have only momentum variations, so its arguments are: $c = u_{j}\varDelta t / h$ and r computed over m.

Appendix 2: Time Integration

If a semi-discrete scheme is built, a time integration is mandatory to solve the problem. Moreover, if the flux computation is based on centered averages, in order to ensure the stability of the scheme, a single forward step is not enough. Let us go through this problem and consider a model problem like the scalar advection equation with the following numerical flux

$$\begin{aligned} f_{j+\frac{1}{2}} = \frac{1}{2}(1+\phi _{j+\frac{1}{2}})au_j^n + \frac{1}{2}(1-\phi _{j+\frac{1}{2}}) a u_{j+1}^n. \end{aligned}$$

(78)

Let $\{u_{j}\}_{j=1}^{N}$ be the vector of unknowns. Since the semi-discrete scheme has this form

$$\begin{aligned} \begin{aligned} \frac{du_{j}}{dt}&= -\frac{1}{2h}\left[ a\frac{u_{j+1} - u_{j-1}}{2} -\phi _{j+\frac{1}{2}} |a |(u_{j+1} - u_{j}) + \phi _{j-\frac{1}{2}} |a |(u_{j} - u_{j-1})\right] \\&= \frac{1}{2h}\left( -a + \phi _{j+\frac{1}{2}} |a |\right) u_{j+1} - \frac{ |a |}{2h}\left( \phi _{j+\frac{1}{2}} + \phi _{j-\frac{1}{2}}\right) u_{j} + \frac{1}{2h}\left( a + \phi _{j-\frac{1}{2}} |a |\right) u_{j-1}, \end{aligned} \end{aligned}$$

under the hypothesis that we are observing a compactly supported wave, we can rewrite the scheme like an ODE system

$$\begin{aligned} \frac{d}{dt} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \\ \vdots \\ u_{N-1} \\ u_{N} \end{bmatrix} = \begin{bmatrix} \alpha&\beta \\ \gamma&\alpha&\beta \\&\gamma&\ddots&\ddots \\&\ddots&\alpha&\beta \\&&\gamma&\alpha&\beta \\&&\gamma&\alpha \end{bmatrix} \begin{bmatrix} u_{1} \\ u_{2} \\ u_{3} \\ \vdots \\ u_{N-1} \\ u_{N} \end{bmatrix}, \end{aligned}$$

(79)

where $\alpha = - |a | / (2h)(\phi _{j + 1 / 2} + \phi _{j - 1 / 2})$, $\beta = 1/ (2h) (a + \phi _{j - 1 / 2} |a |)$ and $\gamma = 1 / (2h) (-a + \phi _{j + 1 / 2} |a |)$. Now, it is possible to study the stability analyzing the eigenvalues of the space discretization operator.

Lemma 1

Let be $A\in {\mathbb {R}}^{m\times m}$ a tridiagonal matrix with the main diagonal made up of a’s, the upper diagonal of b’s and the lower diagonal of c’s. Then it admits m eigenvalues of the form

$$\begin{aligned} \lambda _{s} = a + 2\sqrt{bc}\cos \left( \frac{\pi s}{m + 1}\right) \end{aligned}$$

for $s = 1,\dots , m$.

The proof is a straightforward computation and it is omitted.

Here the operator is a tridiagonal matrix, whose eigenvalues are always

$$\begin{aligned} \lambda _{s} = \alpha + 2\sqrt{\beta \gamma }\cos \left( \frac{\pi s}{N+1}\right) \quad \text {for}\, s=1,\dots ,N \end{aligned}$$

thanks to Lemma 1. We want to localize these eigenvalues on the complex plane, so let us analyze the square root that could give a non-trivial imaginary part. Suppose that a is positive, for negative a it would have been the same, then

$$\begin{aligned} \sqrt{\beta \gamma } = \frac{a}{2h}\sqrt{z(\phi _{j+\frac{1}{2}}, \phi _{j-\frac{1}{2}})} \end{aligned}$$

where $z(u,t) = ut - 1 + u - t$. Since $0 \le \phi \le 1$, z is always not-positive so we have complex eigenvalues, which lie in the image of $\mathopen [0,1\mathclose ]^{2} \times \left\{ 1, \dots , N \right\}$ mapped by the following transformation

$$\begin{aligned} (u,t,\phi ) \rightarrow \frac{a}{2h}\left( -(u + t), \sqrt{-z(u,t)}\cos \left( \frac{\pi s}{N+1}\right) \right) . \end{aligned}$$

(80)

In order to establish whether a time integration scheme is stable or not, we need to find its region of absolute stability.

Let us introduce a Runge–Kutta type TVD² time discretization proposed by Gottlieb and Shu in [10]:

• the second order RK2-TVD is

  $$\begin{aligned} \begin{aligned} u^{(1)}&= u^{n} + \varDelta t {\mathrm {L}}u^{(n)} \\ u^{n+1}&= \frac{1}{2}u^{n} + \frac{1}{2}u^{(1)} + \frac{1}{2}\varDelta t {\mathrm {L}}u^{(1)} \end{aligned} \end{aligned}$$

  (81)

  with a unitary CFL condition,

• the third order RK3-TVD is

  $$\begin{aligned} \begin{aligned} u^{(1)}&= u^{n} + \varDelta t {\mathrm {L}}u^{(n)} \\ u^{(2)}&= \frac{3}{4}u^{n} + \frac{1}{4}u^{(1)} + \frac{1}{4}\varDelta t {\mathrm {L}}u^{(1)} \\ u^{n+1}&= \frac{1}{3}u^{n} + \frac{2}{3}u^{(2)} + \frac{2}{3}\varDelta t {\mathrm {L}}u^{(2)} \end{aligned} \end{aligned}$$

  (82)

  with a unitary CFL condition.

Given the following IVP

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{{\mathrm {d}}}{{\mathrm {d}}t}u(t) = \lambda u(t) \\ u(0) = u_{0} \end{array}\right. } \end{aligned}$$

(83)

where $\lambda \in {\mathbb {C}}^{-}$, whose exact solution tends to zero when t tends to infinity, we can handle the absolute stability of a time integration scheme with this definition.

Definition 1

(Absolute Stability Region) A time integration scheme is said to be absolutely stable if its provided solution for the model problem (83) is such that

$$\begin{aligned} \lim _{t\rightarrow \infty }|u^{n}| = 0. \end{aligned}$$

Rewriting the scheme for the model problem as follows:

$$\begin{aligned} u^{n+1} = R(z)u^{n} \end{aligned}$$

where $z = \lambda \varDelta t$ and R is a function that characterize the scheme; we have that a value of z such that $|R(z) | < 1$ gives a stable scheme. Therefore the absolute stability region is

$$\begin{aligned} {\mathscr {A}} = \left\{ z \in {\mathbb {C}}: |R(z) | < 1 \right\} . \end{aligned}$$

(84)

Proposition 3

The absolute stability regions of the schemes defined above are

$$\begin{aligned} {\mathscr {A}}_{\text {RK2-TVD}}&= \left\{ z\in {\mathbb {C}} : |1 + z + \frac{z^{2}}{2} | < 1 \right\} \end{aligned}$$

(85)

$$\begin{aligned} {\mathscr {A}}_{\text {RK3-TVD}}&= \left\{ z\in {\mathbb {C}} : |1 + z + \frac{z^{2}}{2} + \frac{z^{3}}{6} | < 1 \right\} . \end{aligned}$$

(86)

Proof

Let ${\mathrm {L}}u^{n} = \lambda u^{n}$ be the operator so that the discretization (81) becomes

$$\begin{aligned} \begin{aligned} u^{(1)}&= (1 + z)u^{n} \\ u^{n+1}&= \frac{1}{2} u^{n} + \frac{1}{2} u^{(1)} + \frac{z}{2}u^{(1)} \\&= \frac{1}{2}u^{n} + \frac{1}{2}(1 + z) u^{n} + \frac{z}{2}(1 + z) u^{n} \\&= \left( 1 + z + \frac{z^{2}}{2} \right) u^{n}. \end{aligned} \end{aligned}$$

Similarly for the scheme (82) we get

$$\begin{aligned} u^{n+1} = \left( 1 + z + \frac{z}{2} + \frac{z^{3}}{6} \right) u^{n} \end{aligned}$$

and the proposition is thus proved. $\square$

The purpose of this section is to establish which is the most useful time integration scheme to solve the system (79). Thus, we have to compare the position of each eigenvalue with the absolute stability region shown above. We have seen that the eigenvalues lie into a region delineated by the transformation (80) conveniently scaled by $\varDelta t$, therefore it is sufficient to check whether this region is included into a stability region or not. RK3-TVD includes all the eigenvalues even though they would be all pure imaginary numbers. On the other hand, RK2-TVD does not include the whole eigenvalues region but it spares a lot of computation and has a, not included, tiny region extremely close to the imaginary axis. Thus, the eigenvalues hardly fall outside the RK2-TVD stability region.