Entropic Regularization of the Discontinuous Galerkin Method in Conservative Variables for Two-Dimensional Euler Equations

Abstract

An entropic regularization of the conservative stable discontinuous Galerkin method (DGM) in conservative variables for two-dimensional Euler equations is constructed based on a special slope limiter. This limiter ensures the fulfillment of two-dimensional analogs of the monotonicity conditions and a discrete analog of the entropic inequality. The developed method was tested on two-dimensional model gas-dynamic problems.

1 INTRODUCTION

One of the promising areas of research in the development of numerical methods for solving gas-hydrodynamic problems is the creation of algorithms that are conservative and entropy stable [1–12]. Many different versions of entropically stable schemes have been proposed; however, research in this area is still far from complete.

Among the multitude of numerical methods used in the designated area, the discontinuous Galerkin method (DGM) should be singled out [13–19], which, on the one hand, is characterized by a compact stencil, and, on the other hand, by the potentially highly accurate finite element (FE) approximations. Various modifications of the DGM have been proposed that are conservative and possess the property of entropic stability [13, 20–25]. Nevertheless, there is still a need to further improve the entropically stable versions of the DGM in order to improve the quality of numerical solutions.

Initially, we proposed a variational method for the entropic regularization of the DGM, in which the values of the DGM coefficients are determined as a solution to some variational problem for the conditional minimum [20] with a convex constraint that ensures the fulfillment of the entropic inequality. However, it was soon discovered that the formal fulfillment of the entropic stability condition alone, which consists in the nonnegative production of entropy, is insufficient for the numerical stability of the algorithm [21]. Further, it was found that to ensure the computational stability of the algorithm, some degree of monotonicity of FE approximations is necessary. As a result, the set of constraints in the corresponding variational problem was supplemented by a system of linear inequalities expressing the monotonicity of FE approximations [22]. As a result of this modification, a computationally stable version of the entropically regularized DGM was obtained. When solving problems with two or more spatial variables, for example, multidimensional problems of gas dynamics, there is a need for effective generalizations of the concept of monotonicity. One of the possible approaches was proposed for the DGM in nonconservative variables in the case of the Euler equations [23–25]. In this paper, we construct an entropically stable DGM for the Euler equations in traditional conservative variables using the generalized monotonicity conditions similar to those used in [25]. The need to build a version of the DGM in conservative variables is due to two circumstances. First, conservative variables are traditionally used in numerical methods for problems in continuum mechanics. The study would be incomplete if we restricted ourselves to the case of nonconservative variables, which is not typical. Second, on the same grid, the algorithm in conservative variables turns out to be more computationally efficient. The computational cost of the version in conservative variables for the same problems turns out to be lower by factors of 2 to 3 than the version in nonconservative variables. At the same time, the accuracy of the approximate solution on the selected test problems remains approximately the same.

The material of the work is presented in the following sequence. Section 2 consists of the problem statement; Section 3 derives the DGM equations in conservative variables density, pulse density, and total gas energy; in Section 4, the leading coefficients of FE approximations are limited based on the monotonicity condition generalized to the case of two spatial variables and the entropic stability condition; and in Section 5, the numerical results are discussed.

2 PROBLEM STATEMENT

We consider a system of Euler equations describing two-dimensional gas dynamics in dimensionless variables, for which we set an initial-boundary value problem in a rectangular domain Π:

$$\begin{gathered} \frac{{\partial {\mathbf{U}}}}{{\partial t}} + \frac{{\partial {{{\mathbf{F}}}_{1}}({\mathbf{U}})}}{{\partial {{x}_{1}}}} + \frac{{\partial {{{\mathbf{F}}}_{2}}({\mathbf{U}})}}{{\partial {{x}_{2}}}} = 0,\quad {\mathbf{U}}({\mathbf{x}},0) = {{{\mathbf{U}}}_{0}}({\mathbf{x}}), \\ {\mathbf{x}} = ({{x}_{1}},{{x}_{2}}) \in \Pi = ({{a}_{1}},{{b}_{1}}) \times ({{a}_{2}},{{b}_{2}}),\quad t > 0, \\ \end{gathered} $$

(1)

where ${\mathbf{U}}(x,t) = {{({{U}^{{(1)}}},{{U}^{{(2)}}},{{U}^{{(3)}}},{{U}^{{(4)}}})}^{{\text{T}}}}$ and ${{{\mathbf{F}}}_{j}}({\mathbf{U}}) = {{(F_{j}^{{(1)}},F_{j}^{{(2)}},F_{j}^{{(3)}},F_{j}^{{(4)}})}^{{\text{T}}}}$ are column vectors of conservative variables and fluxes in the direction of axis Ox_j ($j = 1,2$), respectively. Conditions at the boundary $\partial \Pi$ of rectangle Π are given by ghost cells that are adjacent to the boundary $\partial \Pi$ outside (the boundary conditions are specified in Section 5). The ideal gas equation of state

$$p = (\gamma - 1)\rho \varepsilon ,\quad \gamma = {{c}_{P}}{\text{/}}{{c}_{V}}$$

(2)

closes the system of equations (1). In Eqs. (1) and (2), the following notations are used: p is pressure, ρ is density, ε is specific internal energy, $\gamma = 1.4$ is the adiabatic index, and ${{c}_{P}}$ and ${{c}_{V}}$ are the heat capacities of an ideal gas at constant pressure and constant volume, respectively. Taking into account the equation of state (2), for the momentum density and total energy density, respectively, the following designations are used:

$${\mathbf{I}} = ({{I}_{1}},{{I}_{2}}),\quad {{I}_{j}} = \rho {{u}_{j}},\quad E = p{\text{/}}(\gamma - 1) + (I_{1}^{2} + I_{2}^{2}){\text{/}}(2\rho ),$$

(3)

where ${\mathbf{u}} = ({{u}_{1}},{{u}_{2}})$ is the velocity vector. Taking into account (3), the explicit expressions for the column vector of conservative variables and the column vector of fluxes, respectively, take the form

$${\mathbf{U}} = \left( \begin{gathered} \rho \hfill \\ {{I}_{1}} \hfill \\ {{I}_{2}} \hfill \\ E \hfill \\ \end{gathered} \right),\quad {{{\mathbf{F}}}_{j}}({\mathbf{U}}) = \left( \begin{gathered} {{I}_{j}} \\ {{\rho }^{{ - 1}}}{{I}_{1}}{{I}_{j}} + p{{\delta }_{{1,j}}} \\ {{\rho }^{{ - 1}}}{{I}_{2}}{{I}_{j}} + p{{\delta }_{{2,j}}} \\ (E + p){{\rho }^{{ - 1}}}{{I}_{j}} \\ \end{gathered} \right),$$

(4)

where ${{\delta }_{{ij}}}$ is the Kronecker symbol.

Satisfaction of the entropic inequality [26]

$$\iint\limits_\Omega {[(\rho s)(t + \Delta t) - (\rho s)(t)]d{{x}_{1}}d{{x}_{2}}} + \int\limits_t^{t + \Delta t} {d\tau \oint\limits_{\partial \Omega } {s({{I}_{1}}{{\nu }_{1}} + {{I}_{2}}{{\nu }_{2}})d\sigma } } \geqslant 0$$

(5)

in any subdomain $\Omega \subset \Pi$ with a fairly smooth boundary $\partial \Omega$ is an important prerequisite for a physically correct description of gas-dynamic processes, where ${\boldsymbol{\nu }} = ({{\nu }_{1}},{{\nu }_{2}})$ is the outer normal vector to $\partial \Omega$ and s is the entropy density

$$s = \ln p - \gamma \ln \rho .$$

(6)

The numerical solution of problem (1) is based on the entropically regularized DGM, a general description of which is given in [22–25]. In contrast to [25], here we construct the classical DGM in conservative variables ρ, I, and E. The proposed algorithm is characterized by the implementation of discrete analogs of conservation laws and a discrete analog of the entropic inequality in variables ρ, I, and E.

3 EQUATIONS FOR THE DGM COEFFICIENTS

In a closed rectangle $\bar {\Pi } = \Pi \cup \partial \Pi$, we define a Cartesian grid with nodes

$${{{\mathbf{x}}}_{{mn}}} = (x_{1}^{{(m)}},x_{2}^{{(n)}}) = ({{a}_{1}} + m{{h}_{1}},{{a}_{2}} + n{{h}_{2}}),\quad m = 0, \ldots ,M;\quad n = 0, \ldots ,N,$$

(7)

$${{h}_{1}} = ({{b}_{1}} - {{a}_{1}}){\text{/}}M,\quad {{h}_{2}} = ({{b}_{2}} - {{a}_{2}}){\text{/}}N.$$

(8)

Obviously, grid (7) is related to a set of computational cells ${{K}_{{mn}}}$ = $(x_{1}^{{(m - 1)}},x_{1}^{{(m)}})$ × $(x_{2}^{{(n - 1)}},x_{2}^{{(n)}})$. Each cell ${{K}_{{mn}}}$ has four neighbors (on the sides) that are nearest to it: ${{K}_{{m - 1,n}}}$, ${{K}_{{m,n - 1}}}$, ${{K}_{{m + 1,n}}}$, and ${{K}_{{m,n + 1}}}$ with which it exchanges mass, momentum, energy, and entropy.

We introduce the local variables

$${{z}_{j}} = 2h_{j}^{{ - 1}}({{x}_{j}} - x_{j}^{{(m - 1)}}) - 1,\quad j = 1,2$$

(9)

in order to use them to write equations for the DGM coefficients in a single calculation cell ${{K}_{{mn}}}$. Local variables (9) take values in the interval (–1, 1).

To construct an approximate solution within the computational cell ${{K}_{{mn}}}$, we will use the local polynomial basis related to this cell. Let us present an approximate solution of problem (1) ${{{\mathbf{U}}}_{h}}$ in cell ${{K}_{{mn}}}$ as a linear function of local space variables

$${{{\mathbf{U}}}_{h}}({{z}_{1}},{{z}_{2}},t) = {{{\mathbf{U}}}_{0}}(t) + {{{\mathbf{U}}}_{1}}(t){{z}_{1}} + {{{\mathbf{U}}}_{2}}(t){{z}_{2}},$$

(10)

limited formally to the second order of approximation. Using the standard technique for deriving equations for the DGM coefficients [12, 17, 18], we arrive at a system of ordinary differential equations (SODE) with respect to the required coefficients ${{{\mathbf{U}}}_{0}}(t)$, ${{{\mathbf{U}}}_{1}}(t)$, and ${{{\mathbf{U}}}_{2}}(t)$

$$\begin{gathered} \frac{{d{{{\mathbf{U}}}_{0}}(t)}}{{dt}} = {{{\mathbf{B}}}_{0}} = - h_{1}^{{ - 1}}h_{2}^{{ - 1}}\oint\limits_{\partial K} {\sum\limits_{l = 1}^2 {{{\nu }_{l}}{{{\mathbf{F}}}_{l}}} d\sigma } , \\ \frac{{d{{{\mathbf{U}}}_{j}}(t)}}{{dt}} = {{{\mathbf{B}}}_{j}} = - 3h_{1}^{{ - 1}}h_{2}^{{ - 1}}\left[ {\oint\limits_{\partial K} {\sum\limits_{l = 1}^2 {{{\nu }_{l}}{{z}_{j}}{{{\mathbf{F}}}_{l}}} d\sigma } - \int\limits_K {\sum\limits_{l = 1}^2 {\frac{{\partial {{z}_{j}}}}{{\partial {{x}_{l}}}}{{{\mathbf{F}}}_{l}}} d{{x}_{1}}d{{x}_{2}}} } \right], \\ j = 1,2. \\ \end{gathered} $$

(11)

Recall that in Eqs. (11) in integrals over the boundary $\partial K$ of cell K (indices m and n are omitted for brevity) ${{\nu }_{l}}$ denotes the component of the external normal to the boundary $\partial K$ along the axis $O{{x}_{l}}$, and ${{{\mathbf{F}}}_{l}}$ is the corresponding numerical Godunov flux. The right parts of Eqs. (11) are transformed to the following form:

$${{{\mathbf{B}}}_{0}} = - \frac{1}{{2{{h}_{1}}}}\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{1}}(1,{{z}_{2}}) - {{{\mathbf{F}}}_{1}}( - 1,{{z}_{2}})]d{{z}_{2}}} - \frac{1}{{2{{h}_{2}}}}\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{2}}({{z}_{1}},1) - {{{\mathbf{F}}}_{2}}({{z}_{1}}, - 1)]d{{z}_{1}}} ,$$

(12)

$$\begin{gathered} {{{\mathbf{B}}}_{1}} = - \frac{3}{{2{{h}_{1}}}}\left\{ {\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{1}}(1,{{z}_{2}}) + {{{\mathbf{F}}}_{1}}( - 1,{{z}_{2}})]d{{z}_{2}}} } \right. \\ \left. { - \;\frac{{{{h}_{1}}}}{{{{h}_{2}}}}\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{2}}({{z}_{1}},1) - {{{\mathbf{F}}}_{2}}({{z}_{1}}, - 1)]{{z}_{1}}d{{z}_{1}}} - \int\limits_{ - 1}^1 {d{{z}_{1}}\int\limits_{ - 1}^1 {d{{z}_{2}}{{{\mathbf{F}}}_{1}}({{z}_{1}},{{z}_{2}},t)} } } \right\}, \\ \end{gathered} $$

(13)

$$\begin{gathered} {{{\mathbf{B}}}_{2}} = - \frac{3}{{2{{h}_{2}}}}\left\{ {\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{2}}({{z}_{1}},1) + {{{\mathbf{F}}}_{2}}({{z}_{1}}, - 1)]d{{z}_{1}}} } \right. \\ \left. { - \;\frac{{{{h}_{2}}}}{{{{h}_{1}}}}\int\limits_{ - 1}^1 {[{{{\mathbf{F}}}_{1}}(1,{{z}_{2}}) - {{{\mathbf{F}}}_{1}}( - 1,{{z}_{2}})]{{z}_{2}}d{{z}_{2}}} - \int\limits_{ - 1}^1 {d{{z}_{1}}\int\limits_{ - 1}^1 {d{{z}_{2}}{{{\mathbf{F}}}_{2}}({{z}_{1}},{{z}_{2}},t)} } } \right\}, \\ \end{gathered} $$

(14)

where the components of the numerical fluxes ${{{\mathbf{F}}}_{j}}$ are considered as complex functions of local coordi-nates (9):

$${{{\mathbf{F}}}_{j}} = {{{\mathbf{F}}}_{j}}({{z}_{1}},{{z}_{2}},t) = {{{\mathbf{F}}}_{j}}({\mathbf{U}}({{z}_{1}},{{z}_{2}},t)).$$

(15)

Note that any explicit time discretization of SODE (11), generally speaking, does not generate a computationally stable numerical algorithm for the approximate solution of the Euler equations (1). To obtain a stable numerical method, it is necessary to modify system (11) accordingly.

4 ENTROPIC REGULARIZATION OF THE SYSTEM OF EQUATIONS FOR DGM COEFFICIENTS

Let us construct an entropic regularization of fully discrete explicit schemes based on SODE (11). Since the explicit Runge–Kutta methods of high orders of accuracy consist of intermediate steps similar to the Euler scheme, we restrict ourselves to considering the simplest explicit first-order time integration method based on the Euler scheme.

As is known, the explicit numerical integration of SODEs for the DGM coefficients (11) in the case of gas-dynamic problems with strong shock waves turns out to be unstable, and these coefficients require correction [12, 16]. The method of numerical integration must satisfy, at each time step, the discrete analogs of the laws of conservation of mass, momentum, total energy, and entropic inequality. Let us pay attention to the fact that the first equation in Eqs. (11)$d{{{\mathbf{U}}}_{0}}(t){\text{/}}dt = {{{\mathbf{B}}}_{0}}$ just describes the discrete analog of the laws of conservation of mass, momentum, and energy for a cell ${{K}_{{mn}}}$ [20, 22] in continuous time. Approximation of this equation by the explicit Euler method with a time step τ

$${{{\mathbf{U}}}_{0}}(t + \tau ) = {{{\mathbf{U}}}_{0}}(t) + \tau {{{\mathbf{B}}}_{0}}(t)$$

(16)

retains the conservation property. Discrete analogs of the corresponding conservation laws for a cell ${{K}_{{mn}}}$ hold in discrete time $t = 0,\tau ,2\tau , \ldots$. Therefore, the transition rule (16) of zero coefficients to the next time layer remains unchanged.

Due to the orthogonality of the system of functions in the form of a linear combination of which an approximate solution is sought in (10); the equations for the remaining DGM coefficients do not affect the conservation property. Correction (regularization) of other coefficients in order to ensure the stability of the algorithm is usually done by limiting them [12, 16]. As a result, after correction, the numerical scheme for the remaining coefficients takes the form

$${{{\mathbf{U}}}_{j}}(t + \tau ) = {{{\mathbf{U}}}_{j}}(t) + \tau {{{\mathbf{\tilde {B}}}}_{j}}(t),\quad j = 1,2.$$

(17)

Thus, instead of the original SODE (11), some modified system is numerically integrated. In the case of entropic regularization, in addition to stability, it is required to ensure that, at each time step, the discrete analog of the entropic inequality (5) is satisfied for the cell ${{K}_{{mn}}}$.

In the variational approach [22], the values ${{{\mathbf{\tilde {B}}}}_{j}}$ ($j = 1,2$) in (17) can be considered as a solution to the problem of determining the conditional minimum point of the quadratic function

$$\Phi ({{{\mathbf{\tilde {B}}}}_{1}},{{{\mathbf{\tilde {B}}}}_{2}}) = \sum\limits_{j = 1}^2 {{{{\left\| {{{{{\mathbf{\tilde {B}}}}}_{j}}(t) - {{{\mathbf{B}}}_{j}}(t)} \right\|}}^{2}}} $$

(18)

on some convex constraint set including the entropic inequality, where $\left\| {\, \cdot \,} \right\|$ denotes some Euclidean norm.

Since the solution of the conditional minimum problem for function (18) in each finite element ${{K}_{{mn}}}$ at each time step is very computationally expensive, in practice we have to resort to approximate methods for solving problem (18) and simplified approaches, for example, limiting the DGM coefficients [22].

Let us list the conditions that an entropically regularized DGM must satisfy.

(I) FE approximations (10) must satisfy at each time step some generalized monotonicity conditions that ensure the stability of the numerical algorithm.

(II) Some of the gas dynamic parameters, such as gas density ρ, pressure p, and total energy density E, have positive values in their physical sense. The algorithm must ensure that the indicated parameters are positive at each time step.

(III) In each finite element ${{K}_{{mn}}}$ entropic inequality (5) must be satisfied at each time step.

Let us ensure the fulfillment of each of conditions I–III in the version of the DGM proposed here.

Let each cell ${{K}_{{mn}}}$ be put into correspondence with a real number ${{f}_{{mn}}}$; i.e., $f = \{ {{f}_{{mn}}}\}$ can be viewed as a piecewise constant function defined at every cell ${{K}_{{mn}}}$. In addition, we define a finite set of integer two-dimensional vectors (translations) $\Lambda = \{ (\alpha ,\beta )\}$. To each cell ${{K}_{{mn}}} \subset \bar {\Pi }$, we assign a nonnegative number

$$S_{{mn}}^{\Lambda }(f) = \mathop {\min }\limits_{\alpha ,\beta \in \Lambda } \left| {{{f}_{{m + \alpha ,n + \beta }}} - {{f}_{{mn}}}} \right|$$

(19)

and in what follows, as a working set of translations, we choose the set

$$\Lambda = \{ (1,0);( - 1,0);(0,1);(0, - 1)\} .$$

(20)

Let us also define the quantities

$${{D}_{{mn}}}(f) = \frac{\delta }{2}S_{{mn}}^{\Lambda }(f),\quad D_{{mn}}^{ + }(f) = \min \left( {\frac{\delta }{2}S_{{mn}}^{\Lambda }(f),(1 - \sigma ){{f}_{{mn}}}} \right)$$

(21)

for some parameters $\delta \in (0, + \infty )$ and $\sigma \in (0,1)$ and the limiting function

$$\Delta (x,M) = \left\{ \begin{gathered} 1,\quad \left| x \right| \leqslant M, \hfill \\ {{M}^{{ - 1}}}\left| x \right|, \hfill \\ \end{gathered} \right.\quad x \in ( - \infty ,\infty ),\quad M \in (0,\infty ).$$

(22)

For the DGM coefficients U₁ and U₂ in cell ${{K}_{{mn}}}$, let us construct a limitation procedure, the purpose of which is to obtain their values that satisfy requirements I–III formulated above.

Considering the DGM coefficients ${{{\mathbf{U}}}_{0}}(t + \tau )$ already known as a result of applying formula (17) and denoting the set of values of components l of these coefficients for all cells as $\{ U_{0}^{{(l)}}(t + \tau )\}$, we calculate the quantities

$$M = \left\{ \begin{gathered} D_{{mn}}^{ + }(\{ U_{0}^{{(l)}}(t + \tau )\} ),\quad l = 1,4; \hfill \\ {{D}_{{mn}}}(\{ U_{0}^{{(l)}}(t + \tau )\} ),\quad l = 2,3 \hfill \\ \end{gathered} \right.$$

(23)

and

$$r = \left| {U_{1}^{{(l)}}(t) + \tau B_{1}^{{(l)}}(t)} \right| + \left| {U_{2}^{{(l)}}(t) + \tau B_{2}^{{(l)}}(t)} \right|.$$

(24)

Next, we determine the corresponding DGM coefficients using the limiting function (22)

$$U_{j}^{{(l)}}(t + \tau ) = \Delta (r,M)[U_{j}^{{(l)}}(t) + \tau B_{j}^{{(l)}}(t)].$$

(25)

According to (17), we obtain ${{{\mathbf{\tilde {B}}}}_{j}} = {{\tau }^{{ - 1}}}[{{{\mathbf{U}}}_{j}}(t + \tau ) - {{{\mathbf{U}}}_{j}}(t)]$.

The generalized monotonicity conditions consist of applying the method for calculating the DGM coefficients U₁ and U₂ in accordance with (23)–(25). Note that, together with these conditions, the positiveness of the gas density ρ ($l = 1$) and total energy density E ($l = 4$) is also ensured due to the use of the special limiter $D_{{mn}}^{ + }$ defined in (21).

To ensure positive pressure p in cell ${{K}_{{mn}}}$, generally speaking, requires an additional constraint on the DGM coefficients U₁ and U₂ in addition to what has just been discussed. Hence, we introduce an auxiliary function

$$Q(\lambda ) = \frac{1}{2}\frac{{{{{[U_{0}^{{(2)}} + \lambda (\left| {U_{1}^{{(2)}}} \right| + \left| {U_{2}^{{(2)}}} \right|)]}}^{2}} + {{{[U_{0}^{{(3)}} + \lambda (\left| {U_{1}^{{(3)}}} \right| + \left| {U_{2}^{{(3)}}} \right|)]}}^{2}}}}{{U_{0}^{{(1)}} - \lambda (\left| {U_{1}^{{(1)}}} \right| + \left| {U_{2}^{{(1)}}} \right|)}},$$

(26)

dominating the kinetic energy in the cell, and the function

$$P(\lambda ) = (\gamma - 1)[U_{0}^{{(4)}} - \lambda (\left| {U_{1}^{{(4)}}} \right| + \left| {U_{2}^{{(4)}}} \right|) - Q(\lambda )],$$

(27)

which is a lower bound for the pressure p in it.

To ensure that the pressure is positive, we check the condition

$$P(1) - \sigma P(0) \geqslant 0,$$

(28)

where the value of parameter σ is the same as in (21), the DGM coefficients U₁ and U₂ are calculated in accordance with (23)–(25), and the value $P(0)$ is considered positive. If inequality (28) is satisfied, then the values of coefficients U₁ and U₂ are kept the same. Otherwise, the latter are limited by the rule ${{{\mathbf{U}}}_{j}} \to \lambda {{{\mathbf{U}}}_{j}}$, where $\lambda \in [0,1)$ is the root of the equation

$$P(\lambda ) = (1 - \sigma )P(0).$$

(29)

In order to avoid the complication of the notation further, we will keep the old notation U₁ and U₂ for the new values of coefficients $\lambda {{{\mathbf{U}}}_{1}}$ and $\lambda {{{\mathbf{U}}}_{2}}$.

Finally, we will provide fulfillment in the cell ${{K}_{{mn}}}$ of the entropic inequality (5) in the approximate solution (10). To do this, we consider the linear function of local coordinates

$${{{\mathbf{V}}}_{\mu }}({{z}_{1}},{{z}_{2}},t) = {{{\mathbf{U}}}_{0}}(t) + \mu ({{{\mathbf{U}}}_{1}}(t){{z}_{1}} + {{{\mathbf{U}}}_{2}}(t){{z}_{2}}),$$

(30)

where parameter $\mu \in [0,1]$, and define the pressure function for it

$${{p}_{\mu }}({{z}_{1}},{{z}_{2}},t) = (\gamma - 1)\left[ {V_{\mu }^{{(4)}} - \frac{{{{{(V_{\mu }^{{(2)}})}}^{2}} + {{{(V_{\mu }^{{(3)}})}}^{2}}}}{{2V_{\mu }^{{(1)}}}}} \right]$$

(31)

and the entropy function

$$S(\mu ,t) = \frac{{{{h}_{1}}{{h}_{2}}}}{2}\int\limits_{ - 1}^1 {d{{z}_{1}}\int\limits_{ - 1}^1 {d{{z}_{2}}V_{\mu }^{{(1)}}(\ln {{p}_{\mu }} - \gamma \ln V_{\mu }^{{(1)}})} } .$$

(32)

The last term on the left side of (5), which is a flux term, is approximated in accordance with the Euler scheme (16)

$${{\Phi }_{S}}(t) = \oint\limits_{\partial K} {{{{\left. {s({{I}_{1}}{{\nu }_{1}} + {{I}_{2}}{{\nu }_{2}})d\sigma } \right|}}_{t}}} \cong \frac{1}{{\Delta t}}\int\limits_t^{t + \Delta t} {d\tau \oint\limits_{\partial K} {s({{I}_{1}}{{\nu }_{1}} + {{I}_{2}}{{\nu }_{2}}){\kern 1pt} {\kern 1pt} d\sigma } } .$$

(33)

The next approximation of the left side of inequality (5)

$${{P}_{S}}(\mu ) = S(\mu ,t + \tau ) - S(1,t) + \tau {{\Phi }_{S}}(t)$$

(34)

will be called the discrete analog of entropy production in cell $K = {{K}_{{mn}}}$. Next, we choose value $\mu \in [0,1]$ according to the rule

$$\mu = \left\{ \begin{gathered} 1,\quad {{P}_{S}}(1) \geqslant 0; \hfill \\ \mu ^*,\quad {{P}_{S}}(\mu ^*) = 0, \hfill \\ \end{gathered} \right.$$

(35)

and we use expression (30) with the value (35) of the parameter μ as an approximate solution to (10) in cell ${{K}_{{mn}}}$.

5 RESULTS AND DISCUSSION

Let us check the DGM developed in Section 4 with an entropic slope limiter in conservative variables on the two-dimensional Riemann problems from [27]. Each of these problems is posed as follows: at t = 0 the computational domain 0 < x, y < 1 (${{a}_{1}} = {{a}_{2}} = 0$, ${{b}_{1}} = {{b}_{2}} = 1$) is divided into four equal squares (quarters), in each of which some constant density, pressure and velocity are specified (see Table 1). The gas is assumed to be diatomic, γ = 1.4. A symmetry condition is set on the boundary of the computational domain: $\partial {\mathbf{U}}{\text{/}}\partial n = 0$. It is required to determine the gas flow up to the time $t = {{t}_{{\max }}}$ inclusive.

Table 1. Initial gas parameters and values ${{t}_{{\max }}}$ in two-dimensional Riemann problems

Calculations using the regularized DGM will be performed on a grid of 200 × 200 cells with the following parameter values: δ = 2 and σ = 10⁻⁵ with the Courant number 0.2. The integrals are calculated using the Gauss method. In addition, we will make a comparison with the bicompact SDIRK3B4 scheme from [28]. This scheme uses locally one-dimensional splitting and the conservative monotonization method. On smooth solutions, the SDIRK3B4 scheme has the fourth order of approximation in space and the second order in time. The calculations according to the bicompact scheme were performed on the same grid of 200 × 200 cells, with the Courant number 0.8 and monotonization parameter C₁ = 10.

Problem 3. The lines of the initial discontinuities in this problem are shock wave fronts. In the process of their motion, three new shock waves arise. The latter limit the zone, in which the pressure is lower than the initial pressure in the upper right quarter of the computational domain. This difference in pressure causes a mass injection that has the shape of a mushroom in the density isoline pattern. Accordingly, the resolution of this smooth mushroom structure and seven shock waves is of interest.

Figure 1 shows at $t = 0.3$ entropy production rate P_s in the DGM and solutions (parameters p, ρ, u₁, u₂) for the DGM and the bicompact scheme. First of all, we note that the developed method of entropic regularization provides nonnegative entropy production in all cells, as was required. Both schemes resolve all shock waves equally well. However, the bicompact scheme resolves the aforementioned mushroom structure better, which is explained by its higher approximation orders (on smooth solutions).

Fig. 1.

Results in Problem 3, time t = 0.3. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.16 to 1.71 with a step of 0.05), vectors show velocity (with a multiplier of 0.05). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

Problem 4. Similarly to Problem 3, the initial discontinuities are shock waves. As they diverge, they create an area of increased pressure in the form of a lens bounded by two new shock waves. The shock wave propagation velocities in Problem 4 are higher than in Problem 3. In addition, the solution of Problem 4 does not contain structures of other types, which makes it possible to test the schemes on shock waves with greater clarity.

The flow patterns at t = 0.25, obtained by the DGM and the bicompact scheme, are shown in Fig. 2. It can be seen that the DGM smears the shock waves more strongly when compared with the results in Problem 3. Entropy production is nonnegative in all cells; it is maximum in shock waves. The bicompact scheme demonstrates the same spatial resolution as in the previous problem. Note that the solution calculated by the bicompact scheme contains small nonmonotonicities near the convergence points of the shock waves.

Fig. 2.

Results in Problem 4, time t = 0.25. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.52 to 1.92 with a step of 0.05), vectors show velocity (with a multiplier of 0.1). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

Problem 6. This problem tests the resolution of four moving contact discontinuities. In addition to them, the gas also has a diamond-shaped zone of moderate rarefaction.

From the visualizations of the flows at t = 0.3 in Fig. 3, we can conclude that the entropically regularized DGM resolves moving contact discontinuities on 16–20 cells. Excessive numerical dissipation also shows up in the entropy production plot. The bicompact scheme resolves contact discontinuities on 5 to 6 cells, which is typical for schemes of this class and order of accuracy. Nevertheless, the bicompact scheme allows a negative production of entropy on rarefaction waves, but it is small and amounts to $O({{\tau }^{3}})$ at $\tau \to 0$ [29].

Fig. 3.

Results in Problem 6, time t = 0.3. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.25 to 3.05 with a step of 0.1), vectors show velocity (with a multiplier of 0.1). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

Problem 12. The initial jumps in this problem are two shock waves and two stationary contact discontinuities. Moving away from the center of the computational domain, the shock waves leave behind a high-pressure zone bounded by two new shock waves. The fact that the solution of Problem 12 contains both shock waves and contact discontinuities (moreover, stationary) distinguishes it from Problems 3, 4, and 6 considered above.

Figure 4 shows at time t = 0.25 entropy production in the DGM and solutions calculated using the DGM and the bicompact scheme. As before, the new entropic regularization method excludes negative values P_s. Note that the DGM resolves stationary contact discontinuities on one cell, which is explained by the well-known property of the numerical Godunov flux. The bicompact scheme uses the global Lax‒Friedrichs flux splitting, so its resolution of contact discontinuities does not depend on the velocities. The contact discontinuity is smeared by a bicompact scheme into 5 cells, as in Problem 6. As for the resolution of shock waves, it repeats that obtained in Problem 4.

Fig. 4.

Results in Problem 12, time t = 0.25. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.54 to 1.7 with a step of 0.04), vectors show velocity (with a multiplier of 0.1). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

Problem 15. At the initial moment of time, the quarters of the computational domain are separated by two moving contact discontinuities, the shock wave, and the discontinuity, which subsequently turns into a continuous rarefaction wave. As the gas moves, two small-amplitude shock waves additionally appear in it. A feature of this problem (relative to Problems 3, 4, 6, 12) is the presence of all three typical structures: a rarefaction wave, a contact discontinuity, and a shock wave.

Figure 5 shows at t = 0.2 the flow patterns calculated using the DGM and the bicompact scheme, as well as the graph P_s for the DGM. It can be seen that the second law of thermodynamics in the difference form holds in the regularized DGM without any violations. The width of the smearing of contact discontinuities in the DGM is proportional to their velocities, which corresponds to the results in Problems 6 and 12. The bicompact scheme still resolves these structures on five cells. The resolution of the primary shock wave in the DGM and in the bicompact scheme is almost the same.

Fig. 5.

Results in Problem 15, point in time t = 0.2. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.43 to 0.99 with a step of 0.02), vectors show velocity (with a multiplier of 0.1). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

Problem 17 is similar to Problem 15, the difference lies in the immobility of contact discontinuities in its initial condition. In the course of time, however, they begin to move in some of their parts.

The calculation results presented at t = 0.3 in Fig. 6 repeat what was obtained in the other problems. It is curious to note that the moving parts of contact discontinuities appear on the graph P_s due to their numerical dissipation.

Fig. 6.

Results in Problem 17, time t = 0.3. Left: plot of entropy production in the DGM. Center: flow pattern obtained by DGM. Color fill shows pressure, isolines show density (values from 0.54 to 1.99 with a step of 0.05), vectors show velocity (with a multiplier of 0.1). Right: flow pattern obtained by the bicompact scheme. All calculations were performed on a grid of 200 × 200 cells.

6 CONCLUSIONS

A new conservative entropically stable version of the discontinuous Galerkin method in conservative variables for the Euler equations with two space variables is proposed, which can be generalized in an obvious way to equations with three space variables. The advantage of the proposed method is nonnegativity of the entropy production.

A new conservative entropically stable version of the DGM was tested on problems from [27]. Its testing showed satisfactory results. Compared to the version of the DGM in nonconservative variables [25], the new algorithm turns out to be faster by factors of 2 to 3 in the same problems.

The high-order bicompact scheme has a better spatial resolution than the proposed version of the DGM, which is due to the higher order of accuracy of the bicompact scheme both in space and time. Further improvement of the quality of the numerical solution in the entropically stable version of the DGM is possible by increasing its approximation order and improving the procedure for correcting the leading coefficients of the DGM.