A Semi-Lagrangian Godunov-Type Method without Numerical Viscosity for Shocks

Abstract

One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is a numerical viscosity for shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The proposed method belongs to a group of particle-in-cell (PIC) methods; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. This article describes the application of the aforementioned method for the inviscid Burgers’ equation, adiabatic gas dynamics equations, and the one-dimensional scalar hyperbolic conservation law. The numerical analysis results for several test cases (e.g., the standard shock-tube problem of Sod, the Riemann problem of Lax, the double expansion wave problem, the Shu–Osher shock-tube problem) are compared with the exact solution and Harten’s data. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10⁻⁵ to find the pressure difference between the current and previous iterations.

1. Introduction

Simulating compressible flows is challenging owing to the presence of discontinuities in fluid flow, such as shocks, contact waves, and broad-band continuous flow scales. Numerous high-resolution schemes have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously, e.g., the artificial viscosity method [1,2], total variation diminishing (TVD) method [3], essentially non-oscillatory (ENO) scheme [4], and weighted essentially non-oscillatory (WENO) scheme [5].

Harten et al. studied the ENO scheme [4,6,7,8], which was the first successful high-order method to enable the spatial discretization of hyperbolic conservation laws that had the ENO property. This property is considered to be very useful in the numerical simulation of hyperbolic conservation laws, because high-order numerical methods often produce spurious oscillations—especially near shocks or other discontinuities. The finite-volume ENO spatial discretization has been studied [4], where it was shown to have uniform high-order accuracy in capturing the location of any discontinuity in fluid flow. Subsequently, Shu and Osher [9,10] developed the finite-difference ENO scheme. The main idea of the ENO scheme is to choose a stencil of interpolation points that suppresses oscillations, i.e., one chooses the stencil on which the solution varies the smoothest and approximates the flux at the cell boundaries with high-order accuracy, thus avoiding the large, spurious oscillations caused by interpolating data across discontinuities.

WENO schemes have been introduced [5,11,12] to address potential numerical instabilities when choosing ENO stencils. WENO methods use a convex combination of all of the ENO stencils; they achieve a higher order of accuracy than ENO methods in smooth regions, while retaining the ENO property at discontinuities.

The aforementioned methods follow the Eulerian approach. One disadvantage of these methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] is the presence of a numerical viscosity for shocks. Additionally, there are methods [25,26,27,28] using the exact Riemann solver; however, they too are Eulerian methods; moreover, these methods have numerical viscosity for shocks. Godunov et al. [25] proposed using moving grids to eliminate this disadvantage, but this dramatically complicates the problem, particularly in two- or three-dimensional cases. In this paper, we use a Lagrangian Godunov-type method and a fixed homogeneous grid to eliminate the disadvantage of numerical viscosity for shocks via a simplified approach. Additionally, we describe wall conditions. As discussed in the following sections, the proposed method has no numerical viscosity for shocks, as a well-known method of characteristics (MOC) [29,30,31]. In particular [30,31], the water hammer phenomenon modelling by the MOC using the Riemann invariants or solvers is considered. In the MOC, the convective terms are typically neglected, because the Riemann invariants in the water hammer problem are only weakly interdependent [30,31]. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations [29], but we split the equations into parts describing convection and acoustic processes separately. Different time steps are used to solve these equations because the local acoustic velocity (speed of sound) can differ many times from the convective velocity of the flow. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law via the proposed method, we additionally use the iterative Godunov exact solver [25], because the Riemann invariants are non-conserved for moderate and strong shocks. On the other hand, the proposed method is similar to particle-in-cell (PIC) methods. PIC methods are readily available in the literature [32,33,34,35]; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. Finally, it should be mentioned that besides the Eulerian and Lagrangian approaches to the description of shocks, there is another approach based on the variational principle [36]; however, for this approach, when modeling flows, a numerical scheme of Eulerian or Lagrangian type is still needed.

2. Materials and Methods

2.1. A Methodological Concept for the Inviscid Burgers’ Equation

For a better understanding of an idea of the proposed method, we describe an algorithm of the method for the inviscid Burgers’ equation [37]. The inviscid Burgers’ equation is a scalar nonlinear equation given as follows:

$$\frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0,$$

(1)

where $\mathrm{u}$ is the convective velocity of flow, $\mathrm{x}$ is the direction, and $\mathrm{t}$ is the time. Equation (1) can be solved using the proposed Lagrangian scheme. The cells are considered in pairs. We will use two fundamental solutions of the inviscid Burgers’ equation: shock, and rarefaction wave. The cells’ coordinates after transferring are determined as follows:

$$\begin{gathered} {\mathrm{x}}_1^{}={\mathrm{x}}_{\mathrm{i}}^{}+{\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}{}_{ \Delta }\mathrm{t}, \\ {\mathrm{x}}_2^{}={\mathrm{x}}_{\mathrm{i}+1}^{}+{\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}{}_{ \Delta }\mathrm{t}, \end{gathered}$$

(2)

where ${}_{\Delta }\mathrm{t}$ is the time step. Furthermore, we use the following parameters: ${\mathrm{e}}_1={10}^{-4}$, which is the tolerance associated with the comparison of the flow variables; and ${\mathrm{e}}_2={10}^{-8}$, which is the machine arithmetic tolerance associated with the comparison of the $\mathrm{x}$ coordinates.

The solution is sought in the following form:

(1)

If the conditions

$$\left|{\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}}\right|<{\mathrm{e}}_1,$$

(3)

are satisfied, then for all grid cells for which the conditions are satisfied, the solution at the next time moment $\mathrm{k}$ is trivial:

$${\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}=\frac{1}{2}({\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}+{\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}).$$

(4)

(2)

If the conditions

$$({\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}})<-{\mathrm{e}}_1$$

(5)

are satisfied, then we have the shock.

(2.1) if

$$({\mathrm{x}}_2-{\mathrm{x}}_1)>{\mathrm{e}}_2,$$

(6)

then the solution is defined as

$$\begin{gathered} (2.1 \mathrm{a}) \mathrm{if} {\mathrm{x}}_1\le {\mathrm{x}}_{\mathrm{j}}\le ({\mathrm{x}}_1+{\mathrm{x}}_2)/2 \mathrm{then} {\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}, \\ (2.1 \mathrm{b}) \mathrm{if} ({\mathrm{x}}_1+{\mathrm{x}}_2)/2<{\mathrm{x}}_{\mathrm{j}}\le {\mathrm{x}}_2 \mathrm{then} {\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}. \end{gathered}$$

(7)

(2.2) If

$$({\mathrm{x}}_2-{\mathrm{x}}_1)\le {\mathrm{e}}_2,$$

(8)

then the discontinuity boundary is first determined as

$${\mathrm{x}}_{}^{}={\mathrm{x}}_{\mathrm{i}}^{}+\mathrm{h}/2+0.5{({\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}+{\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1})}_{ \Delta }\mathrm{t},$$

(9)

and the solution is assigned to cell $\mathrm{j}$ to the left of the discontinuity boundary

$${\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1},$$

(10)

and in cell $\mathrm{j}+1$ to the right of the discontinuity boundary, the following values are assigned:

$${\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}.$$

(11)

(3)

If

$$({\mathrm{u}}_{\mathrm{i}+1}-{\mathrm{u}}_{\mathrm{i}})\ge {\mathrm{e}}_1,$$

(12)

then we have the rarefaction wave, and the solution for ${\mathrm{x}}_1\le {\mathrm{x}}_{\mathrm{j}}\le {\mathrm{x}}_2$ is determined as follows:

$${\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1).$$

(13)

In this case, the loop over all ${\mathrm{x}}_{\mathrm{i}}$ is performed twice: first, for all ${\mathrm{u}}_{\mathrm{i}}\ge {\mathrm{e}}_1$ from $\mathrm{n}-1$ to $1$, and second, for all ${\mathrm{u}}_{\mathrm{i}}\le -{\mathrm{e}}_1$ from $1$ to $\mathrm{n}-1$. Here, $\mathrm{n}$ is the number of grid cells. When conditions (1)–(3) are satisfied, if it turns out that the solution has already been assigned to the grid cell, then it is replaced by a new one. When assigning the solution to the grid cells, the condition is separately checked so that the propagation of this solution at the convection velocity does not overtake (does not overwrite) the solution of the forward shock, if one exists. For this, at the beginning, the positions of all of the shock boundaries at the next moment in time are determined.

The modeling area is taken to be $\mathrm{x}\in \left[\mathrm{a}, \mathrm{b}\right]$. The grid contains $\mathrm{n}$ cells, and the grid step $\mathrm{h}$ is

$$\mathrm{h}=\frac{\mathrm{b}-\mathrm{a}}{\mathrm{n}},$$

(14)

as shown in Figure 1. The time step was chosen similar to the Courant criterion.

$${}_{\Delta }\mathrm{t}={\mathrm{k}}_{\mathrm{u}}\frac{\mathrm{h}}{{\mathrm{u}}_{\mathrm{s}}},$$

(15)

where ${\mathrm{u}}_{\mathrm{s}}$ is the convective velocity of the shock, and ${\mathrm{k}}_{\mathrm{u}}$ is an integer. For example, if ${\mathrm{k}}_{\mathrm{u}}=4$, then the shock propagates to the four grid cells.

2.2. Basic Terms and a Methodological Concept for Adiabatic Gas Dynamics Equations

Consider the one-dimensional adiabatic gas dynamics equations [37]:

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}+\mathsf{\rho }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0. \end{gathered}$$

(16)

Here, $\mathsf{\rho }$ is the density and $\mathrm{p}$ is the pressure.

The pressure is related to the adiabatic law

$$\frac{\mathrm{p}}{{\mathsf{\rho }}^{\mathsf{\kappa }}}=\frac{{\mathrm{p}}_0}{{\mathsf{\rho }}_0^{\mathsf{\kappa }}}=\mathrm{A}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t},$$

(17)

where $\mathsf{\kappa }$ is the ratio of specific heat coefficients at constant pressure and volume. For air, $\mathsf{\kappa }=1.4$. Values ${\mathsf{\rho }}_0$ and ${\mathrm{p}}_0$ can be chosen from initial conditions of a considered problem.

As previously shown [38], the system of Equation (16) can be represented as a system of wave equations as follows:

$$\begin{gathered} \frac{\partial {\mathrm{w}}_1}{\partial \mathrm{t}}+(\mathrm{u}-\mathrm{c})\frac{\partial {\mathrm{w}}_1}{\partial \mathrm{x}}=0, \\ \frac{\partial {\mathrm{w}}_2}{\partial \mathrm{t}}+(\mathrm{u}+\mathrm{c})\frac{\partial {\mathrm{w}}_2}{\partial \mathrm{x}}=0, \end{gathered}$$

(18)

where $\mathrm{c}$ is the sound velocity, and ${\mathrm{w}}_{\mathrm{i}}$ represents variables that are determined as follows:

$$\begin{gathered} {\mathrm{w}}_1=\mathrm{u}-\tilde{\mathrm{P}}, \\ {\mathrm{w}}_2=\mathrm{u}+\tilde{\mathrm{P}}. \end{gathered}$$

(19)

Here, the function $\tilde{\mathrm{P}}$ is

$$\tilde{\mathrm{P}}=\frac{2{\mathrm{c}}_0}{\mathsf{\kappa }-1}\left({\left(\frac{\mathsf{\rho }}{{\mathsf{\rho }}_0}\right)}^{\frac{\mathsf{\kappa }-1}{2}}-1\right),$$

(20)

where ${\mathrm{c}}_0$ is the sound velocity, which is determined from the initial conditions as follows:

$${\mathrm{c}}_0=\sqrt{\mathsf{\kappa }\frac{{\mathrm{p}}_0}{{\mathsf{\rho }}_0}}.$$

(21)

Equation (18) describes a movement of two waves with velocities $\mathrm{u}-\mathrm{c}$ and $\mathrm{u}+\mathrm{c}$, and their respective transfer values ${\mathrm{w}}_1$ and ${\mathrm{w}}_2$. The system of Equation (18) can be solved using the Lagrangian approach. In this paper, the following method is proposed: The stages of convection and acoustics are considered separately, and splitting into physical processes is performed because the local acoustic velocity (speed of sound) can differ greatly from the convective velocity of the flow. Thus, at the acoustic stage we solve the system

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathsf{\rho }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0, \end{gathered}$$

(22)

obtained from (16) by discarding the convective terms; its corresponding system of wave equations is

$$\begin{gathered} \frac{\partial {\mathrm{w}}_1}{\partial \mathrm{t}}-\mathrm{c}\frac{\partial {\mathrm{w}}_1}{\partial \mathrm{x}}=0, \\ \frac{\partial {\mathrm{w}}_2}{\partial \mathrm{t}}+\mathrm{c}\frac{\partial {\mathrm{w}}_2}{\partial \mathrm{x}}=0, \end{gathered}$$

(23)

The system (23) is obtained from (22) with help of (19).

At the convection stage, we solve the system with the convective terms only:

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0. \end{gathered}$$

(24)

We apply the numerical scheme principle described above for the inviscid Burgers’ equation to find the solution of the systems (23) and (24); for more details, see Appendix A.

The time step at the acoustics stage was chosen according to the Courant criterion.

$${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={\mathrm{k}}_{\mathrm{c}}\frac{\mathrm{h}}{{\mathrm{c}}_{\mathrm{s}}},$$

(25)

where ${\mathrm{c}}_{\mathrm{s}}$ is the acoustic velocity of the shock. For example, if ${\mathrm{k}}_{\mathrm{c}}=3$, then the shock propagates through three grid cells at the acoustics stage.

The time step at the convection stage was chosen similar to that of (25):

$${}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={\mathrm{k}}_{\mathrm{u}}\frac{\mathrm{h}}{{\mathrm{u}}_{\mathrm{s}}},$$

(26)

where ${\mathrm{u}}_{\mathrm{s}}$ is the convective velocity of the shock. For example, if ${\mathrm{k}}_{\mathrm{u}}=2$, then the shock propagates to two grid cells at the convection stage. If ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}\ne {}_{\Delta }\mathrm{t}{}_{\mathrm{u}}$, then the marching time step ${}_{\Delta }\mathrm{t}$ is chosen so that

$$\begin{gathered} {}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={\mathrm{n}}_{\mathrm{c}}{}_{\Delta }\mathrm{t}, \\ {}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={\mathrm{n}}_{\mathrm{u}}{}_{\Delta }\mathrm{t}, \end{gathered}$$

(27)

where ${\mathrm{n}}_{\mathrm{c}}$ and ${\mathrm{n}}_{\mathrm{u}}$ are integers.

2.3. Basic Terms and a Methodological Concept for the One-Dimensional Scalar Hyperbolic Conservation Law

Consider the one-dimensional scalar hyperbolic conservation law [13]:

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}+\mathsf{\rho }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\epsilon }}{\partial \mathrm{x}}+\frac{\mathrm{p}}{\mathsf{\rho }}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0. \end{gathered}$$

(28)

Here, $\mathsf{\epsilon }$ is the internal specific energy.

The pressure is related to thermodynamic state variables $\mathsf{\rho }$ and $\mathsf{\epsilon }$ through the following state equation:

$$\mathrm{p}=(\mathsf{\kappa }-1) \mathsf{\rho } \mathsf{\epsilon }.$$

(29)

Godunov et al. [25] proposed the iterative exact Riemann solver implemented in a finite-difference scheme. Instead of this method, other researchers later used a control volume method, while we used the Lagrangian approach. In this paper, the following method is proposed: The stages of convection and acoustics are considered separately, and splitting into physical processes is performed because the local acoustic velocity (speed of sound) can differ greatly from the convective velocity of the flow. Thus, at the acoustic stage we solve the system

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathsf{\rho }\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\frac{1}{\mathsf{\rho }}\frac{\partial \mathrm{p}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{t}}+\frac{\mathrm{p}}{\mathsf{\rho }}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0. \end{gathered}$$

(30)

and the Equation (29). The system (30) is obtained from (28) by discarding the convective terms, and it can be represented in a matrix form as follows:

$$\left\{\begin{array}{c}\frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}\\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}\\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{t}}\end{array}\right\}=\left[\mathrm{A}\right]\left\{\begin{array}{c}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}\\ \frac{\partial \mathrm{u}}{\partial \mathrm{x}}\\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{x}}\end{array}\right\}$$

(31)

where $\left[\mathrm{A}\right]$ is the Jacobian matrix

$$\left[\mathrm{A}\right]=\left[\begin{array}{ccc}0& \mathsf{\rho }& 0\\ \frac{(\mathsf{\kappa }-1)\mathsf{\epsilon }}{\mathsf{\rho }}& 0& \mathsf{\kappa }-1\\ 0& (\mathsf{\kappa }-1)\mathsf{\epsilon }& 0\end{array}\right].$$

(32)

The matrix $\left[\mathrm{A}\right]$ can be expressed as

$$\left[\mathrm{A}\right]=\left[\mathrm{R}\right]\left[\mathsf{\Lambda }\right]\left[\mathrm{L}\right]$$

(33)

in terms of a diagonal matrix $[\mathsf{\Lambda }]$ of eigenvalues of the Jacobian matrix

$$\left[\mathsf{\Lambda }\right]=\left[\begin{array}{ccc}-\mathrm{c}& 0& 0\\ 0& 0& 0\\ 0& 0& \mathrm{c}\end{array}\right],$$

(34)

a matrix of corresponding left eigenvectors $[\mathrm{L}]$, determined as

$$\left[\mathrm{L}\right]=\left[\begin{array}{ccc}-\frac{\mathrm{c}}{\mathsf{\kappa }\mathsf{\rho }}& 1& -\frac{\mathsf{\kappa }-1}{\mathrm{c}}\\ 1& 0& -\frac{\mathsf{\rho }\mathsf{\kappa }}{{\mathrm{c}}^2}\\ \frac{\mathrm{c}}{\mathsf{\kappa }\mathsf{\rho }}& 1& \frac{\mathsf{\kappa }-1}{\mathrm{c}}\end{array}\right],$$

(35)

and a matrix of corresponding right eigenvectors $[\mathrm{R}]$, defined as

$$\left[\mathrm{R}\right]=\left[\begin{array}{ccc}-\frac{\mathsf{\rho }}{2\mathrm{c}}& \frac{\mathsf{\kappa }-1}{\mathsf{\kappa }}& \frac{\mathsf{\rho }}{2\mathrm{c}}\\ \frac{1}{2}& 0& \frac{1}{2}\\ -\frac{\mathrm{c}}{2\mathsf{\kappa }}& -\frac{(\mathsf{\kappa }-1)\mathsf{\epsilon }}{\mathsf{\kappa }\mathsf{\rho }}& \frac{\mathrm{c}}{2\mathsf{\kappa }}\end{array}\right].$$

(36)

Since

$$\left[\mathrm{L}\right]\left[\mathrm{R}\right]=\left[\mathrm{E}\right],$$

(37)

where $\left[\mathrm{E}\right]$ is a diagonal identity matrix, the multiplication of Equation (31) from the left by $\left[\mathrm{L}\right]$ and the use of (37) gives

$$\left[\mathrm{L}\right]\left\{\begin{array}{c}\frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}\\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}\\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{t}}\end{array}\right\}=\left[\mathsf{\Lambda }\right]\left[\mathrm{L}\right]\left\{\begin{array}{c}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}\\ \frac{\partial \mathrm{u}}{\partial \mathrm{x}}\\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{x}}\end{array}\right\}.$$

(38)

If $\left[\mathrm{L}\right]=[\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}]$, we can obtain from (38) the characteristic form

$$\begin{gathered} \frac{\partial {\mathrm{w}}_1}{\partial \mathrm{t}}-\mathrm{c}\frac{\partial {\mathrm{w}}_1}{\partial \mathrm{x}}=0, \\ \frac{\partial {\mathrm{w}}_2}{\partial \mathrm{t}}=0, \\ \frac{\partial {\mathrm{w}}_3}{\partial \mathrm{t}}+\mathrm{c}\frac{\partial {\mathrm{w}}_3}{\partial \mathrm{x}}=0, \end{gathered}$$

(39)

where

$$\begin{gathered} {\mathrm{w}}_1=-\frac{\mathrm{c}*}{\mathsf{\kappa }\mathsf{\rho }*}\mathsf{\rho }+\mathrm{u}-\frac{\mathsf{\kappa }-1}{\mathrm{c}*}\mathsf{\epsilon }, \\ {\mathrm{w}}_2=\mathsf{\rho }-\frac{\mathsf{\kappa }\mathsf{\rho }*}{\mathrm{c}{*}^2}\mathsf{\epsilon }, \\ {\mathrm{w}}_3=\frac{\mathrm{c}*}{\mathsf{\kappa }\mathsf{\rho }*}\mathsf{\rho }+\mathrm{u}+\frac{\mathsf{\kappa }-1}{\mathrm{c}*}\mathsf{\epsilon }. \end{gathered}$$

(40)

Here, an asterisk indicates constant values. Equation (39) describes the movement of two waves with velocities $-\mathrm{c}$ and $\mathrm{c}$, and their respective transfer values, ${\mathrm{w}}_1$ and ${\mathrm{w}}_3$ (40).

The system of Equation (39) can be solved using the proposed Lagrangian approach. However, in this paper, we use the Godunov exact solver for the solution of the full nonlinear system (28), (29), and then subtract the convective velocity from the solution to obtain the solution of the Equation (30). For more details, see Appendix B.

At the convection stage, we solve the system with the convective terms only:

$$\begin{gathered} \frac{\partial \mathsf{\rho }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\rho }}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{x}}=0, \\ \frac{\partial \mathsf{\epsilon }}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathsf{\epsilon }}{\partial \mathrm{x}}=0, \end{gathered}$$

(41)

as well as Equation (29).

The time step ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}$ at the acoustics stage was chosen according to the Courant criterion (25). The time step ${}_{\Delta }\mathrm{t}{}_{\mathrm{u}}$ at the convection stage was chosen as (26).

3. Results

In this section, we apply the proposed method to solve several problems with different initial and boundary conditions.

3.1. Results for the Inviscid Burgers’ Equation

Firstly, we considered the initial value problem [39] for the inviscid Burgers’ equation with the initial conditions

$$\begin{gathered} {\mathrm{u}}_{\mathrm{j}}^0=0, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<0, \\ {\mathrm{u}}_{\mathrm{j}}^0=1, \mathrm{if} 0\le {\mathrm{x}}_{\mathrm{j}}\le 1, \\ {\mathrm{u}}_{\mathrm{j}}^0=0, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}>1. \end{gathered}$$

(42)

The exact solution of the shock–rarefaction wave problem (42) was determined as follows:

$$\mathrm{u}(\mathrm{t},\mathrm{x})=\{\begin{array}{c}0, \mathrm{i}\mathrm{f} \mathrm{x}<0\\ \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{t}$}\right., \mathrm{i}\mathrm{f} 0\le \mathrm{x}\le \mathrm{t}\\ 1, \mathrm{i}\mathrm{f} \mathrm{t}<\mathrm{x}\le \raisebox{1ex}{$\mathrm{t}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1\\ 0, \mathrm{i}\mathrm{f} \mathrm{x}>\raisebox{1ex}{$\mathrm{t}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.+1\end{array}, \mathrm{for} \mathrm{t}\le 2,$$

(43)

and

$$\mathrm{u}(\mathrm{t},\mathrm{x})=\{\begin{array}{c}0, \mathrm{i}\mathrm{f} \mathrm{x}<0\\ \raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{t}$}\right., \mathrm{i}\mathrm{f} 0\le \mathrm{x}\le \sqrt{2\mathrm{t}}\\ 0, \mathrm{i}\mathrm{f} \mathrm{x}>\sqrt{2\mathrm{t}}\end{array}, \mathrm{for} \mathrm{t}>2.$$

(44)

The modeling area was taken to be $\mathrm{x}\in \left[-1, 4\right]$. The grid contained 50 cells, and the grid step was $\mathrm{h}=0.1$. The time step was chosen according to the Courant criterion (15), where ${\mathrm{u}}_{\mathrm{s}}=0.5$ is the convective velocity of the shock in (42), where ${\mathrm{k}}_{\mathrm{u}}=4$, i.e., the shock propagates to the four grid cells. The time step was ${}_{\Delta }\mathrm{t}=0.8$. It should be noted that the rarefaction wave (at its right boundary) has the convective velocity ${\mathrm{u}}_{\mathrm{r}}=1.0$ in (43), and propagates to the eight grid cells. At the time $\mathrm{t}=2$, the rarefaction wave catches up with the shock, but is not able to overtake it (44).

The results after two and five time steps are shown in Figure 2 and Figure 3 in comparison to the exact solutions (43) and (44), respectively. We can see the good agreement between the numerical and exact solutions, and that the proposed method does not have numerical viscosity for shocks. However, owing to the accumulation of the roundoff error, the shock in Figure 3 overtakes the exact solution by one grid cell at the shown moment in time.

Secondly, we considered the well-known test problem [40] with the periodic initial function

$${\mathrm{u}}_{\mathrm{j}}^0=0.5\sin (-\mathsf{\pi } \mathrm{x}).$$

(45)

The modeling area was taken to be $\mathrm{x}\in \left[-1, 1\right]$. The grid contained 80 cells, and the grid step was $\mathrm{h}=0.025$. The time step was chosen according to the Courant criterion (15), where ${\mathrm{u}}_{\mathrm{s}}=0.5$ is the convective velocity of the shock in (45), where ${\mathrm{k}}_{\mathrm{u}}=2$, i.e., the shock propagates to the two grid cells. The time step was ${}_{\Delta }\mathrm{t}=0.1$.

The result after 11 time steps is shown in Figure 4 in comparison to the exact solution [40]. We can see the good agreement between numerical and exact solutions, and that the proposed method does not have numerical viscosity for shocks.

3.2. Results for the Adiabatic Gas Dynamics Equations

The first example for the adiabatic gas dynamics equations is the standard shock-tube problem with the following initial conditions:

$$\begin{gathered} {\mathrm{p}}_{\mathrm{j}}^0=1, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=1, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<0, \\ {\mathrm{p}}_{\mathrm{j}}^0=0.378929142, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=0.5, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}\ge 0. \end{gathered}$$

(46)

The modeling area was taken to be $\mathrm{x}\in \left[-1,1\right]$. The grid contained 200 cells, and the grid step was $\mathrm{h}=0.01$. The time step at the acoustics stage ${}_{\Delta }{\mathrm{t}}_{\mathrm{c}}=0.054218535053$ was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=6$, i.e., the shock propagates through six grid cells at the acoustics stage. The time step at the convection stage ${}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.052230695168$ was chosen according to a similar criterion (26), where ${\mathrm{k}}_{\mathrm{u}}=2$, i.e., the shock propagates to two grid cells at the convection stage. The marching time step was chosen as ${}_{\Delta }\mathrm{t}{=}_{\Delta }{\mathrm{t}}_{\mathrm{c}}{=}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.052230695168$.

The results after 10 marching time steps $\mathrm{t}=0.52230695168$ are shown in Figure 5, Figure 6 and Figure 7 in comparison to the exact solution. Figure 5, Figure 6 and Figure 7 show the absence of the numerical viscosity for the shock. However, owing to the accumulation of the roundoff error, the shock overtakes the exact solution by two grid cells at the shown moment in time.

The second example for the adiabatic gas dynamics equations is the double expansion wave problem [41] with the following initial conditions:

$$\begin{gathered} {\mathrm{p}}_{\mathrm{j}}^0={10}^5, {\mathrm{u}}_{\mathrm{j}}^0=-100, {\mathsf{\rho }}_{\mathrm{j}}^0=1.2, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<0, \\ {\mathrm{p}}_{\mathrm{j}}^0=10^5, {\mathrm{u}}_{\mathrm{j}}^0=100, {\mathsf{\rho }}_{\mathrm{j}}^0=1.2, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}\ge 0. \end{gathered}$$

(47)

The modeling area was taken to be $\mathrm{x}\in \left[-0.5,0.5\right]$. The grid contained 100 cells, and the grid step was $\mathrm{h}=0.01$. The time step at the acoustics stage ${}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.000204939017$ was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=7$, i.e., the shock propagates through seven grid cells at the acoustics stage. The time step at the convection stage ${}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.0002$ was chosen according to a similar criterion (26), where ${\mathrm{k}}_{\mathrm{u}}=2$, i.e., the shock propagates to two grid cells at the convection stage. The marching time step was chosen as ${}_{\Delta }\mathrm{t}{=}_{\Delta }{\mathrm{t}}_{\mathrm{c}}{=}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.0002$.

The results after four marching time steps $\mathrm{t}=0.0008$ are shown in Figure 8, Figure 9 and Figure 10 in comparison to the exact solution. Figure 8, Figure 9 and Figure 10 show a good agreement between the exact and numerical solutions for the double expansion wave problem.

3.3. Results for the One-Dimensional Scalar Hyperbolic Conservation Law

The first example is the standard shock-tube problem of Sod [42], with the following initial conditions:

$$\begin{gathered} {\mathrm{p}}_{\mathrm{j}}^0=1, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=1, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<0, \\ {\mathrm{p}}_{\mathrm{j}}^0=0.1, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=0.125, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}\ge 0. \end{gathered}$$

(48)

The modeling area was taken to be $\mathrm{x}\in \left[-4.5, 5.5\right]$. The grid contained 100 cells, and the grid step was $\mathrm{h}=0.1$. The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=1$, i.e., the shock propagates through one grid cell at the acoustics stage. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=2$, i.e., the shock propagates to two grid cells at the convection stage. The marching time step was ${}_{\Delta }{\mathrm{t}}_{\mathrm{u}}=0.02020929$. The acoustics stage was performed in 6 time steps ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}=6{}_{\Delta }\mathrm{t}$, and the convection stage was performed after 10 steps ${}_{\Delta }\mathrm{t}{}_{\mathrm{u}}=10{}_{\Delta }\mathrm{t}$.

The results after 110 marching time steps are shown in Figure 11, Figure 12 and Figure 13 in comparison to the exact solution and the data obtained by Harten [3] via the ULT1C scheme. Note that more modern numerical schemes, such as WENO [14,24], have even higher numerical viscosity for shocks. Therefore, we chose the ULT1C scheme for comparison. Figure 11, Figure 12 and Figure 13 show the absence of the numerical viscosity for the shock, in contrast to the Harten method. However, owing to the accumulation of the roundoff error, the shock is delayed by one grid cell at the shown moment in time.

The second example is the Riemann problem of Lax [43], with the following initial conditions:

$$\begin{gathered} {\mathrm{p}}_{\mathrm{j}}^0=3.52773, {\mathrm{u}}_{\mathrm{j}}^0=0.69888, {\mathsf{\rho }}_{\mathrm{j}}^0=0.445, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<0, \\ {\mathrm{p}}_{\mathrm{j}}^0=0.571, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=0.5, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}\ge 0. \end{gathered}$$

(49)

The modeling area was taken as $\mathrm{x}\in \left[-8, 6\right]$. The grid contained 140 cells, and the grid step was $\mathrm{h}=0.1$. The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=2$, i.e., the shock propagates through two grid cells at the acoustics stage. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=3$, i.e., the shock propagates through three grid cells at the convection stage. The marching time step was ${}_{\Delta }\mathrm{t}=0.2$. The acoustics and convection stages were performed at each marching time step, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={}_{\Delta }\mathrm{t}$.

The results are shown in Figure 14, Figure 15 and Figure 16 in comparison with the exact solution and the data obtained by Harten [3] using the ULT1C scheme at the time $\mathrm{t}=2.0$. Figure 14, Figure 15 and Figure 16 show the absence of the numerical viscosity for the shock, in contrast to the Harten method. However, owing to the accumulation of the roundoff error, the shear wave is delayed by one grid cell at the shown moment in time.

The third example for the one-dimensional scalar hyperbolic conservation law is the double expansion wave problem [41] with the initial conditions given by (47).

The modeling area was taken to be $\mathrm{x}\in \left[-0.5,0.5\right]$. Two kinds of grids were used: a coarse grid (100 grid points), and a finer grid (1000 grid points); the grid step was $\mathrm{h}=0.01$ and $\mathrm{h}=0.001$, respectively. The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=7$—i.e., the shock propagates through seven grid cells at the acoustics stage—for both the coarse grid and the finer grid. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=2$—i.e., the shock propagates through two grid cells at the convection stage—for both grids. The marching time step was ${}_{\Delta }\mathrm{t}=0.0002$ for the coarse grid and ${}_{\Delta }\mathrm{t}=0.00002$ for the finer grid. The acoustics and convection stages were performed at each marching time step, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={}_{\Delta }\mathrm{t}$.

The results after 4 marching time steps for the coarse grid and after 40 marching time steps for the finer grid ($\mathrm{t}=0.0008$) are shown in Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 in comparison to the exact solution. Figure 17, Figure 18 and Figure 19 show a good agreement and Figure 20, Figure 21 and Figure 22 show a better agreement between the exact and numerical solutions for the double expansion wave problem.

The fourth example is the Shu–Osher shock-tube problem.

The Shu–Osher problem [44] simulates a normal shock front moving inside a one-dimensional inviscid flow with artificial density fluctuations. The initial conditions for the simulation are

$$\begin{gathered} {\mathrm{p}}_{\mathrm{j}}^0=10.3333, {\mathrm{u}}_{\mathrm{j}}^0=2.629369, {\mathsf{\rho }}_{\mathrm{j}}^0=3.857143, \mathrm{if} {\mathrm{x}}_{\mathrm{j}}<1/8, \\ {\mathrm{p}}_{\mathrm{j}}^0=1.0, {\mathrm{u}}_{\mathrm{j}}^0=0, {\mathsf{\rho }}_{\mathrm{j}}^0=1+0.2\sin (16\mathsf{\pi }\mathrm{x}), \mathrm{if} {\mathrm{x}}_{\mathrm{j}}\ge 1/8. \end{gathered}$$

(50)

The modeling area was taken to be $\mathrm{x}\in \left[0,1\right]$. Three kinds of grids were used: a coarse grid (192 grid points), a fine grid (384 grid points), and a finer grid (1000 grid points). The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=2$—i.e., the shock propagates through two grid cells at the acoustics stage—for all grids. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=5$—i.e., the shock propagates through five grid cells at the convection stage—for all grids. The marching time step was ${}_{\Delta }\mathrm{t}=0.010270978683$ for the coarse grid, ${}_{\Delta }\mathrm{t}=0.005135489341$ for the fine grid, and ${}_{\Delta }\mathrm{t}=0.001972027907$ for the finer grid. The acoustics and convection stages were performed at each marching time step, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={}_{\Delta }\mathrm{t}$. To avoid additional fluctuations in the solution, we calculated the rarefaction wave at the acoustics stage instead of (A50) as

$$\begin{gathered} {\mathrm{U}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{U}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathrm{U}}_{\mathrm{R}}{}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathrm{U}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathrm{E}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{E}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathrm{E}}_{\mathrm{R}}{}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathrm{E}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathrm{R}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{R}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathrm{R}}_{\mathrm{R}}{}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathrm{R}}_{\mathrm{R}}{}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathrm{P}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}}=(\mathsf{\kappa }-1){\mathrm{R}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}} {\mathrm{E}}_{\mathrm{R}}{}_{\mathrm{j}}^{\mathrm{k}}, \end{gathered}$$

(51)

and at the convection stage instead of (A67) as

$$\begin{gathered} {\mathrm{u}}_{\mathrm{j}}^{\mathrm{k}}={\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathrm{u}}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathrm{u}}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathsf{\epsilon }}_{\mathrm{j}}^{\mathrm{k}}={\mathsf{\epsilon }}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathsf{\epsilon }}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathsf{\epsilon }}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathsf{\rho }}_{\mathrm{j}}^{\mathrm{k}}={\mathsf{\rho }}_{\mathrm{i}}^{\mathrm{k}-1}+\frac{{\mathsf{\rho }}_{\mathrm{i}+1}^{\mathrm{k}-1}-{\mathsf{\rho }}_{\mathrm{i}}^{\mathrm{k}-1}}{{\mathrm{x}}_2-{\mathrm{x}}_1}({\mathrm{x}}_{\mathrm{j}}-{\mathrm{x}}_1), \\ {\mathrm{p}}_{\mathrm{j}}^{\mathrm{k}}=(\mathsf{\kappa }-1){\mathsf{\rho }}_{\mathrm{j}}^{\mathrm{k}} {\mathsf{\epsilon }}_{\mathrm{j}}^{\mathrm{k}}. \end{gathered}$$

(52)

The results are shown in Figure 23, Figure 24 and Figure 25 in comparison with the “exact” solution [44] at the time $\mathrm{t}=0.178$. The results show that the fine grid and the finer grid are better than the coarse grid. However, owing to the accumulation of the roundoff error, the shock is delayed by a few grid cells for the coarse grid results at the shown moment in time.

The fifth example is a one-dimensional problem with initial conditions

$${\mathrm{p}}_{\mathrm{j}}^0=101325.0, {\mathrm{u}}_{\mathrm{j}}^0=100.0, {\mathsf{\rho }}_{\mathrm{j}}^0=1.25.$$

(53)

In this case, at the time moment $\mathrm{t}=0$, a thin wall was lowered in the channel section $\mathrm{x}=0$.

The modeling area was taken to be $\mathrm{x}\in \left[-5, 6\right]$. The grid contained 110 cells, and the grid step was $\mathrm{h}=0.1$. The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=7$, i.e., the shock propagates through seven grid cells at the acoustics stage. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=6$, i.e., the shock propagates through six grid cells at the convection stage. The marching time step was ${}_{\Delta }\mathrm{t}=0.002$. The acoustics stage was performed at each marching time step, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={}_{\Delta }\mathrm{t}$; the convection stage was performed after three steps, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{u}}=3{}_{\Delta }\mathrm{t}$.

The results are shown in Figure 26, Figure 27 and Figure 28 in comparison with the exact solution at the time $\mathrm{t}=0.012$. Figure 26, Figure 27 and Figure 28 show the absence of numerical viscosity for the shock. In this case, there is no delay caused by the shock.

The sixth example is a one-dimensional problem with the initial conditions

$${\mathrm{p}}_{\mathrm{j}}^0=101325.0, {\mathrm{u}}_{\mathrm{j}}^0=600.0, {\mathsf{\rho }}_{\mathrm{j}}^0=1.25.$$

(54)

In this case, as in the third problem, at the time $\mathrm{t}=0$, a thin wall was lowered in the channel section $\mathrm{x}=0$.

The modeling area was taken to be $\mathrm{x}\in \left[-5, 5\right]$. The grid contained 100 cells, and the grid step was $\mathrm{h}=0.1$. The time step at the acoustics stage was chosen according to the Courant criterion (25), where ${\mathrm{k}}_{\mathrm{c}}=2$, i.e., the shock propagates through two grid cells at the acoustics stage. The time step at the convection stage was chosen according to (26), where ${\mathrm{k}}_{\mathrm{u}}=5$, i.e., the shock propagates through five grid cells at the convection stage. The marching time step was ${}_{\Delta }\mathrm{t}=0.00079$. The acoustics and convection stages were performed at each marching time step, i.e., ${}_{\Delta }\mathrm{t}{}_{\mathrm{c}}={}_{\Delta }\mathrm{t}{}_{\mathrm{u}}={}_{\Delta }\mathrm{t}$.

The results are shown in Figure 29, Figure 30 and Figure 31 in comparison with the exact solution at the time $\mathrm{t}=0.00474$. Figure 29, Figure 30 and Figure 31 show the absence of numerical viscosity for the shock. In this case, there is no delay caused by the shock.

4. Discussion

The numerical solution results for the aforementioned well-known test cases were compared with the exact solutions and Harten’s data. Based on the results obtained, the following conclusions can be drawn:

(1)

The advantage of the proposed method in comparison with the known methods of TVD, ENO, and WENO is the absence of numerical viscosity (diffusion) for shocks;

(2)

The convective terms are not neglected in comparison with the MOC methods, which is important in gas flow modeling;

(3)

Over time, shocks or rarefaction waves can propagate somewhat slower or faster than in the exact solution. This can be attributed to the rounding off of the position of the wave fronts to the accuracy of the grid cell, owing to the use of the fixed homogeneous grid.

The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10⁻⁵ to find the pressure difference between the current and previous iterations.

The directions for further research are connected with two-dimensional modeling and the study of complex flow structures caused by fluid–solid interactions.