Approximation of the Interactions of Rarefaction Waves by the Wave Front Tracking Method

Abstract

The interaction of two simple delta shock waves for a pressureless gas dynamic system is considered. The result of the interaction is a delta shock wave with constant speed. This interaction is approximated by letting the perturbed parameter in the Euler equations for isentropic fluids go to zero. Each delta shock wave is approximated by two shock waves of the first and second family when the perturbed parameter goes to zero. These shock waves are solutions of two Riemann problems at time $t=0$. The solution of the Riemann problem for $t>0$ can also contain rarefaction waves. If the perturbed parameter approaches 0, the strength of the rarefaction waves increases and the number of interactions of the rarefaction waves increases, as well. When two split rarefaction waves interact, the number of Riemann problems to be solved is $m_1·m_2$, where $m_i$ is the number of ith rarefaction waves. The main topic of this paper is to develop an algorithm that reduces the number of these Riemann problems. The algorithm is based on the determination of the intermediate states that make the Rankine–Hugoniot deficit small. The approximated wave front tracking algorithm was used for the numerical verification of these interactions. The theoretical background was the concept of the shadow wave solution.

1. Introduction

Fluid dynamics problems contain conservation laws. Theoretical background can be found in [1,2]. The author in [2] introduced the linear theory, the reaction–diffusion equations, the shock wave theory, and the Conley index. In addition, the properties and simple one-dimensional wave equation, Holmgren’s theorem, distribution theory, eliptic and parabolic PDE, shock wave theory, and weak solutions have been considered. The book [1] contains nonlinear equations, quasilinear systems, hyperbolic systems, and the theory of characteristics. In addition, Riemann invariants and simple waves were generalized with the theory of one-dimensional gas motion. Shock waves and weak solutions were introduced. Discontinuous solutions (shocks) were treated in a very general context. A detailed analysis of shallow water waves was also given.

The analysis of the conservation laws can be found in the books [3,4]. They contain the classical theory of semilinear and quasilinear systems and describe the method of characteristics for nonlinear scalar equations. Weak solutions, basic definitions, and various conditions for the admissibility of entropy were given. In addition, the concepts of genuinely nonlinear and linear degenerate characteristics were introduced. The authors constructed the standard self-similar solution of the Riemann problem with centered rarefaction, shocks, and contact discontinuity waves. Wave front tracking algorithms were chosen to construct the global solutions of a scalar conservation law. The Cauchy problem for $n\times n$ systems of conservation laws with a detailed description of the front-tracking algorithm was also presented with various applications.

In everyday life, humans face shock waves and rarefaction largely in their surroundings. It is therefore necessary to know the behavior of these waves in order to protect oneself from destructive effects. The propagation of shock and rarefaction waves in various dynamics from solving nonlinear hyperbolic inviscid Burgers equation numerically was observed in [5]. Burgers’ equation is explained as a mathematical model for turbulent flow. In recent years, the Burgers equation continued to draw the attention of researchers. It is used as a model to test several numerical methods since it includes a convection term $u{u}_x$ and a viscosity term $\theta u_{xx}$. As $\theta \to 0$, Burgers equation $u_t+u{u}_x=\theta u_{xx}$ becomes a hyperbolic equation, called the inviscid Burgers equation, $u_t+u{u}_x=0$. The authors used the method of characteristics to find the exact solution of the inviscid Burgers equation. The first-order explicit upwind scheme and second-order Lax–Wendroff scheme were used to solve this equation to improve the understanding of numerical diffusion (smearing) and oscillations that can occur when using such schemes. Numerical solutions were investigated for different initial conditions, and the shock and rarefaction waves were studied for the Riemann problem.

A Cauchy problem for hyperbolic conservation laws has two problems. The first is that the solution does not remain continuous in finite time. The second problem is the choice of an admissible solution among several weak solutions. A unique solution to the Cauchy problem is obtained by introducing terms such as admissible wave, physically irrelevant solution, and so on. The admissibility of the weak solution and the complete theoretical basis of the conservation laws are given by Lax [6].

A Cauchy problem (initial value problem) with piecewise constant initial data is called a Riemann problem. For the Euler equations of gas dynamics, one can find a detailed construction of the solution of the Riemann problem in [7,8]. There are many papers in which one can find examples of systems that do not have a classical solution to the Riemann problem. Most of these solutions are delta waves (see [9,10,11]) or singular shock waves (see [12,13]), and they can be considered as special cases of a new type of solution—shadow waves (hereafter referred to as SDW [14]). There are several conditions under which the authors search for a solution for these systems. Here, overcompressibility was a criterion for the admissibility of the wave. A special problem is when the interaction involves singular solutions.

The wave front tracking (WFT) method was generalized by Bressan [15] and Risebro [16]. There was no restriction on the number of generalized equations for genuinely nonlinear systems. In Bressan’s algorithm, a non-physical wave was proposed in the approximate Riemannian solution, whereas Risebro had the idea that the waves merge into one of the two main waves involved in the interaction.

Solving conservation laws with the help of numerical tools is described in [4,17,18]. For example, Glimm’s difference scheme and the WFT method can be used to solve problems in fluid dynamics (see [19,20,21]).

In this paper, the numerical verification of two simple SDW interactions is performed using the WFT method. The algorithms presented by Asakura [22] and Bressan [3] are used. The original problem was modified by introducing the perturbation. Some properties of SDW are exploited to check the relevance of the obtained numerical solution. The specific object of this research is to create the algorithm for approximating the interaction of rarefaction waves to reduce the computational time. Numerical examples showed that, after application of new algorithm, the accuracy remained acceptable while the runtime was drastically reduced.

2. The Pressureless Gas Dynamics Model

The system of hyperbolic PDEs governing the motion of a compressible gas in the absence of viscosity is considered in this manuscript. These consist of conservation laws for mass and momentum. Together they are referred to as the compressible Euler equations or, simply, the Euler equations. The one-dimensional Euler small pressure gas dynamics system is

$$\begin{array}{ccc}\hfill {\partial }_t\rho +{\partial }_x(\rho u)& =& 0\hfill \\ \hfill {\displaystyle {\partial }_t(\rho u)+{\partial }_x(\rho u^2+\epsilon {\rho }^{\gamma }/\gamma )}& =& 0,\hfill \end{array}$$

(1)

where $\rho \ge 0$ is the density, u is the velocity, $\epsilon >0$ is a small parameter, and $\gamma =\gamma (\epsilon )>1$ is a parameter, also. Here, $p(\rho )={\rho }^{\gamma }/\gamma$ is the pressure and $m=\rho u$ moment.

As stated in [9], a solution to the Riemann problem for (1) converges to the generalized solution to the weakly hyperbolic system

$$\begin{array}{ccc}\hfill {\partial }_t\rho +{\partial }_x(\rho u)& =& 0\hfill \\ \hfill {\displaystyle {\partial }_t(\rho u)+{\partial }_x(\rho u^2)}& =& 0,(x,t)\in \mathbb{R}\times {\mathbb{R}}_{+},\hfill \end{array}$$

(2)

with the same initial data as $\epsilon \to 0$. In the literature, the system (2) can be found as the pressureless gas dynamics model (PGD model [23]). Having the initial data

$$\rho (x,0)=\left\{\begin{array}{cc}{\rho }_0,& x<0,\\ {\rho }_1,& x>0,\end{array}\right.u(x,0)=\left\{\begin{array}{cc}{u}_0,& x<0,\\ u_1,& x>0,\end{array}\right.$$

(3)

it is well known that there is no classical solution to the Riemann problem (2,3) if $u_0>u_1$. To be more precise, any two-shock Riemann solution to the Euler equations for isentropic fluids (1) tends to a $\delta$-shock solution (4), viewed as a distribution,

$$(\rho (x,t),u(x,t))=\left\{\begin{array}{cc}({\rho }_0,u_0),& x<ct\\ ({\rho }_1,u_1),& x>ct\end{array}\right.+\sigma {\delta }_{x=ct}$$

(4)

to the Euler equations for pressureless fluids (2) as the pressure vanishes. Here, $c=\frac{1}{\left[\rho \right]}(\left[\rho u\right]-\left[u\right]\sqrt{{\rho }_0{\rho }_1})\mathrm{and}\sigma =(-\left[u\right]\sqrt{{\rho }_0{\rho }_1},0)$ with $\left[x\right]=x_1-x_0$ (see [24]).

Nishida and Smoller ([20]) showed that a global weak solution exists for (1,3) and a bounded total variation on each line $t=\mathrm{const}.\ge 0$, provided that $\epsilon$ times the total variation of the initial data was sufficiently small. If one replaces ${\kappa }^2$ from [20,22] with $\epsilon /(1+2\epsilon )$, an exact calculation shows that the strength of the wave is now limited by $\sqrt{\epsilon }$ instead of $\epsilon$ [24]. This allows us to take the initial data with large total variation and set $\epsilon \to 0$ after solving (1).

The interaction problem of the delta shock for the pressureless system was solved in [14,24]. The authors used the SDW solution concept. The result of the interaction of two simple delta shocks is a delta shock with constant speed. Here, the limit of numerical WFT results when the small pressure term vanishes was compared with the SDW solution concept.

Now we present two ideas. The first one is the following: for $\epsilon >0$ small enough, two Riemann problems (1,3) are set such that each solution consists of shock waves of the first and second family (this approximates the interaction of two delta shock waves). Next, we apply the WFT method for each $\epsilon$ and let $\epsilon \to 0$. The solution results when there are no further interactions. The second idea is to use the SDW solution concept. Finally, a limit for a numerical WFT is obtained when the small pressure term vanishes and the SDW solution has been compared. The one-dimensional Euler gas dynamics system with low pressure (1) was used to approximate the unpressurized gas dynamics model (2) as $\epsilon \to 0$, i.e., $\epsilon {\rho }^{\gamma }/\gamma \to 0$ as $\epsilon \to 0$. The delta shock wave as a solution of (2) was approximated by two shock waves of the first and second family, which are the solutions (together with the rarefaction waves) of the Riemann problem for (1).

Elementary Waves

As in [20], from now on, we choose $\gamma =1+2\epsilon$. Throughout the whole paper [20], authors set $\epsilon$ to be a small positive constant. Here, $\epsilon$ is not a constant since we let $\epsilon \to 0$. To prevent a misunderstanding, note that $x^{\epsilon }$ denotes exponent. The eigenvalues of system (1) are ${\lambda }_1=u-\sqrt{\epsilon }{\rho }^{\epsilon }$ and ${\lambda }_2=u+\sqrt{\epsilon }{\rho }^{\epsilon }$, and the corresponding eigenvectors are $r_1=(-1,-u+\sqrt{\epsilon }{\rho }^{\epsilon })$ and $r_2=(1,u+\sqrt{\epsilon }{\rho }^{\epsilon })$. The rarefaction curves of (1) through the point $({\rho }_0,u_0)$ are

$$\begin{array}{c}u-u_0=-\frac{1}{\sqrt{\epsilon }}({\rho }^{\epsilon }-{\rho }_0^{\epsilon }),0\le \rho \le {\rho }_0:1\text{-}\mathrm{rarefaction} \mathrm{curve},\hfill \\ u-u_0=\frac{1}{\sqrt{\epsilon }}({\rho }^{\epsilon }-{\rho }_0^{\epsilon }),\rho \ge {\rho }_0:2\text{-}\mathrm{rarefaction} \mathrm{curve}.\hfill \end{array}$$

(5)

The shock curves of (1) through the point $({\rho }_0,u_0)$ are

$$\begin{array}{c}{\displaystyle u-u_0=-\frac{\sqrt{\epsilon }}{\sqrt{1+2\epsilon }}·\sqrt{\frac{{\rho }^{1+2\epsilon }-{\rho }_0^{1+2\epsilon }}{{\rho }_0\rho (\rho -{\rho }_0)}}(\rho -{\rho }_0),\rho >{\rho }_0:1\text{-}\mathrm{shock} \mathrm{curve},}\hfill \\ {\displaystyle u-u_0=\frac{\sqrt{\epsilon }}{\sqrt{1+2\epsilon }}·\sqrt{\frac{{\rho }^{1+2\epsilon }-{\rho }_0^{1+2\epsilon }}{{\rho }_0\rho (\rho -{\rho }_0)}}(\rho -{\rho }_0),0<\rho <{\rho }_0:2\text{-}\mathrm{shock} \mathrm{curve}.}\hfill \end{array}$$

Shock and rarefaction curves are presented in Figure 1.

The Riemann problem for (1) with initial data (3) was solved by Riemann [25]. A detailed explanation of the wave front tracking algorithm can be found in [24]. We will use the following convention: $C_{\ast }$, independent of $\epsilon$, denotes the maximum of the constants $C_{\ast }$ and $C_{\ast \ast }$ from Theorems 2.4 and 2.5 in [24], respectively. As in [24] (Lemma 2.7), $C^{\ast }$ is a constant, also independent of $\epsilon$.

Theorem 1

([24]). For arbitrary initial data with the bounded total variation, there exists ${\epsilon }_0$ such that for each $\epsilon <{\epsilon }_0$, the WFT scheme remains stable and provides a global in time solution for (1). For each $t>0$, the sum of strengths of shock waves does not exceed $\frac{1}{C_{\ast }\sqrt{\epsilon }}min\left\{1/2,C^{\ast }/4\right\}$.

A detailed explanation of the modified wave front tracking algorithm can be found in [3,24] and will be omitted here. The following parameters are used for the construction of the modified WFT algorithm:

• $\widehat{\lambda }$—a fixed speed of a non-physical wave,
• $\widehat{\delta }>0$—a constant that controls rarefaction waves strength;
• $n_g\in \mathbb{N}$—a number determining which type of Riemann solver is going to be used (accurate or approximative).

3. SDW Solutions to the Pressureless Gas Dynamics Model

The solution concept was taken from [14,24] (somewhat simplified). Its consistency with the numerical WFT results from Theorem 1 by letting $\epsilon \to 0$ is given in [24]. We take a conservation law system (2).

Definition 1

([14,24]). Vector valued function of the form

$$({\rho }_{\epsilon },u_{\epsilon })(x,t)=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x<ct-a_{\epsilon }t-x_{1,\epsilon }\hfill \\ ({\rho }_{1,\epsilon },u_{1,\epsilon }),\hfill & ct-a_{\epsilon }t-x_{1,\epsilon }<x<ct\hfill \\ ({\rho }_{2,\epsilon },u_{2,\epsilon }),\hfill & ct<x<ct+b_{\epsilon }t+x_{2,\epsilon }\hfill \\ ({\rho }_1,u_1),\hfill & x>ct+b_{\epsilon }t+x_{2,\epsilon }\hfill \end{array}\right.$$

(6)

is called a constant SDW (Figure 2). Further, ${\sigma }_{\epsilon }(t):=(a_{\epsilon }t+x_{1,\epsilon })({\rho }_{1,\epsilon },u_{1,\epsilon })+(b_{\epsilon }t+x_{2,\epsilon })({\rho }_{2,\epsilon },u_{2,\epsilon })$ is called the strength, and c is called the speed of the SDW. The SDW central line is given by $x=ct$, and $x=ct-a_{\epsilon }t-x_{1,\epsilon }$ and $x=ct+b_{\epsilon }t+x_{2,\epsilon }$ are called the external SDW lines. The values $x_{1,\epsilon }$ and $x_{2,\epsilon }$ are called the shifts, and $({\rho }_{1,\epsilon },u_{1,\epsilon })$ and $({\rho }_{2,\epsilon },u_{2,\epsilon })$ are called the intermediate states of a given SDW. Here, $x_{1,\epsilon },x_{2,\epsilon }\sim \epsilon$, and $a_{\epsilon }$ and $b_{\epsilon }$ are smooth functions equal zero at $t=0$ with growth order less or equal to ε. If $x_{1,\epsilon }=x_{2,\epsilon }=0$, then SDW is called simple.

Assumption 1

([14,24]). The component ${\rho }_{\epsilon }$ of an SDW satisfies $\parallel {\rho }_{\epsilon }{\parallel }_{{\mathbf{L}}^{\infty }}=\mathcal{O}({\epsilon }^{-1})$. It is called the major component ($\epsilon {\rho }_{\epsilon }\to C>0$ when $\epsilon \to 0$); $\parallel u_{\epsilon }{\parallel }_{{\mathbf{L}}^{\infty }}=o({\epsilon }^{-1})$, and this component is called the minor component ($\epsilon u_{\epsilon }\to 0$ when $\epsilon \to 0$).

Definition 2

([14]). Delta shock is SDW associated with a δ distribution, with all minor components having finite limits as $\epsilon \to 0$.

The solution to system (2) is given in the following theorem.

Theorem 2

([14,24]). Let $u_0>u_1$. Then, there exists an unique simple SDW (6) (for $x_{1,\epsilon }=x_{2,\epsilon }=0$) solution of the Riemann problem (2).

Now, we give the conditions when the interaction of two simple SDW gives a constant SDW. Let us assume that two simple SDW solutions for (2) interact:

$$({\tilde{\rho }}_{\epsilon },{\tilde{u}}_{\epsilon })(x,t)=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x-a<(\tilde{c}-{\tilde{a}}_{\epsilon })t,\hfill \\ ({\tilde{\rho }}_{1,\epsilon },{\tilde{u}}_{1,\epsilon }),\hfill & (\tilde{c}-{\tilde{a}}_{\epsilon })t<x-a<\tilde{c}t,\hfill \\ ({\tilde{\rho }}_{2,\epsilon },{\tilde{u}}_{2,\epsilon }),\hfill & \tilde{c}t<x-a<(\tilde{c}+{\tilde{b}}_{\epsilon })t,\hfill \\ ({\rho }_1,u_1),\hfill & x-a>(\tilde{c}+{\tilde{b}}_{\epsilon })t\hfill \end{array}\right.$$

and

$$({\widehat{\rho }}_{\epsilon },{\widehat{u}}_{\epsilon })(x,t)=\left\{\begin{array}{cc}({\rho }_1,u_1),\hfill & x-b<(\widehat{c}-{\widehat{a}}_{\epsilon })t,\hfill \\ ({\widehat{\rho }}_{1,\epsilon },{\widehat{u}}_{1,\epsilon }),\hfill & (\widehat{c}-{\widehat{a}}_{\epsilon })t<x-b<\widehat{c}t,\hfill \\ ({\widehat{\rho }}_{2,\epsilon },{\widehat{u}}_{2,\epsilon }),\hfill & \widehat{c}t<x-b<(\widehat{c}+{\widehat{b}}_{\epsilon })t,\hfill \\ ({\rho }_2,u_2),\hfill & x-b>(\widehat{c}+{\widehat{b}}_{\epsilon })t,\hfill \end{array}\right.$$

where $\tilde{c}>\widehat{c}$ and $a<b$. Then, as in Figure 3, $(X,T)$ is the interaction point of external SDW lines $x=a+(\tilde{c}+{\tilde{b}}_{\epsilon })t$ and $x=b+(\widehat{c}-{\widehat{a}}_{\epsilon })t$, i.e., $X=a+(\tilde{c}+{\tilde{b}}_{\epsilon })T=b+(\widehat{c}-{\widehat{a}}_{\epsilon })T$ and $T=(b-a)/(\tilde{c}-\widehat{c}+{\widehat{a}}_{\epsilon }+{\tilde{b}}_{\epsilon })$. For $t=T$, a distributional limit of solution is a sum of a classical piecewise constant function and a delta function supported by the interaction point. Therefore, it is natural to ask ourselves a question: When does the interaction produces a shadow wave solution for $t>T$?

Assumption 2

([14,24]). There exists a shadow wave solution (6) for $x_{1,\epsilon }=x_{1,\epsilon }=0$ to the system (2) with the initial data

$$(\rho ,u)(x,0)=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x<0,\hfill \\ ({\rho }_2,u_2),\hfill & x>0.\hfill \end{array}\right.$$

Assumption 3

([14,24]). Denote by $\tilde{\underline{\kappa }}$, $\widehat{\underline{\kappa }}$, $\underline{\kappa }\in {\mathbb{R}}^2$ the Rankine–Hugoniot deficits corresponding to $(\tilde{\rho },\tilde{u})$, $(\widehat{\rho },\widehat{u})$, and $(\rho ,u)$, respectively. It is possible to chose $\alpha \ge 0$ such that $\alpha \underline{\kappa }=T(\tilde{\underline{\kappa }}+\widehat{\underline{\kappa }})$.

These two assumptions were analyzed in [14,24]. Let $x_{1,\epsilon },x_{2,\epsilon }=\mathcal{O}(\epsilon )$ be non-negative real numbers for $\epsilon$ small enough. Using Lemma 3.2 from [24], it can be concluded that the function

$$({\rho }_{\epsilon },u_{\epsilon })(x,t)=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x<(c-a_{\epsilon })t-x_{1,\epsilon },\hfill \\ ({\rho }_{1,\epsilon },u_{1,\epsilon }),\hfill & (c-a_{\epsilon })t-x_{1,\epsilon }<x<ct,\hfill \\ ({\rho }_{2,\epsilon },u_{2,\epsilon }),\hfill & ct<x<(c+b_{\epsilon })t+x_{2,\epsilon },\hfill \\ ({\rho }_2,u_2),\hfill & x>(c+b_{\epsilon })t+x_{2,\epsilon },\hfill \end{array}\right.$$

is actually an SDW solution if $x_{1,\epsilon },x_{2,\epsilon }=\alpha (a_{\epsilon },b_{\epsilon })$ holds.

Define

$$({\rho }_{\epsilon },u_{\epsilon })(x,t)=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x-X<(c-a_{\epsilon })(t-T)-x_{1,\epsilon },t>T,\hfill \\ ({\rho }_{1,\epsilon },u_{1,\epsilon }),\hfill & (c-a_{\epsilon })(t-T)-x_{1,\epsilon }<x-X<c(t-T),t>T,\hfill \\ ({\rho }_{2,\epsilon },u_{2,\epsilon }),\hfill & c(t-T)<x-X<(c+b_{\epsilon })(t-T)+x_{2,\epsilon },t>T,\hfill \\ ({\rho }_2,u_2),\hfill & x-X>(c+b_{\epsilon })(t-T)+x_{2,\epsilon },t>T.\hfill \end{array}\right.$$

(7)

The anticipated solution $V_{\epsilon }$ is obtained by gluing the solution before the interaction (for $t<T$, Figure 3) with the one defined by (7) for $t>T$:

$$V_{\epsilon }(x,t)=\left\{\begin{array}{cc}(({\tilde{\rho }}_{\epsilon },{\tilde{u}}_{\epsilon })\wedge ({\widehat{\rho }}_{\epsilon },{\widehat{u}}_{\epsilon }))(x,t),\hfill & t<T,\hfill \\ ({\rho }_{\epsilon },u_{\epsilon })(x,t),\hfill & t>T.\hfill \end{array}\right.$$

(8)

Based on Assumption 3, there exists $\alpha \in \mathbb{R}$ such that

$$\alpha \left[\begin{array}{c}{\underline{\kappa }}_1\\ c{\underline{\kappa }}_1\end{array}\right]=T\left[\begin{array}{c}{\tilde{\underline{\kappa }}}_1+{\widehat{\underline{\kappa }}}_1\\ \tilde{c}{\tilde{\underline{\kappa }}}_1+\widehat{c}{\widehat{\underline{\kappa }}}_1\end{array}\right].$$

(9)

Now, following expression (9), the speed of the SDW after the interaction can be expressed as

$$c=\frac{\tilde{c}{\tilde{\underline{\kappa }}}_1+\widehat{c}{\widehat{\underline{\kappa }}}_1}{{\tilde{\underline{\kappa }}}_1+{\widehat{\underline{\kappa }}}_1}=\frac{\tilde{c}({\tilde{\xi }}_1+{\tilde{\xi }}_2)+\widehat{c}({\widehat{\xi }}_1+{\widehat{\xi }}_2)}{{\tilde{\xi }}_1+{\tilde{\xi }}_2+{\widehat{\xi }}_1+{\widehat{\xi }}_2}=:{\xi }_{\pi }.$$

(10)

The condition when the interaction of two simple SDW gives a constant SDW is ([24]) $\frac{{\rho }_2{u}_2-{\rho }_0{u}_0+(u_0-u_2)\sqrt{{\rho }_0{\rho }_2}}{{\rho }_2-{\rho }_0}={\xi }_{\pi },{\rho }_2\ne {\rho }_0.$

4. Algorithm of a New Concept of Rarefaction Wave Interactions Approximation

In the general case, the solution of Riemann problem (1,3) may contain rarefaction waves as well as shock waves. The expression $R_i$ denotes the rarefaction curve of the ith family, where $i=1,2$. Let the $R_2$ curve start from $({\rho }_0,u_0)$ and $R_1$ from $({\rho }_1,u_1)$. Based on (5), there exists $s>0$, such that

$${(\rho ,u)}^T:=R_1(s)({\rho }_1,u_1)=\left[\begin{array}{c}{\rho }_1-s\hfill \\ u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-s)}^{\epsilon }-{\rho }_1^{\epsilon })\hfill \end{array}\right]$$

(11)

and

$${(\rho ,u)}^T:=R_2(s)({\rho }_0,u_0)=\left[\begin{array}{c}{\rho }_0+s\hfill \\ u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+s)}^{\epsilon }-{\rho }_0^{\epsilon })\hfill \end{array}\right].$$

(12)

Suppose that $R_2$ connects state $w_2=({\rho }_0,u_0)$ from the left and state $({\rho }_1,u_1)$ from the right, i.e., there exists $s_2>0$ such that $({\rho }_1,u_1)=R_2(s_2)({\rho }_0,u_0)$. Similarly, let $R_1$ join state $w_1=({\rho }_1,u_1)$ from the left and state $({\rho }_2,u_2)$ from the right. As in [3], let ${\widehat{\delta }}_2>0$, ${\widehat{\delta }}_1>0$ be the small given constants. Consider the integers

$$p_2=1+\lceil s_2/{\widehat{\delta }}_2\rceil \mathrm{and}{p}_1=1+\lceil s_1/{\widehat{\delta }}_1\rceil ,$$

where $\lceil x\rceil$ stands for the largest integer $\le x$. For $j=1,2,\dots ,p_k$, $k=1,2$, we define the intermediate states and characteristics as follows:

$$w_{k,j}=R_k(j{s}_k/p_k)(w_k)\mathrm{and}{x}_{k,j}(t)=\overline{x}+(t-\overline{t}){\lambda }_k(w_{k,j}).$$

(13)

Applying (13) (for $k=2$) on (12), we have $w_{2,p_2}=({\rho }_1,u_1)$ and

$$w_{2,j}=\left[\begin{array}{c}{\rho }_{2,j}\hfill \\ u_{2,j}\hfill \end{array}\right]=\left[\begin{array}{c}{\rho }_0+j{s}_2/p_2\hfill \\ u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+j{s}_2/p_2)}^{\epsilon }-{\rho }_0^{\epsilon })\hfill \end{array}\right].$$

Applying (13) (for $k=1$) on (11), we have $w_{1,p_1}=({\rho }_2,u_2)$ and

$$w_{1,j}=\left[\begin{array}{c}{\rho }_{1,j}\hfill \\ u_{1,j}\hfill \end{array}\right]=\left[\begin{array}{c}{\rho }_1-j{s}_1/p_1\hfill \\ u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-j{s}_1/p_1)}^{\epsilon }-{\rho }_1^{\epsilon })\hfill \end{array}\right].$$

For $j=1,2,\dots ,p_k$, $k=1,2$,

$${\lambda }_2(w_{2,j})=u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+j{s}_2/p_2)}^{\epsilon }-{\rho }_0^{\epsilon })+\sqrt{\epsilon }{({\rho }_0+j{s}_2/p_2)}^{\epsilon }$$

(14)

and

$${\lambda }_1(w_{1,j})=u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-j{s}_1/p_1)}^{\epsilon }-{\rho }_1^{\epsilon })-\sqrt{\epsilon }{({\rho }_1-j{s}_1/p_1)}^{\epsilon }.$$

(15)

For ${\overline{\theta }}_j=j{s}_2/p_2$, (14) yields

$${\lambda }_2(w_{2,j})=u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+{\overline{\theta }}_j)}^{\epsilon }-{\rho }_0^{\epsilon })+\sqrt{\epsilon }{({\rho }_0+{\overline{\theta }}_j)}^{\epsilon }$$

and then

$${\overline{\theta }}_j={\left(\frac{\sqrt{\epsilon }({\lambda }_2(w_{2,j})-u_0+\frac{1}{\sqrt{\epsilon }}{\rho }_0^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon }-{\rho }_0$$

(16)

holds. Similarly, for ${\theta }_j=j{s}_1/p_1$, (15) yields

$${\lambda }_1(w_{1,j})=u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-{\theta }_j)}^{\epsilon }-{\rho }_1^{\epsilon })-\sqrt{\epsilon }{({\rho }_1-{\theta }_j)}^{\epsilon }$$

and then

$${\theta }_j={\rho }_1-{\left(\frac{-\sqrt{\epsilon }({\lambda }_1(w_{1,j})-u_1-\frac{1}{\sqrt{\epsilon }}{\rho }_1^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon }$$

(17)

holds. Expressions (16,17) will be useful later.

Now, the main idea is to use intermediate states and characteristics of $R_2$ as we defined in (13). For $R_1$, we define new intermediate states and characteristics as follows (see Figure 4):

• Find the intersection point of the two half-lines $x_{2,p_2}(t)$ and $x_{1,1}(t)$, solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\lambda }_2(w_{2,p_2}),\frac{x-\widehat{x}}{t-\widehat{t}}={\lambda }_1(w_{1,1}).$$
  We mark the solution as $(x_1,t_1)$.
• Solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\lambda }_2(w_{2,p_2-1}),\frac{x-\widehat{x}}{t-\widehat{t}}={\lambda }_1(w_{1,1}),$$

  we find the intersection between $x_{2,p_2-1}(t)$ and $x_{1,1}(t)$. Let us denote the solution of this system as $(x_1^{\ast },t_1^{\ast })$.
• Choose $\Delta t_1$ to make the RH deficit as small as possible for $t_2=t_1^{\ast }+\Delta t_1$. To be more precise, after $\Delta t_1$ is chosen, we check if

  $$\begin{array}{c}|c_{1,2}({\rho }_{1,1}-{\rho }_{2,p_2-1})-({\rho }_{1,1}{u}_{1,1}-{\rho }_{2,p_2-1}{u}_{2,p_2-1})|<{\delta }_3\\ |c_{1,2}({\rho }_{1,1}{u}_{1,1}-{\rho }_{2,p_2-1}{u}_{2,p_2-1})-({\rho }_{1,1}{u}_{1,1}^2+\epsilon {\rho }_{1,1}^{\gamma }/\gamma -{\rho }_{2,p_2-1}{u}_{2,p_2-1}^2-\epsilon {\rho }_{2,p_2-1}^{\gamma }/\gamma )|<{\delta }_3\end{array}$$

  holds for $0<{\delta }_3\ll 1$. If it does not hold, we choose $t_2=t_1^{\ast }+\Delta t_1/2$ and check again, and so on. This stops after $n_1$ steps, even if the inequalities are not satisfied, and we accept new time $t_2=t_1^{\ast }+\Delta t_1/(2n_1)$. Here,

  $$\begin{array}{c}{\tilde{w}}_{1,1}=({\rho }_{1,1},u_{1,1}),\\ x_2={\lambda }_2(w_{2,p_2-1})(t_2-\overline{t})+\overline{x},\\ c_{1,2}=(x_2-x_1)/(t_2-t_1),\\ {\tilde{\lambda }}_{1,2}=(x_2-\widehat{x})/(t_2-\widehat{t}),\\ {\displaystyle {\tilde{x}}_{1,2}(t)={\tilde{\lambda }}_{1,2}(t-\widehat{t})+\widehat{x},}\\ {\displaystyle {\tilde{s}}_{1,2}={\rho }_1-{\left(\frac{-\sqrt{\epsilon }({\tilde{\lambda }}_{1,2}-u_1-\frac{1}{\sqrt{\epsilon }}{\rho }_1^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon },}\\ {\tilde{\rho }}_{1,2}={\rho }_1-{\tilde{s}}_{1,2},\\ {\displaystyle {\tilde{u}}_{1,2}=u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-{\tilde{s}}_{1,2})}^{\epsilon }-{\rho }_1^{\epsilon }),}\\ {\tilde{w}}_{1,2}=({\tilde{\rho }}_{1,2},{\tilde{u}}_{1,2})\end{array}$$

  holds, based on (17).
• Now, for each $j=2,3,\dots ,p_2-1$ we do the following. Find the intersection between $x_{2,p_2-j}(t)$ and ${\tilde{x}}_{1,j}(t)$, solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\lambda }_2(w_{2,p_2-j}),\frac{x-\widehat{x}}{t-\widehat{t}}={\tilde{\lambda }}_{1,j}.$$
  We denote the solution as $(x_j^{\ast },t_j^{\ast })$. Then, choose $\Delta t_j$ to make the RH deficit as small as possible for $t_{j+1}=t_j^{\ast }+\Delta t_j$. After $\Delta t_j$ is chosen, we check if

  $$\begin{array}{c}|c_{j,j+1}({\tilde{\rho }}_{1,j}-{\rho }_{2,p_2-j})-({\tilde{\rho }}_{1,j}{\tilde{u}}_{1,j}-{\rho }_{2,p_2-j}{u}_{2,p_2-j})|<{\delta }_3\\ |c_{j,j+1}({\tilde{\rho }}_{1,j}{\tilde{u}}_{1,j}-{\rho }_{2,p_2-j}{u}_{2,p_2-j})-({\tilde{\rho }}_{1,j}{\tilde{u}}_{1,j}^2+\epsilon {\tilde{\rho }}_{1,j}^{\gamma }/\gamma -{\rho }_{2,p_2-j}{u}_{2,p_2-j}^2-\epsilon {\rho }_{2,p_2-j}^{\gamma }/\gamma )|<{\delta }_3\end{array}$$

  holds. If it does not hold, we choose $t_{j+1}=t_j^{\ast }+\Delta t_j/2$ or $t_{j+1}=t_j^{\ast }+\Delta t_j/4$,…until the upper inequalities are satisfied. If it is not a case, then after $n_j$ steps, we accept new time $t_{j+1}=t_j^{\ast }+\Delta t_j/(2n_j)$. Here,

  $$\begin{array}{c}{x}_{j+1}={\lambda }_2(w_{2,p_2-j})(t_{j+1}-\overline{t})+\overline{x},\\ c_{j,j+1}=(x_{j+1}-x_j)/(t_{j+1}-t_j),\\ {\tilde{\lambda }}_{1,j+1}=(x_{j+1}-\widehat{x})/(t_{j+1}-\widehat{t}),\\ {\displaystyle {\tilde{x}}_{1,j+1}(t)={\tilde{\lambda }}_{1,j+1}(t-\widehat{t})+\widehat{x}}\\ {\displaystyle {\tilde{s}}_{1,j+1}={\rho }_1-{\left(\frac{-\sqrt{\epsilon }({\tilde{\lambda }}_{1,j+1}-u_1-\frac{1}{\sqrt{\epsilon }}{\rho }_1^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon }}\\ {\tilde{\rho }}_{1,j+1}={\rho }_1-{\tilde{s}}_{1,j+1}\\ {\displaystyle {\tilde{u}}_{1,j+1}=u_1-\frac{1}{\sqrt{\epsilon }}({({\rho }_1-{\tilde{s}}_{1,j+1})}^{\epsilon }-{\rho }_1^{\epsilon })}\\ {\tilde{w}}_{1,j+1}=({\tilde{\rho }}_{1,j+1},{\tilde{u}}_{1,j+1})\end{array}$$
• The half-lines that we keep during WFT approximation are $x_{2,1}(t)$, ${\tilde{x}}_{1,p_2}(t)$, $x_{1,p_1}(t)$.

Let us repeat the above procedure, but now we use intermediate states and characteristics of $R_1$ as we defined in (13). For $R_2$, we define new intermediate states and characteristics as follows (see Figure 5):

• Find the intersection point of the two half-lines $x_{2,p_2}(t)$ and $x_{1,1}(t)$, solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\lambda }_2(w_{2,p_2}),\frac{x-\widehat{x}}{t-\widehat{t}}={\lambda }_1(w_{1,1}).$$
  We mark the solution as $(x_1,t_1)$.
• Solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\lambda }_2(w_{2,p_2}),\frac{x-\widehat{x}}{t-\widehat{t}}={\lambda }_1(w_{1,2}),$$

  we find the intersection between $x_{2,p_2}(t)$ and $x_{1,2}(t)$. Let us denote the solution of this system as $(x_1^{\ast },t_1^{\ast })$.
• Choose $\Delta t_1$ to make the RH deficit as small as possible for $t_2=t_1^{\ast }+\Delta t_1$. To be more precise, after $\Delta t_1$ is chosen, we check if

  $$\begin{array}{c}|c_{1,2}({\rho }_{1,1}-{\tilde{\rho }}_{2,p_2-1})-({\rho }_{1,1}{u}_{1,1}-{\tilde{\rho }}_{2,p_2-1}{\tilde{u}}_{2,p_2-1})|<{\delta }_3\\ |c_{1,2}({\rho }_{1,1}{u}_{1,1}-{\tilde{\rho }}_{2,p_2-1}{\tilde{u}}_{2,p_2-1})-({\rho }_{1,1}{u}_{1,1}^2+\epsilon {\rho }_{1,1}^{\gamma }/\gamma -{\tilde{\rho }}_{2,p_2-1}{\tilde{u}}_{2,p_2-1}^2-\epsilon {\tilde{\rho }}_{2,p_2-1}^{\gamma }/\gamma )|<{\delta }_3\end{array}$$

  holds for $0<{\delta }_3\ll 1$. If it does not hold, we choose $t_2=t_1^{\ast }+\Delta t_1/2$ and check again, and so on. This stops after $n_1$ steps, even if the inequalities are not satisfied, and we accept new time $t_2=t_1^{\ast }+\Delta t_1/(2n_1)$. Here,

  $$\begin{array}{c}{x}_2={\lambda }_1(w_{1,2})(t_2-\widehat{t})+\widehat{x},\\ c_{1,2}=(x_2-x_1)/(t_2-t_1),\\ {\tilde{\lambda }}_{2,p_2-1}=(x_2-\overline{x})/(t_2-\overline{t}),\\ {\displaystyle {\tilde{x}}_{2,p_2-1}(t)={\tilde{\lambda }}_{2,p_2-1}(t-\overline{t})+\overline{x},}\\ {\displaystyle {\tilde{s}}_{2,p_2-1}={\left(\frac{\sqrt{\epsilon }({\tilde{\lambda }}_{2,p_2-1}-u_0+\frac{1}{\sqrt{\epsilon }}{\rho }_0^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon }-{\rho }_0,}\\ {\tilde{\rho }}_{2,p_2-1}={\rho }_0+{\tilde{s}}_{2,p_2-1},\\ {\displaystyle {\tilde{u}}_{2,p_2-1}=u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+{\tilde{s}}_{2,p_2-1})}^{\epsilon }-{\rho }_0^{\epsilon }),}\\ {\tilde{w}}_{2,p_2-1}=({\tilde{\rho }}_{2,p_2-1},{\tilde{u}}_{2,p_2-1})\end{array}$$

  holds, based on (16).
• Now, for each $j=2,3,\dots ,p_1-1$ we do the following. Find the intersection between ${\tilde{x}}_{2,p_2-j+1}(t)$ and $x_{1,j+1}(t)$, solving the system

  $$\frac{x-\overline{x}}{t-\overline{t}}={\tilde{\lambda }}_{2,p_2-j+1},\frac{x-\widehat{x}}{t-\widehat{t}}={\lambda }_1(w_{1,j+1}).$$
  We denote the solution as $(x_j^{\ast },t_j^{\ast })$. Then, choose $\Delta t_j$ to make the RH deficit as small as possible for $t_{j+1}=t_j^{\ast }+\Delta t_j$. To be more precise, after $\Delta t_j$ is chosen, we check if

  $$\begin{array}{c}|c_{j,j+1}({\rho }_{1,j}-{\tilde{\rho }}_{2,p_2-j})-({\rho }_{1,j}{u}_{1,j}-{\tilde{\rho }}_{2,p_2-j}{\tilde{u}}_{2,p_2-j})|<{\delta }_3\\ |c_{j,j+1}({\rho }_{1,j}{u}_{1,j}-{\tilde{\rho }}_{2,p_2-j}{\tilde{u}}_{2,p_2-j})-({\rho }_{1,j}{u}_{1,j}^2+\epsilon {\rho }_{1,j}^{\gamma }/\gamma -{\tilde{\rho }}_{2,p_2-j}{\tilde{u}}_{2,p_2-j}^2-\epsilon {\tilde{\rho }}_{2,p_2-j}^{\gamma }/\gamma )|<{\delta }_3\end{array}$$

  holds. If it does not hold, we choose $t_{j+1}=t_j^{\ast }+\Delta t_j/2$ or $t_{j+1}=t_j^{\ast }+\Delta t_j/4$,…until the upper inequalities are satisfied. If it is not a case, then after $n_j$ steps, we accept new time $t_{j+1}=t_j^{\ast }+\Delta t_j/(2n_j)$. Here,

  $$\begin{array}{c}{x}_{j+1}={\lambda }_1(w_{1,j+1})(t_{j+1}-\widehat{t})+\widehat{x},\\ c_{j,j+1}=(x_{j+1}-x_j)/(t_{j+1}-t_j),\\ {\tilde{\lambda }}_{2,p_2-j}=(x_{j+1}-\overline{x})/(t_{j+1}-\overline{t}),\\ {\displaystyle {\tilde{x}}_{2,p_2-j}(t)={\tilde{\lambda }}_{2,p_2-j}(t-\overline{t})+\overline{x}}\\ {\displaystyle {\tilde{s}}_{2,p_2-j}={\left(\frac{\sqrt{\epsilon }({\tilde{\lambda }}_{2,p_2-j}-u_0+\frac{1}{\sqrt{\epsilon }}{\rho }_0^{\epsilon })}{\epsilon +1}\right)}^{1/\epsilon }-{\rho }_0}\\ {\tilde{\rho }}_{2,p_2-j}={\rho }_0+{\tilde{s}}_{2,p_2-j}\\ {\displaystyle {\tilde{u}}_{2,p_2-j}=u_0+\frac{1}{\sqrt{\epsilon }}({({\rho }_0+{\tilde{s}}_{2,p_2-j})}^{\epsilon }-{\rho }_0^{\epsilon })}\\ {\tilde{w}}_{2,p_2-j}=({\tilde{\rho }}_{2,p_2-j},{\tilde{u}}_{2,p_2-j})\end{array}$$
• In addition to half-lines $x_{2,1}(t)$, ${\tilde{x}}_{1,p_2}(t)$, and $x_{1,p_1}(t)$, in WFT we introduce one more half-line ${\tilde{x}}_{2,p_2-p_1+1}(t)$ (Figure 6) and continue with the wave front tracking method as in [3,22].

5. Numerical Results

All numerical results were obtained using Mathematica 12.3 software ([26]). The consistency of the theoretical and numerical approaches will are shown. The algorithm of the rarefaction wave interactions approximation are applied in Examples 4 and 5. Let us consider (1) with the initial data

$$(\rho ,u){|}_{t=0}=\left\{\begin{array}{cc}({\rho }_0,u_0),\hfill & x<a_1,\hfill \\ ({\rho }_1,u_1),\hfill & a_1<x<a_2,\hfill \\ ({\rho }_2,u_2),\hfill & x>a_2,\hfill \end{array}\right.$$

(18)

where $u_0>u_1>u_2$, $a_1<a_2$. For $\epsilon$ small enough (see [9]), there exist $({\rho }_{1,\epsilon },u_{1,\epsilon })\in {\mathbb{R}}_{+}\times \mathbb{R}$ and $({\rho }_{2,\epsilon },u_{2,\epsilon })\in {\mathbb{R}}_{+}\times \mathbb{R}$, such that:

• $({\rho }_0,u_0)$ and $({\rho }_{1,\epsilon },u_{1,\epsilon })$ are connected by an 1-shock;
• $({\rho }_{1,\epsilon },u_{1,\epsilon })$ and $({\rho }_1,u_1)$ are connected by a 2-shock;
• $({\rho }_1,u_1)$ and $({\rho }_{2,\epsilon },u_{2,\epsilon })$ are connected by an 1-shock;
• $({\rho }_{2,\epsilon },u_{2,\epsilon })$ and $({\rho }_2,u_2)$ are connected by a 2-shock.

In order to verify two simple delta shock interactions, the WFT algorithm was used to obtain the numerical solutions. The assumption is that $({\rho }_0,u_0)$ can be connected to $({\rho }_2,u_2)$ by a simple SDW; ${\rho }_2$ must be chosen such that condition (10) is fulfilled. The result of the interaction is a single SDW with a constant speed.

In the following two examples, (1,18) is considered with $a_1=0$, $a_2=2$, $({\rho }_0,u_0)=(1,1)$, $({\rho }_1,u_1)=(1.1,0.7)$, and $u_2=0.6$. Two SDWs will interact at the point $(X,T)=(8.62431,10.1891)$. A single constant SDW as a solution to the interaction problem will exist for ${\rho }_2=1.087912$. After the interaction, the correct speed of the resulting wave equals $c_{\delta }=0.79579$, obtained from (10). Here, ${\rho }_0<{\rho }_1$.

• Example 1

Table 1. The intermediate states $({\rho }_{\epsilon },u_{\epsilon })$ after solving the problem (1,18) for different $\epsilon$, $\widehat{\delta }=\sqrt{\epsilon }$, and $n_g=6$. The initial data are $({\rho }_0,u_0)=(1,1)$, $({\rho }_1,u_1)=(1.1,0.7)$, and $({\rho }_2,u_2)=(1.087912,0.6)$, with $a_1=0$, $a_2=2$; $c_1$ is the speed of the first left shock $S_1$, and $c_2$ is the speed of the last right shock $S_2$; $|E{q}_1|$ and $|E{q}_2|$ are the left-hand sides of the integral form of the first and second equation in (1), respectively; $c_{\delta }=0.79579$. There are no interactions after $t_{end}$.

$\mathit{\epsilon }$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{u}}_{\mathit{\epsilon }}$	${\mathit{c}}_1$	${\mathit{c}}_2$	$|{\mathit{Eq}}_1|$	$|{\mathit{Eq}}_2|$	${\mathit{t}}_{\mathit{end}}$
0.1	1.90065	0.78587	0.54812	1.03467	0.00014	0.00029	51539.2
0.05	2.42960	0.78960	0.64243	0.94334	0.00046	0.00091	44283.9
0.01	5.97852	0.79403	0.75266	0.83719	0.00197	0.01341	1280.45
0.009	6.45010	0.79419	0.75643	0.83359	0.00235	0.01775	1047.91
0.007	7.78950	0.79453	0.76426	0.82610	0.00152	0.02834	663.973
0.005	10.1816	0.79492	0.77259	0.81822	0.00037	0.04625	379.533
0.003	15.6887	0.79570	0.78179	0.81020	0.05231	0.05667	182.345

shows the intermediate states $({\rho }_{\epsilon },u_{\epsilon })$ for different $\epsilon$ as $\epsilon \to 0$. The numerical simulations are shown in Figure 7 and Figure 8. It can be seen that the non-constant region becomes narrower as $\epsilon \to 0$. The strength of each rarefaction front is bounded by $\widehat{\delta }=\sqrt{\epsilon }$. With g we mark the generation order of a wave, or how many interactions are needed to produce a wave. For $g\ge 6=:n_g$, the simplified Riemann solver is going to be used in this example. The accurate Riemann solver will be applied for $g<6$.

From Table 1 it can be seen that the accuracy remains acceptable when $\epsilon \to 0$ and $u_{\epsilon }\to c_{\delta }$. Let us now explain Figure 7. For each $\epsilon$, two piecewise linear half-lines are created. The left one starts from the point $(x,t)=(a_1,0)$, the right one from the point $(x,t)=(a_2,0)$. The ith linear segment of this half-line can be written in the form $x=c_{i,j}(t-t_i)+x_i$, $i\ge 1$, $j=1,2$, $x_i\le x\le x_{i+1}$, $t_i\le t\le t_{i+1}$, where $c_{i,1}$ stands for the velocity of the first ($S_1$) wave on the left-hand side in the phase plane, and $c_{i,2}$ stands for the velocity of the last ($S_2$) wave on the left-hand side in the phase plane at every ith segment. The interaction of the waves takes place at the points $(x_i,t_i),i\ge 1$. After the interaction of two delta shocks, the resulting delta shock centerline in Figure 7 (dashed line) starts at $(X,T)$, and it is explicitly calculated from (2).

The interaction points are shown in Figure 8 for $\epsilon =0.005$. Each interaction point represents a spatial and time variable where an interaction between two waves occurred. As can be seen from Table 1, the last interaction took place at $t=379.533$.

Solutions to the Riemann problem (1,18) for different $\epsilon$ at $t=\mathrm{51,540}$ are presented in Figure 9 and Figure 10. For $rh{o}_0=1$ (left line, Figure 9) and ${\rho }_2=1.087912$ (right line, Figure 9), as $\epsilon \to 0$, ${\rho }_{\epsilon }$ (centered line, Figure 9) becomes a large number such that $\epsilon {\rho }_{\epsilon }\to const$.

For $u_0=1$ (left line, Figure 10) and $u_2=0.6$ (right line, Figure 10), as $\epsilon \to 0$, $u_{\epsilon }$ (centered line, Figure 10) goes to $c_{\delta }=0.79579$.

• Example 2

Now, we fix $\epsilon =0.009$, the strength of rarefaction waves are bounded by $\sqrt{\epsilon }$, and $n_g$ is changing.

Table 2. The intermediate states $({\rho }_{\epsilon },u_{\epsilon })$ after solving the problem (1,18) for $\epsilon =0.009$ and $\widehat{\delta }=\sqrt{\epsilon }$. The initial data are $({\rho }_0,u_0)=(1,1)$, $({\rho }_1,u_1)=(1.1,0.7)$, and $({\rho }_2,u_2)=(1.087912,0.6)$, with $a_1=0$, $a_2=2$; $c_1$ is the speed of the first left shock $S_1$, $c_2$ is the speed of the last right shock $S_2$; $|E{q}_1|$ and $|E{q}_2|$ are the left-hand sides of the integral form of the first and second equation in (1), respectively; $c_{\delta }=0.79579$. There are no interactions after $t_{end}$.

${\mathit{n}}_{\mathit{g}}$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{u}}_{\mathit{\epsilon }}$	${\mathit{c}}_1$	${\mathit{c}}_2$	$|{\mathit{Eq}}_1|$	$|{\mathit{Eq}}_2|$	${\mathit{t}}_{\mathit{end}}$
6	6.45010	0.79419	0.75643	0.83359	0.00235	0.01775	1047.91
7	6.45068	0.79418	0.75642	0.83357	0.00109	0.00696	2381.43
8	6.45058	0.79418	0.75642	0.83357	0.00029	0.00251	5703.15
9	6.45060	0.79418	0.75641	0.83356	0.00005	0.00203	12960.5
10	6.45059	0.79417	0.75641	0.83356	0.00001	0.00029	31037.1

shows, as $n_g$ increases, accuracy is better and $t_{end}$ is greater. In addition, as we increase the number $n_g$, we expect to obtain a more precise solution.

• Example 3

Here ${\rho }_0>{\rho }_1$, $\widehat{\delta }=\sqrt{\epsilon }$, $g\ge 6=:n_g$. As one can see from

Table 3. The intermediate states $({\rho }_{\epsilon },u_{\epsilon })$ after solving the problem (1,18) for different $\epsilon$, $\widehat{\delta }=\sqrt{\epsilon }$, and $n_g=6$. The initial data are $({\rho }_0,u_0)=(1,1)$, $({\rho }_1,u_1)=(0.9,0.7)$, and $({\rho }_2,u_2)=(0.91011,0.6)$, with $a_1=0$, $a_2=2$; $c_1$ is the speed of the first left shock $S_1$, $c_2$ is the speed of the last right shock $S_2$; $|E{q}_1|$ and $|E{q}_2|$ are the left-hand sides of the integral form of the first and second equation in (1), respectively; $c_{\delta }=0.80471$. There are no interactions after $t_{end}$.

$\mathit{\epsilon }$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{u}}_{\mathit{\epsilon }}$	${\mathit{c}}_1$	${\mathit{c}}_2$	$|{\mathit{Eq}}_1|$	$|{\mathit{Eq}}_2|$	${\mathit{t}}_{\mathit{end}}$
0.1	1.74716	0.81567	0.56896	1.05016	0.00016	0.00004	37815.5
0.05	2.22987	0.81158	0.65838	0.95749	0.00042	0.00007	39901.9
0.01	5.47442	0.80668	0.76348	0.84789	0.00792	0.00948	1179.15
0.009	5.90553	0.80652	0.76708	0.84414	0.01313	0.01241	965.899
0.007	7.15678	0.80598	0.76987	0.83954	0.02456	0.02547	635.784
0.005	9.47854	0.80512	0.77259	0.82874	0.00037	0.04625	379.533
0.003	13.7598	0.80475	0.78815	0.81687	0.04255	0.04113	153.487

, as $\epsilon \to 0$, accuracy remains acceptable and $u_{\epsilon }\to c_{\delta }$.

In the following two examples, the algorithm of rarefaction wave interactions approximation will be applied.

• Example 4

Let $a_1=0$, $a_2=2$, $({\rho }_0,u_0)=(1,1)$, $({\rho }_1,u_1)=(1.2,0.8)$, and $u_2=0.7$. Now, for ${\rho }_2=1.14286$, a single simple SDW exists as a solution to the interaction problem. With such data, two SDWs will interact in a point $(X,T)=(12.365,13.809)$. After interaction, the speed of the resulting wave is $c_{\delta }=0.844994$. The solution of the observed Riemann problem is presented in

Table 4. $({\rho }_0,u_0)=(1,1),({\rho }_1,u_1)=(1.2,0.8),({\rho }_2,u_2)=(1.14286,0.7)$, $n_g=6$, $\widehat{\delta }=\sqrt{\epsilon }$.

$\mathit{\epsilon }$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{u}}_{\mathit{\epsilon }}$	${\mathit{c}}_1$	${\mathit{c}}_2$	$|{\mathit{Eq}}_1|$	$|{\mathit{Eq}}_2|$	Runtime	${(\mathit{x},\mathit{t})}_{\mathit{end}}$
0.1	1.68612	0.828059	0.577459	1.097462	0.00014	0.00020	0.9 s	(54,245, 93,937)
0.05	2.03792	0.834141	0.674341	1.005426	0.00037	0.00054	1.1 s	(62,255, 61,919)
0.01	4.22713	0.841636	0.792563	0.894116	0.02266	0.00402	4.8 s	(5127.9, 5734.4)
0.005	6.71266	0.843121	0.815659	0.872483	0.00501	0.02155	4477 s	(1229.6, 1408.6)
0.003	9.95854	0.844183	0.826443	0.863428	0.01212	0.00688	≈168 h	(812.54, 1008.5)
0.005 *	6.71283	0.843118	0.815657	0.872480	0.00660	0.03910	15.3 s	(2598.4, 2977.4)
0.003 *	9.94664	0.843848	0.826450	0.862552	0.09700	0.05000	15.5 s	(894.2, 1036.1)

* presented algorithm for approximations of $RW$ interactions was applied.

for different choices of $\epsilon$. The strength of each divided rarefaction wave is bounded by $\widehat{\delta }=\sqrt{\epsilon }$, and we set $n_g=6$. After point ${(x,t)}_{end}$, there are no more interactions of the waves.

As one can see from Table 4, the applied algorithm drastically reduced runtime, and the accuracy is acceptable.

• Example 5

Initial data are the same as in Example 4, but now we fix $\epsilon =0.01$ and look for a solution to Riemann problem (1,18) for different choices of rarefaction wave strengths $\widehat{\delta }$ (see

Table 5. $({\rho }_0,u_0)=(1,1),({\rho }_1,u_1)=(1.2,0.8),({\rho }_2,u_2)=(1.14286,0.7)$, $n_g=6$, $\epsilon =0.01$.

$\widehat{\mathit{\delta }}$	${\mathit{\rho }}_{\mathit{\epsilon }}$	${\mathit{u}}_{\mathit{\epsilon }}$	${\mathit{c}}_1$	${\mathit{c}}_2$	$|{\mathit{Eq}}_1|$	$|{\mathit{Eq}}_2|$	Runtime	${(\mathit{x},\mathit{t})}_{\mathit{end}}$
0.1	4.22713	0.841636	0.792563	0.894116	0.02266	0.00402	13 s	(5127.9, 5734.4)
0.05	4.22717	0.841634	0.792562	0.894115	0.00061	0.00786	25 s	(5122.8, 5728.6)
0.04	4.22717	0.841634	0.792562	0.894115	0.00059	0.00690	39 s	(5123.0, 5728.9)
0.03	4.22716	0.841634	0.792562	0.894115	0.00160	0.00691	289 s	(5121.2, 5726.9)
0.02	4.22716	0.841635	0.792562	0.894115	0.00266	0.00641	1890 s	(5120.4, 5726.0)
0.01	4.22716	0.841635	0.792562	0.894115	0.00352	0.00638	≈96 h	(5119.2, 5724.5)
0.02 *	4.22717	0.841634	0.792562	0.894115	0.00819	0.00969	5.5 s	(14,096, 15,765)
0.01 *	4.22717	0.841634	0.792562	0.894115	0.00690	0.01020	4.8 s	(14,104, 15,773)
0.005 *	4.22717	0.841634	0.792562	0.894115	0.00600	0.009801	4.83 s	(14,116, 15,787)
0.001 *	4.22717	0.841634	0.792562	0.894115	0.004487	0.014761	10.9 s	(14,260, 15,948)
0.0001 *	4.22717	0.841634	0.792562	0.894115	0.01231	0.02341	62.1 s	(14,347, 16,845)

* presented algorithm for approximations of $RW$ interactions was applied.

).

It should be noticed that, for $\widehat{\delta }=0.02$, runtime is 1890 seconds, whereas for $\widehat{\delta }=0.01$, it is nearly 96 h. After the given algorithm was applied, runtime decreased, accuracy remained acceptable even when number of constant intermediate states arising from rarefaction waves increased (i.e., $\widehat{\delta }$ decreased).

6. Conclusions

In the first part of the manuscript, a solution concept of shadow waves for the Euler equations for pressureless fluids is introduced together with the numerical wave front tracking algorithm. As the perturbed parameter tends to zero in the Euler equations for isentropic fluids, two simple delta shock interactions produce a delta shock with constant speed. From the given numerical examples, it can be seen that, as $\epsilon \to 0$, the density of the intermediate state in obtained solutions increases as $1/\epsilon$, and the velocity is closer to the calculated value. In the second part of this investigation, the rarefaction wave interactions were approximated and simplified by the proposed new algorithm. The further numerical examples showed that the accuracy remained acceptable while the runtime was drastically decreased. For $\epsilon =0.003$, the proposed algorithm provides the solution almost 40,000 times faster. Acceptable accuracy and decreased runtime are the key features of the proposed algorithm.