## Abstract

We present a theoretical aspect of conservation laws by using simplest scalar models with essential properties. We start by rewriting the general scalar conservation law as a quasilinear partial differential equation and solve it by method of characteristics. Here we come across with the notion of strong and weak solutions depending on the initial value of the problem. Taking into account a special initial data for the left and right side of a discontinuity point, we get the related Riemann problem. An illustration of this problem is provided by some examples. In the remaining part of the chapter, we extend this analysis to the gas dynamics given in the Euler system of equations in one dimension. The transformations of this system into the Lagrangian coordinates follow by applying a suitable change of coordinates which is one of the main issues of this section. We next introduce a first-order Godunov finite volume scheme for scalar conservation laws which leads us to write Godunov schemes in both Eulerian and Lagrangian coordinates in one dimension where, in particular, the Lagrangian scheme is reformulated as a finite volume method. Finally, we end up the chapter by providing a comparison of Eulerian and Lagrangian approaches.

### Keywords

- conservation laws
- Burgers’ equation
- shock and rarefaction waves
- weak and strong solutions
- Riemann problem
- Euler system
- Godunov schemes
- Eulerian coordinates
- Lagrangian coordinates

## 1. Introduction

We present a general form of scalar conservation laws with further properties including some basic models and provide examples of computational methods for them. The equations described by

in one dimension are known as scalar conservation laws where

### 1.1 Conservation laws: integral form and differential form

We start by investigating the relation of the equations in gas dynamics with conservation laws. We take into account the equation of conservation of mass in one dimension. The density and the velocity are assumed to be constant in the tube where

The last equation is called integral form of conservation law. Integrating this expression in time from

Using the fundamental theorem of calculus after reduction of Eq. (3), it follows that

As a result, we get

Here the end points of the integrations are arbitrary; that is, for any

which is said to be the differential form of the conservation law.

### 1.2 A first-order quasilinear partial differential equations

A general solution to a quasilinear partial differential equation of the form

where

By applying a parametrization of

In addition to these equations, if an initial condition

Observe that the scalar conservation law (1) is a particular example of Eq. (7) if we assign

This means, the quantity

### 1.3 Strong (classical) solutions

We consider the initial value problem

where the initial data is assumed to be continuously differentiable, that is,

where we define characteristic curves of Eq. (12) to be the solution of *u* be a strong solution and the initial data *u*_{0} be differentiable. Observe that (12) is equivalent to a quasilinear form:

with

Along this characteristic curve,

is satisfied, that is,

Hence we can define smooth solutions by

### 1.4 Linear advection equation

The basic example of the scalar conservation law is the linear advection equation. It can be obtained by setting

is a linear advection equation. Similar to Eqs. (11) and (12), an initial value problem for linear advection equation is described by

Applying the method of characteristics, it follows that

where

Here

### 1.5 Burgers’ equation

Burgers’ equation is the simplest nonlinear partial differential equation and is the one of the most common models used in the scalar conservation laws and fluid dynamics. The classical Burgers’ equation is described by

where

where

Plugging these terms in Eq. (21), we get

Taking integration with respect to

Rewriting Eq. (25) by

it follows that

As a result the explicit form of traveling wave solution of Eq. (21) becomes

where

*Remark.* If the initial data is smooth and very small, then the

### 1.6 Inviscid Burgers’ equation

Whenever

Observe that

Recall that the characteristic speed

and differentiating equation (30) with respect to

Thus, substituting Eqs. (31) and (32) in (29), we can recover the inviscid Burgers’ equation. Consequently, the relations (31) and (32) imply that the solutions of Eq. (1) and particularly of Eq. (29) depend on the initial value

### 1.7 Shock waves

Let the constants *u**L* and *uR* are given with a linear function,

is a simple example of discontinuous solution of the conservation law (11). If

where

### 1.8 Rarefaction waves

A rarefaction wave is a strong solution which is a union of characteristic lines. A rarefaction fan is a collection of rarefaction waves. These waves are constant on the characteristic line

If, for instance, f is convex, then the rarefaction waves are increasing. If we consider again the inviscid Burgers’ equation with the initial values, then the region without characteristics in Figure 2 will be covered by rarefaction solution which is described by

An illustration of rarefaction waves and rarefaction fan in Eq. (36) is given in Figure 3.

*Remark*. Whenever characteristics intersect, we may have multiple valued solution or no solution; but we have no more classical (strong) solution. To get rid of this situation, we introduce a more wide-ranging notion of solution, the weak solution, in the next part. By this arrangement, we may have non-differentiable and even discontinuous solutions.

### 1.9 Weak solution

Weak solutions occur whenever there is no smooth (classical) solution. These solutions may not be differentiable or even not continuous. Considering

Putting the initial condition

Observe that there are no more derivatives of

### 1.10 Riemann problem

The Riemann problem is a Cauchy problem with a particular initial value which consists a conservation law together with piecewise constant data having a single discontinuity. We consider the Riemann problem for a convex flux described by

The solution is a set of shock and rarefaction waves depending on the relation between

Case 1:

is a shock wave satisfying the shock speed

Case 2: (

### 1.11 Rankine-Hugoniot jump condition

A jump discontinuity along the characteristic line is controlled by the Rankine-Hugoniot jump condition. Integrating the scalar conservation law (1) in

Suppose that there is a discontinuity at the point

By the fundamental theorem of calculus, the relations (41) and (42) yield

Taking the limit whenever

The relation (44) is said to be the Rankine-Hugoniot jump condition. Geometrical meaning of the Rankine-Hugoniot jump condition is that the shock speed is the slope of the secant line through the points

### 1.12 Entropy functions

Entropy and entropy flux are defined for attaining physically meaningful solutions. If

is satisfied for continuously differentiable functions

which looks like to the scalar conservation law (1). Indeed, if we multiply Eq. (1) by

It follows that Eqs. (46) and (47) are equivalent with

holds for all convex entropy pairs

### 1.13 Entropy condition

Weak solutions to conservation laws may contain discontinuities as a result of a discontinuity in the initial data or of characteristics that cross each other or because of the jump conditions which are satisfied across the discontinuities. Although the Rankine-Hugoniot jump condition is satisfied, the uniqueness of the solution may always not be guaranteed. In order to eliminate the nonphysical solutions among the weak solutions, we need an additional condition, so-called entropy condition. It is described by the following: A discontinuity propagating with the characteristic speed

**Example 1.1.** The weak solutions to conservation laws need not be unique. If we write the inviscid Burgers’ equation in quasilinear form and multiply by

The inviscid Burgers’ equation and Eq. (49) have exactly the same smooth solutions. But their weak solutions are different. A shock traveling speed for the inviscid Burgers’ equation is

**Example 1.2.** We first consider the initial value problem for

Applying the method of characteristics for

Next if we integrate Eq. (51) with respect to

where

which satisfies both the jump condition and the entropy condition as

**Example 1.3.** We now interchange the roles of

By the method of characteristics, we obtain a solution

which is a classical (strong) solution on both sides of the characteristic line

which satisfies both jump and entropy conditions. Here we can observe the rarefaction fan arising on the interval

## 2. The gas dynamic equations in one dimension

The equation of fluid dynamics can be represented in Eulerian and Lagrangian forms. Eulerian coordinates are related to the coordinates of a fixed observer. On the other hand, Lagrangian coordinates are in usual related to the local flow velocity. That is, due to the velocity taking different values in different parts of the fluid, the change of coordinates is different from one point to another one.

### 2.1 Eulerian coordinates

The equations of gas dynamics in Eulerian coordinates can be written in the following conservative forms:

where we ignored the heat conduction. If we denote

then Eq. (57) can be written by

where

### 2.2 Hyperbolicity of the Euler system

If we do not neglect the heat conduction, then the

where

where

which are real, and the eigenvectors are linearly independent implying that the Euler equations for perfect gases are hyperbolic.

### 2.3 Rankine-Hugoniot conditions for the Euler system

Using the results in the previous part, the Rankine-Hugoniot jump conditions for the Euler system will be of the form

where the indices

### 2.4 Riemann problem for the Euler system

The Riemann problem for the one-dimensional Euler equation (57) is represented by

The reader is addressed to the references [18, 24] for further details.

### 2.5 Lagrangian coordinates

We aim to transform the equations of gas dynamics (57) given in the Eulerian coordinates into the Lagrangian coordinates for one-dimensional case. We start denoting by

We set the following change of coordinates from Euler coordinates to Lagrange coordinates for space and time as

It follows that

which gives

It follows by some algebraic manipulations that the gas dynamic equations become

In order to derive a more convenient form of the system (69), we derive firstly the equation of conservation of mass:

where

which yields

Hence the second and third equations of Eq. (69) become

Moreover, we define a mass variable

Finally, using Eqs. (69) and (73), the Euler system (57) can be written in Lagrangian coordinates with the mass variable in the form

where

which is strictly hyperbolic. This can be verified by checking the Jacobian matrix of the flux calculated with respect to the variables

with

In fact there are different versions of the gas dynamics in Lagrangian coordinates. In this part we followed the approaches stated in [9, 10, 12]. For further details we cite these works with references therein.

### 2.6 Rankine-Hugoniot conditions for the Lagrangian system

Similarly as in the Euler system, the Rankine-Hugoniot jump conditions for the Lagrangian system (79) are of the form

where

*Remark*. The Eulerian and Lagrangian Rankine-Hugoniot relations are equivalent. Moreover, Eulerian entropy relations are equivalent to all Lagrangian entropy relations (see [9] for further detail).

**Example 2.1.** For simplicity of notation, we take

is a one-dimensional isentropic gas dynamics in Lagrangian coordinates which is also known as

The system (79) is equivalent to

where

has two real distinct eigenvalues

## 3. Godunov schemes

The Godunov scheme deals with solving the Riemann problem forward in time for each grid cell and then taking the mean value over these cells. The Riemann problem is solved per mesh point at each time step iteratively. If there are no strong shock discontinuities, this process may cost much and will not be effective. To get rid of such a situation, we establish approximate Riemann solvers that are easier to implement and also low cost to use. Eulerian and Lagrangian Godunov schemes are current Godunov scheme in literature. Both have advantages and disadvantages depending on the structure of the problem. A brief comparison of the method for these two approaches is presented in the last part of the chapter. In this work we will not go further in numerical examples and details of these methods; instead, we aim to present a general form of Godunov schemes for gas dynamics in Eulerian and Lagrangian coordinate. Before introducing these, we present a first-order Godunov scheme for scalar conservation laws.

### 3.1 First-order Godunov scheme

Consider the scalar conservation law (1). Godunov scheme is a numerical scheme which takes advantage of analytical solutions of the Riemann problem for the conservation law (1). The numerical flux functions are evaluated at the spatial steps

respectively. These two solutions to the Riemann problem will be the numerical solution

Proceeding this process, we define the solution

Dividing both parts by

Thus, Godunov method is a conservative numerical scheme. It can be restated in an alternative form. Assigning the constant value of

Therefore, a first-order Godunov method takes the form

Here the constant value of

The Godunov method is *consistent* with the exact solution of the Riemann problem for the conservation law (1). If we suppose that

for each

where

For numerical illustration of Godunov schemes, we cite the articles [14, 20, 27].

### 3.2 Godunov method in Eulerian coordinates

We consider Eq. (59) with (60). The eigenvalues of

### 3.3 Godunov method in Lagrangian coordinates

Consider the initial condition for a quantity

The eigenvalues satisfy

Then Godunov scheme for the Lagrangian coordinates takes the form

where

If we now consider the moving coordinates, Godunov scheme can also be derived equivalently by the following. Setting

Next we deduce

by a simple induction process. Hence the Lagrangian Godunov schemes become

with

Notice that the Lagrangian Godunov schemes can be reformulated as a finite volume method. Equation (100) can be written in conservative form:

If we integrate these equations on

Here we omit the dependency of

Moreover, if

provided

### 3.4 Comparison of Eulerian and Lagrangian schemes

In the literature there are two types of Godunov schemes: the Eulerian and Lagrangian. To compare one with the other, both have advantages and disadvantages. These are briefly listed in the following:

#### 3.4.1 Eulerian approach

It is more nature; that is the properties of a flow field are described as functions of the coordinates which are in the natural physical space and time. The flow is determined by examining the behavior of the functions. Eulerian coordinates correspond to the coordinates of a fixed observer. This approach is ease of implementation and computation. The computational grids derived from the geometry constraints are generated in advance. The computational cells are fixed in space, and the fluid particles move across the cell interfaces. Since the Eulerian schemes consider the implementation at the nodes of a fixed grid, this may lead to spurious oscillations for the problems like diffusion-dominated transport equations. By adding artificial diffusion, one can get rid of these oscillations; however the nature of the problem may differ from the original one. Besides, refining the grids may also lead to remove numerical oscillations, but this process may augment the computation cost. Besides, while refining the grids, it may cause restriction of the size of time step which is limited by CFL condition. This restriction does not occur in Lagrangian case.

#### 3.4.2 Lagrangian approach

It is based on the notion of mass coordinate denoted by

Apart from the two main approaches, there is another method which is a combination of both, so-called Eulerian-Lagrangian methods. It combines the advantages and eliminates disadvantages of both approaches to get a more efficient method. For further details we address the reader to the reference in the next part.

### Notes

We have tried to present only the theoretical aspects of scalar conservation laws with some basic models and provide some examples of computational methods for the scalar models. There are plenty of contributors to the subject; however, we just cite some important of these and the references therein. Scalar conservation laws are thoroughly studied in particular in [12]; for a more general introduction including systems, see [13, 15, 18, 19, 22] and the references therein. There are some important works related to the concept of entropy provided by [7, 15, 16]. A more precise study of the shock and rarefaction waves can be found in [23]. A simple analysis for inviscid Burgers’ equation is done by [21]. The readers who are deeply interested in systems of conservation laws and the Riemann problem should see [8, 13, 15, 22, 24]. A well-ordered work of the propagation and the interaction of nonlinear waves are provided by [26]. We refer the reader to the papers [1, 17] for the theory of hyperbolic conservation laws on spacetime geometries and finite volume analysis with different aspects. A widely introductory material for finite difference and finite volume schemes to scalar conservation laws can be found in [18]. In this chapter we have studied the one-dimensional gas dynamics on the Eulerian and Lagrangian coordinates. For the detail on the Lagrangian conservation laws, we refer [10]; moreover for both Eulerian and Lagrangian conservation laws, we cite [11]. The proof of the equivalency of the Euler and Lagrangian equations for weak solutions is given in [25]. There are several numerical works for Lagrangian approach; some of the basic works on Lagrangian schemes are given in [2, 3, 4, 5, 6]. We refer the reader to the book [7] for a detailed analysis of the mathematical standpoint of compressible flows. Moreover Godunov-type schemes are precisely analyzed in [14, 27]; whereas, Lagrangian Godunov schemes can be found in [2, 12, 20]. As a last word, we must cite [9] as a recent and more general book consisting of scalar and system approaches of both Eulerian and Lagrangian conservation laws with theoretical and numerical parts which can be a basic source for the curious readers.