Open access peer-reviewed chapter

Scalar Conservation Laws

By Baver Okutmuştur

Submitted: April 24th 2018Reviewed: December 18th 2018Published: April 2nd 2019

DOI: 10.5772/intechopen.83637

Downloaded: 256

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

tu+xfu=0,t>0,xRE1

in one dimension are known as scalar conservation laws where u=utxis the conserved quantity and f=fuis the associated flux function depending on tand x. Whenever an initial condition u0x=u0xis attached to Eq. (1), the problem is called the Cauchy problem the solution of which is a content of this chapter. The outlook of chapter is as follows. We introduce basic concepts and provide particular examples of scalar conservation laws in the first part. The equation of gas dynamics in Eulerian coordinates in one dimension is the main issue of the second part. After providing further instruction for these equations, we provide a transformation of the Eulerian equations in the Lagrangian coordinates. In the final part, we give as an example of computational methods for conservation laws, the Godunov schemes for the Eulerian, and the Lagrangian coordinates, respectively.

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 xis the distance and ρtxis the density at the time tand at the point x. Then if we integrate the density on x1x2, we get total mass x1x2ρtxdxat time t. Assigning the velocity by utx, then mass flux at becomes ρtxutx.It follows that the rate of change of the mass in x1x2is

ddtx1x2ρtxdx=ρtx1utx1ρtx2utx2.E2

The last equation is called integral form of conservation law. Integrating this expression in time from t1to t2,we get

x1x2ρt2xdxx1x2ρt1xdx=t1t2ρtx1utx1dtt1t2ρtx2utx2dt.E3

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

ρtx2utx2ρtx1utx1=x1x2xρtxutxdx.E4

As a result, we get

t1t2x1x2tρtx+xρ(tx)u(tx)dxdt=0.E5

Here the end points of the integrations are arbitrary; that is, for any x1x2and t1t2, the integrant must be zero. It follows that the conservation of mass yields

tρ+x=0,E6

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

atxutu+btxuxu=ctxuE7

where a,b,care non-zero and smooth on a given domain DR3follows by the characteristic method where the characteristic curves are defined by

dtatxu=dxbtxu=ductxu.E8

By applying a parametrization of c, the relation (8) is transformed to a system of ordinary differential equation (ODE):

dtdc=atxu,dxdc=btxu,dudc=ctxu.E9

In addition to these equations, if an initial condition u0=ux0is also given, then by the existence theorem of ODE, there is a unique characteristic curve passing from each point t0x0u0leading to an integral surface which is the solution to Eq. (7).

Observe that the scalar conservation law (1) is a particular example of Eq. (7) if we assign atxu=1, btxuux=fux, and ctxu=0. The conserved quantity can be observed by integrating equation (1) over x0x1. Indeed

ddtx0x1utxdx=x0x1tutxdx=x0x1futxxdx=futx1futx0=inflowatthepointx1outflowatthepointx0.E10

This means, the quantity utxis conserved so that it depends on the difference of the flux functions between the points x0and x1.

1.3 Strong (classical) solutions

We consider the initial value problem

tu+xfu=0,t>0,xRu0x=u0x,xRE11

where the initial data is assumed to be continuously differentiable, that is, u0xC1R. Applying the chain rule to the relation (11), it follows that

tu+fuxu=0,t>0,xR,u0x=u0x,xR,E12

where we define characteristic curves of Eq. (12) to be the solution of ddtxt=futxt=fu. Then a solution to the system (12) in a domain ΩRis said to be a strong (or classical) solution if it satisfies Eq. (11), and it is continuously differentiable on a domain ΩR.Let u be a strong solution and the initial data u0 be differentiable. Observe that (12) is equivalent to a quasilinear form:

tu+λuxu=0,E13

with λu=fu. Applying the method of characteristics to Eq. (13), the partial differential equation is transformed to a system of ordinary differential equations. We consider the characteristic curve passing through the point 0x0:

tx=λutxtx0=x0.E14

Along this characteristic curve,

tutxt=tutxt+txxutxt=tu+λuxu=0E15

is satisfied, that is, uis constant. Hence, the characteristic curves are straight lines satisfying

x=x0+λu0x0t=0.E16

Hence we can define smooth solutions by utx=u0x0. If the slope of the characteristics is mchar=1λu0xi,then depending on the behavior of λ, the solution takes different forms. If λu0xis increasing, then the slopes of the characteristics are decreasing. As a result, the characteristics do not intersect, and thus solution can be defined for all twhich is greater than zero. On the other hand, if λu0xis decreasing, then the slopes of the characteristics will be increasing which implies that the characteristics intersect at some point. But at the intersection point, solution cannot take both values u0x1and u0x2. Therefore, we cannot define the strong solution for all t>0.

1.4 Linear advection equation

The basic example of the scalar conservation law is the linear advection equation. It can be obtained by setting atxu=1, btxu=λ, and ctxu=0in Eq. (7). The flux function takes the form fu=λuwhere λis a constant. Then the following quasilinear partial differential equation

tu+λxu=0E17

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

tu+xfu=0,<x<,t0,u0x=u0x=fx0,<x<.E18

Applying the method of characteristics, it follows that dt1=dxλ=du0or equivalently

u=c1,dxdt=λ=c1,x=c1t+c2,E19

where c1and c2are constant and xλt=c2.As a conclusion, the solution is

utx=u0xλt,t0.E20

Here λis the wave speed, and the characteristic lines xλt=c2are wavefronts which are constants.

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

tu+uxu=νxxu,E21

where νxxuis the viscosity term. Equation (21) can be considered as a combination of nonlinear wave motion and linear diffusion term so that it is balance between time evolution, nonlinearity, and diffusion. The term uxuis a convection term that may have an effect to wave breaking, and the term νxxuis a diffusion term that may cause to efface the wave breaking and to flatten discontinuities, and thus we expect to achieve a smooth solution. We try to find a traveling wave solution of Eq. (21) of the form

utx=gξ=gxλt,withξ=xλt,E22

where gand λare to be determined. Applying the chain rule, we get

tu=λgξ,xu=gξ,xxu=gξ.E23

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

λgξ+gξgξνgξ=0.E24

Taking integration with respect to ξgives

λg+12g2νg=C,C:constant.E25

Rewriting Eq. (25) by

gg1gg2=g22λg2C=2νdg/,E26

it follows that g1,2=λ±λ2+2C. Supposing that g1,g2are real implies g1>g2. Using separation of variable and then integrating equation (26), we get

gξ=g1+g2eg1g22νξ1+eg1g22νξ=g1+g22g1g22tanhg1g24νξE27

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

utx=λg1g22tanh14νg1g2xλtE28

where λ=g1+g22is the wave speed. We can observe that limξgξ=g1and limξgξ=g2with gξ<0for all ξ. This means the solution gξdecreases monotonically with ξfrom the value g1to g2. At ξ=0, u=g1+g22=λ,that is the wave form gξtravels from left to right with speed λequal to the average value of its asymptotic values. The solution resembles to a shock form as it connects the asymptotic states g1and g2. Without the viscosity term, the solutions to Burgers equation allow shock forms which finally break. The diffusion term prevents incrementally deformation of the wave and its breaking by withstanding the nonlinearity. As a conclusion, there exists a balance between nonlinear advection term and the linear diffusion term. The wave form is notably affected by the diffusion coefficient ν. If νis smaller, then the transition layer between two asymptotic values of solution is sharper. In the limit ν0, the solutions converge to the step shock wave solutions to the inviscid Burgers’ equation.

Remark. If the initial data is smooth and very small, then the uxxterm is negligible compared to other terms before the beginning of wave breaking. As the wave breaking starts, the uxxterm raises faster than uxterm. After a while, the term uxxbecomes comparable to the other terms so that it keeps the solution smooth, giving rise to avoid breakdown solutions.

1.6 Inviscid Burgers’ equation

Whenever ν=0, Eq. (21) is called the inviscid Burgers’ equation. This equation can be obtained by substituting fu=u2/2in the scalar conservation law (1), that is

tu+xu2/2=tu+uxu=0.E29

Observe that fuis a nonlinear function of u; thus, the inviscid Burgers’ equation is a nonlinear equation. Equation (29) is now equivalent to Eq. (17) with λ=u. We know the solution of Eq. (17); so, plugging λ=uinto the relation (20) implies that the solution of Eq. (29) is

utx=fxut=u0xut.E30

Recall that the characteristic speed λis constant for linear advection equation; that is, the characteristic curves become parallel for Eq. (17). In contrast, for the inviscid Burgers’ equation (29), the characteristic speed λ=udepends on u. As a result the characteristic lines are not parallel. If we apply the implicit function theorem to Eq. (29), the solution can be written as a function of tand xas u0is differentiable. More particularly, differentiating Eq. (30) with respect to t, we get

tu=u0utt+utu=u0u1+u0t;E31

and differentiating equation (30) with respect to x, we get

xu=u01uxtxu=u01+u0t.E32

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 u0. It can be observed that whenever u0x>0, then by Eq. (32), xudecreases in time because 1+u0t>0for t>0. In other words, the profile of the wave flattens as time increases. On the other hand, whenever u0x<0, then xuincreases in time as 1+u0t<0.Hence uxin Eq. (32) tends to as 1+u0tapproaches to zero. As a result, wave profile become sharp after some time. For further details on the Burgers’ equations, we refer the reader to [12, 13, 22] and the references therein.

1.7 Shock waves

Let the constants uL and uR are given with a linear function, φt=λt. Then

utx=uRifx>λt,uLifx<λt,E33

is a simple example of discontinuous solution of the conservation law (11). If uLuR, the relation (33) is called a shock wave connecting uLto uRwith shock speed λ. As an example, if we take into account the characteristics of the inviscid Burgers’ equations which are of the form dxdt=utx,it follows that

xt=u0x0t+x0E34

where u0x=u0xand x0=x0; thus, the characteristics are straight lines. Depending on the behavior of these characteristics, we have two cases. If uL>uR,characteristics intersect, the solution will have an infinite slope, and the wave will break; as a result a shock is obtained. This is illustrated in Figure 1. On the other hand, if uR>uL, the characteristics do not intersect, and hence a region without characteristic will appear which is physically unacceptable. This is shown in Figure 2. We get rid of this by introducing the rarefaction waves.

Figure 1.

For the initial value uL>uR, characteristics, and shock wave.

Figure 2.

For the initial value uR>uL, characteristics and rarefaction waves.

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 xx0=αt. Here αfuLfuRwhere uLand uRare the values of uat the edge of the rarefaction wave fan. If moreover fis invertible, then the solution u=utxsatisfies

uxt=f1xx0t.E35

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

utx=uLifx/tfuL,f1x/tiffuLx/tfuR,uRiffuRx/t.E36

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

Figure 3.

Rarefaction fan.

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 ϕ:R×R+Ras a smooth test function with a compact support and multiplying the scalar conservation law (1) by this test function ϕ, it follows after integration by parts that

0ϕtu+ϕxfudxdt=ϕu0dx0utϕdxdt+0ϕfudt0fuxϕdxdt=0utϕdxdt0fuxϕdxdtt=0dx.E37

Putting the initial condition u0x=u0xto the above relation, it follows that

0uϕt+fuϕxdxdt+u0xϕxdx=0.E38

Observe that there are no more derivatives of uand fwhich may lead less smoothness. In other words, the smoothness requirement is reduced for finding a solution. Thus, the function utxis said to be the weak solution of the initial value problem (11) if the relation (38) satisfied for all test function ϕ.Here it is significant to note that uneeds not be smooth or continuous to satisfy Eq. (38). Consequently, by weak solutions, we extend the solutions so that discontinuous solutions may also be covered. However, in general weak solutions are not unique. We can also notice that strong solutions are also weak solutions and a weak solution which is continuous and piecewise differentiable is also strong solution.

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

tu+xfu=0,xR,tR+,u0x=uLifx<0,uRifx>0.E39

The solution is a set of shock and rarefaction waves depending on the relation between uLand uR.There are two cases to investigate:

Case 1: uL>uRA shock is obtained because the left-hand side wave moves faster than the right-hand side one. Thus the solution

utx=uLifx/t<λ,uRifx/t>λ,E40

is a shock wave satisfying the shock speed λ=fuRfuLuRuL.

Case 2: (uL<uR) The solution given in Case 1 is also a solution for this case. In addition, we have rarefaction solutions of the form (36) illustrated by Figure 3.

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 x1x2, it follows that

ddtx1x2utxdx+fux1x2=0.E41

Suppose that there is a discontinuity at the point x=ξtx1x2where uand uare continuous on the x1ξtand ξtx2, respectively. Suppose also that whenever x1ξtand x2ξt+, their limits exist. Next, Eq. (41) can be rewritten as

ddtx1ξtutxdx+ddtξtx2utxdx=ftx2ftx1.E42

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

uξxξtuξ+xξt+ddtx1ξtuttxdx+ddtξtx2uttxdx.E43

Taking the limit whenever x1ξtand x2ξt+, it follows that

ξtx2x1=fx2fx1λ=ξt=fx2fx1x2x1.E44

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 uLfuLand uRfuRon the graph of f.

1.12 Entropy functions

Entropy and entropy flux are defined for attaining physically meaningful solutions. If uis the smooth solution of the conservation law (1), then the relation

tGu+xFu=0E45

is satisfied for continuously differentiable functions Gand Fwhere the pair GFis called as entropy pair so that Gis entropy and Fis entropy flux. If in addition uis smooth, then Eq. (45) becomes

Gutu+Fuxu=0E46

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

Gutu+Gufuxu=0.E47

It follows that Eqs. (46) and (47) are equivalent with Fu=Gufu.Here the function utxis said to be the entropy solution of Eq. (1) if

tGu+xFu0

holds for all convex entropy pairs GuFu.

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 λgiven by the Rankine-Hugoniot jump condition satisfies the entropy condition if holds.

fuL>λ>fuRE48

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 2u, we obtain 2utu+2u2xu=0. In conservative form it becomes

tu2+x23u3=0,withfu2=23u23/2.E49

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 λ1=uL+uR/2; however for Eq. (49), we have λ2=(23uL3uR3uL2uR2. That is λ1λ2whenever uLuR, and thus these two equations have different weak solutions.

Example 1.2. We first consider the initial value problem for uL>uRgiven by

tu+xu2/2=0,u0=1ifx0,0ifx>0.E50

Applying the method of characteristics for t>0, it follows that

dudt=0,dxdt=1ifx0,0ifx>0.E51

Next if we integrate Eq. (51) with respect to t, we get the characteristic curves

x=tcifx0,bifx>0,E52

where c>0and bare constants. Due to the discontinuity at the point x=0, there is no strong (classical) solution. The speed of propagation is λ=uL+uR2=0.5.Moreover, the weak solution for tλ=0.5becomes

utx=1ifxt0.50ifxt>0.5,E53

which satisfies both the jump condition and the entropy condition as uL=1>uR=0. The characteristic curves can be observed in Figure 4.

Figure 4.

For initial value uL>uR, the characteristic solutions.

Example 1.3. We now interchange the roles of uLand uRof the Example 1.2 so that uL<uRto get an initial value problem:

tu+xu2/2=0,u0=0ifx0,1ifx>0.E54

By the method of characteristics, we obtain a solution

u1tx=0ifxt11ifxt>1E55

which is a classical (strong) solution on both sides of the characteristic line xt=1. Since it satisfies the Rankine-Hugoniot jump condition along the discontinuity curve, it is a weak solution. However, the entropy condition is not satisfied. It yields an empty region between the characteristic lines shown in Figure 4. In order to cover this empty state, we consider another solution described by

u2tx=0ifx0,xtif0xt1,1ifxt1E56

which satisfies both jump and entropy conditions. Here we can observe the rarefaction fan arising on the interval 0xt1. An illustration of this solution is supplied in Figure 5.

Figure 5.

For initial value uL<uR, characteristic solutions u1tx and u2xt with rarefaction fan.

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:

tρ+xρu=0,tρu+xρu2+p=0,tρe+xρe+pu=0E57

where we ignored the heat conduction. If we denote

U=ρρuρe,FU=ρeρu2+pρeu+pu,E58

then Eq. (57) can be written by

tU+xFU=0E59

where ρis density, pis pressure, uis velocity, and eis the specific internal energy.

2.2 Hyperbolicity of the Euler system

If we do not neglect the heat conduction, then the Uand Fterms in Eq. (59) become

U=ρρuEandF=ρuρu2+pE+pu,E60

where Eis total energy such that E=12ρu2+ρe,e=pδ1ρ, and for perfect gases δ=cp/cvis the ratio of specific heats. Rewriting Eq. (59) in quasilinear form, we get

tU+AUxU=0,E61

where AU=FUis the Jacobian matrix. The eigenvalues of AUthen are λ1=u,λ2=ua,λ3=u+awhere ais the sound speed given by a=δpρ. Moreover the corresponding eigenvectors are

E1=1u12u2,E2=1uaHua,E3=1u+aH+uaE62

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

sρ1ρ2=m2m1,sm2m1=m22ρ2+p2m12ρ1p1,sρ2E2ρ1E1=m2H2H1m1,E63

where the indices 1and 2refer to the left and right of the shock, respectively, and sdenotes the wave speed.

2.4 Riemann problem for the Euler system

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

tU+xFU=0,xR,t>0,U0x=U0x=ULifx<0,URifx>0.E64

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 u=utxthe velocity field of the fluid flow and consider the differential system

dxdt=utx.E65

We set the following change of coordinates from Euler coordinates to Lagrange coordinates for space and time as txtξwhere ξ=ξ1ξ2ξ3R3so that

t=t,xtξt=utxtξ,x0=x0ξ=ξ.E66

It follows that tξ=tξ1ξ2ξ3are the Lagrangian coordinates associated with the velocity field u. We set

Jtξ=detxiξjtξ,E67

which gives

Jttξ=Jtξdivutxtξ,where,divu=j=13ujxj.E68

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

tρJ=0,Conservationofmass,tρuJ+ξp=0,Conservationofmomentum,tρeJ+ξpu=0,Conservationofenergy.E69

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

ρJ=ρ0=ρ0ξE70

where ρ0ξ=ρ0ξ.Assuming that ρ>0, we introduce the specific volume τ=1/ρ, and by using Eq. (68) we get

J=ρ0τ,andtJ=Jxu=ξuE71

which yields

ρ0tτξu=0.E72

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

ρ0tu+ξp=0,Conservationofmomentum,ρ0te+ξpu=0,Conservationofenergy.E73

Moreover, we define a mass variable mby

mξ=0ξρ0ydy,orequivalently,dm=ρ0ξ=ρ0.E74

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

tτmu=0,tu+mp=0,te+mpu=0,E75

where p=pτξ=pτeu2/2. If we set V=τue,FV=uppuwith τ>0,uR,eu2/2>0,we obtain a scalar conservation law of the form

tV+mFV=0E76

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

010pτupεpεupτpu2pεupεE77

with e=ε+12u2. The eigenvalues are σ1=pτppε<σ2=0<σ3=pτppεso that they are all distinct, and thus the system is strictly hyperbolic.

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

στ1τ0=u1u0,σu1u0=p1p0,σe1e0=p1u1p0u0,E78

where σdenotes the speed of propagation of the discontinuity with respect to the mass variable.

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 txas the Lagrangian coordinates. Then the system of equations

tτxu=0,tu+xpτ=0,E79

is a one-dimensional isentropic gas dynamics in Lagrangian coordinates which is also known as p-system. It is the simplest nontrivial example of a nonlinear system of conservation laws. Here τis the specific volume, uis the velocity, and the pressure p=pτis given as a function of τby

pτ=κτγ,γ>0,κ=γ124γ.E80

The system (79) is equivalent to

tV+xfV=0,withV=τu,fV=upτ,E81

where τ>0and τuR2.If we assume that pτ<0, it follows that the Jacobian matrix of f

Jf=01pτ0E82

has two real distinct eigenvalues σ1=(pτ<σ2=pτ. In other words, the system (81) is strictly hyperbolic. On the other hand, for the case pτ>0, it becomes elliptic. Moreover, one can verify that the solutions of the p-system (79) and the Euler system (57) are equivalent.

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 xj1/2and xj+1/2by handling the solutions of the Riemann problem. On each grid cell Ii=xj1/2xj+1/2,we have a piecewise constant function. The Riemann problem for (1) for the left and right sides of Iiare described by

uLx=uj1n;x<0,ujn;x>0,uRx=ujn;x<0,uj+1n;x>0,E83

respectively. These two solutions to the Riemann problem will be the numerical solution u˜tx. Once establishing the solution over the mesh tntn+1, we approximate the solution at the next time step tn+1by the average value

Ujn+1=1Δxxj1/2xj+1/2u˜xtn+1dx.E84

Proceeding this process, we define the solution u˜xtn+1iteratively. Then Ujn+1can be calculated by using the integral form of the conservation law (1) in the following way: We integrate (1) for utxover each grid cell tntn+1×Ij:

xj1/2xj+1/2u˜nxtn+1dxxj1/2xj+1/2u˜nxtndx=tntn+1fu˜j1/2ndttntn+1fu˜j+1/2ndt.E85

Dividing both parts by Δxand using the fact that u˜xtn=ujnis constant at the end points xj1/2and xj+1/2, we get

ujn+1=ujnΔtΔxfu˜j1/2nfu˜j+1/2n.E86

Thus, Godunov method is a conservative numerical scheme. It can be restated in an alternative form. Assigning the constant value of ujnat the points xj1/2and xj+1/2by u(Uj1n,Ujn)and u(Ujn,Uj+1n), respectively, the numerical flux functions become

fu˜j1/2n=fuUj1nUjn=FUj1nUjn,fu˜j+1/2n=fuUjnUj+1n=FUjnUj+1n.E87

Therefore, a first-order Godunov method takes the form

Ujn+1=UjnΔtΔxFUjnUj+1nFUj1nUjn.E88

Here the constant value of u˜ndepends on the initial data. In other words, the Godunov method considers the Riemann problem as constant in each grid interval Ii. It follows that, at the subsequent time stage, the exact solutions of the problem are picked as the numerical fluxes at the grid boundary.

The Godunov method is consistent with the exact solution of the Riemann problem for the conservation law (1). If we suppose that ujn=ujn+1=u¯, then u˜j+1/2n=u¯and Fu¯u¯=fu¯. For the stability, CFL condition requires that

supxR,t>0futxΔtΔx1E89

for each ujn. Next, if assigning uas the intermediate value over the grid Iiin the Riemann solution, it implies that

uuLuR=uL,λ>0,uR,λ<0,E90

where λis the wave propagation speed. Hence the numerical flux for Godunov’s method can be generalized by

fuLuR=minuLuuRfu,ifuLuR,maxuLuuRfu,ifuR<uL.E91

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 FUare σ1=uc<σ2=u<σ3=u+c. Then the Riemann problem at the point xi+1/2between the states Uiand Ui+1which is solved by the Godunov scheme can be written by

ρin+1=ρinΔtΔxiρui+1/2nρui1/2nρuin+1=ρuinΔtΔxiρu2+pi+1/2nρu2+pi1/2n.ρein+1=ρeinΔtΔxiρe+pui+1/2nρe+pui1/2nE92

3.3 Godunov method in Lagrangian coordinates

Consider the initial condition for a quantity vgiven by the mean value

vi0=1Δξiξi1/2ξi+1/2vξ0.E93

The eigenvalues satisfy σ1<σ2=0<σ3.Setting ui+1/2and pi+1/2as the values of uand pat the contact discontinuity between Vinand Vi+1n, it follows that

FwR0VinVi+1n=ui+j/2npi+j/2npui+j/2nT.E94

Then Godunov scheme for the Lagrangian coordinates takes the form

τin+1=τin+ΔtΔmiui+1/2nui1/2nuin+1=uinΔtΔmipi+1/2npi1/2nein+1=einΔtΔmipui+1/2npui1/2nE95

where

Δmi=ρi0Δξi,pin=pτinεin,εin=einuin22.E96

If we now consider the moving coordinates, Godunov scheme can also be derived equivalently by the following. Setting xi+1/2=ξi+1/2with the approximation of u=dx/dt, it follows that the Eulerian coordinate xi+1/2of the interface ξi+1/2at tnis upgraded with respect to

xi+1/2n+1=xi+1/2n+Δtui+1/2n.E97

Next we deduce

ρinxi+1/2nxi1/2n=ΔmiE98

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

Δmi=ρi0xi+1/20xi1/20xi+1/2n+1=xi+1/2n+Δtui+1/2nE99

with

ρin+1=xi+1/2n+1xi1/2n+11Δmiuin+1=uinΔtΔmipi+1/2npi1/2n.ein+1=einΔtΔmipui+1/2npui1/2nE100

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

tφJ+ξf=0.E101

If we integrate these equations on ξi1/2ξi1/2it follows that

ddtxi1/2xi+1/2φdξ+fi+1/2fi1/2=0.E102

Here we omit the dependency of f,φand xon t. Moreover, if we suppose that φis constant in each cell ξi1/2ξi1/2, it follows by an explicit one-step method that is

Δxin+1φin+1=ΔxinφinΔtfi+1/2nfi1/2n.E103

Moreover, if ρueare constant in each cell with v=u, we get the Godunov scheme:

Δxinρin=ΔmiΔmin+1uin+1=ΔminuinΔtpi+1/2npi1/2nΔmin+1ein+1=ΔmineinΔtpui+1/2npui1/2nE104

provided ui+1/2npi+1/2nare determined by the solution of the Riemann problem, which is the desired result.

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 mξ. An important feature of the mass coordinate is that two segments have the same length if the mass contained in these segments is the same. This leads to face with a disadvantage; that is, at each iteration time step, the problem has to be converted from the natural coordinate system to the mass coordinate system. Once the solution at the next step is known, it has to be remapped into the natural coordinate system. As a result, this process raises the cost of the computation. Lagrangian coordinates are associated to the local flow velocity. In other words, as the velocity has different values in different parts of the fluid, then the change of coordinates is different from one point to another one in Lagrangian coordinates. Thus Lagrangian coordinates are equivalent to the Eulerian coordinates at another time. Lagrangian description states the motions and properties of the given fluid particles as they travel to different locations. Hence the computational grid points are precisely fluid particles. Since the particle paths in steady flow coincide with the streamlines, no fluid particles will cross the streamlines. Hence, there is no convective flux across cell boundaries, and the numerical diffusion is minimized. As a result, Godunov method in a Lagrangian grid is easier to handle. Moreover, in the case of higher schemes, the subsonic character of the flow makes the transformation much easier than in Eulerian schemes. Lagrangian schemes consider the implementation in a grid that moves with the flow which is an advantage for the problems like the transport equations since the advective and diffusion terms can separately be examined.

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.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Baver Okutmuştur (April 2nd 2019). Scalar Conservation Laws, Advanced Computational Fluid Dynamics for Emerging Engineering Processes - Eulerian vs. Lagrangian, Albert S. Kim, IntechOpen, DOI: 10.5772/intechopen.83637. Available from:

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