Burgers equation

burgers equation

Mikel Landajuela BCAM Internship - Summer 2011

Abstract

In this paper we present the Burgers equation in its viscous and non-viscous version. After submitting, as a motivation, some applications of this paradigmatic equations, we continue with the mathematical analysis of them. This will lead us to confront one of the main problems linked to non-linear pde: The appearance of shocks. Finally, the paper concludes presenting several numerical schemes for solving these equations and their corresponding implementation in matlab.

Contents

1 Introduction

2

2 Burgers Model

2

2.1 Navier Stokes equations simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.2 Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Mathematical analysis

4

3.1 Inviscid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Characteristic method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Breaking Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

The Rankine-Hugoniot jump condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Note on weak solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Entropy condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Vanishing viscosity approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Viscid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

The Cole-Hopf transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Heat equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Numerical methods

12

4.1 Inviscid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Up-wind nonconservative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Up-wind conservative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Lax-Friedrichs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Lax-Wendroff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

MacCormack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Godunov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Viscid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Parabolic Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1 INTRODUCTION

2

1 Introduction

In this paper we will consider the viscid Burgers equation to be the nonlinear parabolic pde

ut + uux = uxx

(1)

where > 0 is the constant of viscosity. This is the simplest pde combining both nonlinear

propagation effects and diffusive effects. When the right term is removed from (1) we obtain the

hiperbolic pde

ut + uux = 0.

(2)

We will refer to (2) as the inviscid Burgers equation. Note that equation (2) can be rewritten in

the form

u2

ut + [f (u)]x = 0

with

f (u) = , 2

(3)

where is easily recognizable the structure of a scalar hiperbolic conservation law. Many of the

ideas presented in this paper (relating to mathematical treatment, numerical methods,...) can be

formulated in the framework of the theory of scalar hiperbolic conservation laws so for general-

ity we will often refer to (2) in the form (3) and the developments will be valid for a general f in (3).

There is an important connection between the above equations: Equation (2) is the limit as 0 of (1). This is true from the formal point of view but also in the most rigorous sense. This fact will be studied in more detail in the paragraph called Vanishing viscosity approach.

2 Burgers Model

Burgers equations appear often as a simplification of a more complex and sophisticated model. Hence it is usually thought as a toy model, namely, a tool that is used to understand some of the inside behavior of the general problem. Here we will present two examples.

2.1 Navier Stokes equations simplification

Consider the Navier Stokes equations

? v = 0,

(4.1)

(v)t + ? (vv) + p - ?2v = 0. (4.2)

(4)

It is well known that when is consider to be the density, p the pression, v the velocity and ? the viscosity of a fluid, equations (4) describe the dynamics of a divergence free (4.1) incompressible (t = 0) flow where gravitational effects are negligible.

Simplification in (4.2) of the x componet of the velocity vector, which we will call vx, gives

vx

+

vx vx

+

vy vx

+

vz

vx

+

p

-

?

t

x

y

z x

2vx 2vx 2vx x2 + y2 + z2

= 0.

If we consider a 1D problem with no pressure gradient, the above equation reduces to

vx

t

+ vx vx x

2vx - ? x2

=

0.

(5)

If we use now the traditional variable u rather than vx and take to be the kinematic viscosity,

i.e,

=

?

,

then

the

last

equation

becomes

just

the

viscid

Burgers

equation

as

it

has

been

presented

2 BURGERS MODEL

3

in (1).

When the viscosity ? of the fluid is almost zero, one could think, as an idealization, to simply remove the second-derivative term in (5). This would lead to

vx

+ vx vx

=

0

(6)

t

x

which, after making u = vx and dividing by , becomes the inviscid Burgers equation as it is shown in (2). It turns out that, in order to use (6) as a model for the dynamics of an inviscous fluid, it has to be supplemented with other physical conditions (section 3.1) which will prevent equation (6) from developing physical meaningless solutions. This extension is worth because working with (6) is much easier than dealing with (5).

2.2 Traffic Flow

Consider the flow of cars on a highway and let (x, t) denote the density of cars and f (x, t) the traffic flow. We will also consider to be the restriction of to a certain range, 0 max, where max is the value at which cars are bumper to bumper.

Since cars are conserved, the density of cars and the flow must be related by the continuity

equation

f

t + x = 0.

(7)

Obviously, the first expression in which one thinks for the flow is f = v where v is the velocity.

However, it turns out that in order to reflect the fact that drivers will reduce their speed to account

for an increasing density ahead we should suppose that f is a function of the density gradient as

well. A simple assumption is to take

f

()

=

v()

-

D

x

where D is a constant.

(8)

We are assuming also that the velocity v is a given function of : On a highway we would optimally like to drive at some speed vmax (the speed limit perhaps) but with heavy traffic we slow down, with velocity decreasing as density increases. The simplest relation that is aware of this is

v()

=

vmax max

(max

- ).

(9)

Putting (8) and (9) into (7) leads to

d t + dx

vmax max

(max

-

)

2 = D x2 .

(10)

Scaling

through

vmax

=

x0 t0

,

= max,

x = x0x

and

t = t0t

results

in

D

t + [(1 - )]x = xx with

=

and 0 1.

vmaxx0

(11)

The transformation u = 2 - 1 leads to the viscid Burgers equation as it is shown in (1) with the conditions -1 u 1.

3 MATHEMATICAL ANALYSIS

4

3 Mathematical analysis

From the mathematical point of view Burgers equations are a very interesting and suggestive topic: It turns out that a study of them leads to most of the ideas that arises in the field of nonlinear hiperbolyc waves.

3.1 Inviscid

We will focus first on equation (2). Specifically, we will deal with the initial value problem

ut + uux = 0, x R, t > 0, u(x, 0) = u0(x), x R.

(12)

As it as has been suggested previously, although (12) seems to be a very innocent problem a priori it hides many unexpected phenomena.

Characteristic method. The similarity with the advection equation suggests considering, as a first approach to solve (12), the characteristic method. In this case the characteristic equation would be

x (t) = u(x(t), t), t > 0, x(0) = x0.

(13)

If x(t) and u(x, t) ( C1) are solutions of (13) and (12) respectively, then

d dt [u(x(t), t)] = ut(x(t), t) + x (t)ux(x(t), t) = ut(x(t), t) + u(x(t), t)ux(x(t), t) = 0, i.e, u is constant along the characteristic curve x(t) and therefore

u(x(t), t) = u(x(0), 0) = u0(x0),

(14)

which considering the sistem (13), leads to conclude that the characteristic curves are straight lines

determined by initial data :

x = x0 + u0(x0)t, t > 0.

(15)

In principle, one could invert (15) to obtain x0 = x0(x, t). Then, using (14), one would obtain the solution u(x, t) = u0(x0(x, t)). However, as the required inversion usually can not be accomplished analytically, one could use a symbolic calculation program to construct a discretized solution of

(12) by dragging the initial data u0(x0) along the characteristic line (15). This strategy is followed by the following program in mathematica:

Clear[f0, fval, x, f] f0[x_] = Exp[-(2 (x - 1))^2]; (*Condicio?n inicial*) x[t_, x0_] = x0 + f0[x0] t; f[t_, x0_] = f0[x0]; fval[t_] := Table[{x[t, x0], f[t, x0]}, {x0, -.5, 3, .1}]

(*Dibujo de las caracter?isticas*) Plot[Table[x[t, x0], {x0, -.5, 3, .1}], {t, 0, 2},AxesLabel -> {Text[Style["t", Italic, 23]],

Text[Style["x", Italic, 23]]}]

(*Dibujo de la soluci?on u(x,t)*) ListPlot3D[{Table[Table[{x[t, x0], t, f[t, x0]}, {x0, -.5, 3, .1}], {t, 0, 2, 0.1}]},

3 MATHEMATICAL ANALYSIS

5

ColorFunction -> "DarkRainbow", PlotStyle -> Directive[Opacity[0.9]], MeshFunctions -> {#2 &}, Mesh -> 5, PlotRange -> All, Axes -> {True, True, True} ,

Boxed -> False, ImageSize -> 600,AxesLabel -> {Text[Style["x", Italic, 23]], Text[Style["t", Italic, 23]], Text[Style["u", Italic, 23]]}]

Figure 1: Output of the programme for u0(x) = e-(2(x-1))2 . Some characteristic lines are presented in the first picture.

Looking to this example one quickly finds that problem (12) exibits (under certain initial conditions) what is called the wavebreaking phenomenon: The peak of the pulse moves the fastest because wave speed increases with increasing amplitude. This can also be understood by looking to the corresponding slope of the characteristic line in the (x, t) plane. Eventually the peak overtakes the rest of the pulse, or the characteristic cross with another one, and the solution becomes multiple valued. The time value tB at which this happens for the first time is called the breaking time.

Breaking Time. In order to determine the breaking time, let us consider two characteristics that arise from initial conditions x1 and x2 = x1 + x. According to (15), these characteristics will cross when

x1 + u0(x1)t = x2 + u0(x2)t.

Solving for t leads to

t = - x1 - x2

=

x .

(16)

u0(x1) - u0(x2) u0(x1) - u0(x1 + x)

When

x

0

then

the

time

in

(16)

converges

to

t

=

-

u0

1 (x1

)

.

To

find

the

breaking

time

tB

we

find the minimum (positive) value for t,

1

tB = min

xR

- u0(x)

.

Although the solution obtained via the characteristic method seems to be valid for t < tB, it is clear that mathematically multiple valued solutions in a region is unphysical in most of the situations (think of u as density) so it can not be accepted for t > tB. What is then the solution

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