a particular hopf-cole solution to burgers' equation

Verifying that the basic Hopf−Cole transformation, w = 2 u-1 ux

where ut = uxx , satisfies the homogeneous Burgers’ equation wt - wxx - wwx = 0 per Equation World’s description

In[2]:= CleanSlate@D;

HCleanSlateL Contexts purged: 8Global‘<HCleanSlateL Approximate kernel memory recovered: 1 Kb

In[1]:= w@x_, t_D := ��2

u@x, tD D@u@x, tD, xD;

In[2]:= terms = Expand@8D@w@x, tD, tD,-D@w@x, tD, 8x, 2<D,-w@x, tD D@w@x, tD, xD<D;

MatrixForm@termsDOut[3]//MatrixForm=

i

k

jjjjjjjjjjjjjjjjjjjjj

- ��2 uH0,1L@x,tD uH1,0L@x,tD

u@x,tD2 + ��2 uH1,1L@x,tD

u@x,tD

- ��4 uH1,0L@x,tD3

u@x,tD3 + ��6 uH1,0L@x,tD uH2,0L@x,tD

u@x,tD2 - ��2 uH3,0L@x,tD

u@x,tD

��4 uH1,0L@x,tD3

u@x,tD3 - ��4 uH1,0L@x,tD uH2,0L@x,tD

u@x,tD2

y

{

zzzzzzzzzzzzzzzzzzzzz

In[4]:= transformedTerms = Expand@terms ��. 8D@u@x, tD, 8x, 2<D ® D@u@x, tD, tD,D@u@x, tD, 8x, 3<D ® D@u@x, tD, 8x, 1<, 8t, 1<D<D;

MatrixForm@transformedTermsDOut[5]//MatrixForm=

i

k



u@x,tD2 + ��2 uH1,1L@x,tD

u@x,tD

��6 uH0,1L@x,tD uH1,0L@x,tD

u@x,tD2 - ��4 uH1,0L@x,tD3

u@x,tD3 - ��2 uH1,1L@x,tD

u@x,tD


u@x,tD2 + ��4 uH1,0L@x,tD3

u@x,tD3

y

{


In[6]:= residual = Total@transformedTermsD

Out[6]= 0

Verifying the Hopf−Cole transformation, w = -2 Νu-1 ux where

ut = Νuxx , satisfies the homogenous Burgers’ variant wt - Νwxx + wwx = 0 with constant Ν



In[1]:= w@x_, t_D := ��-2 Ν


In[2]:= terms = Expand@8D@w@x, tD, tD,-Ν D@w@x, tD, 8x, 2<D,w@x, tD D@w@x, tD, xD<D;


i

k


��2 Ν uH0,1L@x,tD uH1,0L@x,tD

u@x,tD2 - ��2 Ν uH1,1L@x,tD

u@x,tD

��4 Ν2 uH1,0L@x,tD3

u@x,tD3 - ��6 Ν2 uH1,0L@x,tD uH2,0L@x,tD

u@x,tD2 + ��2 Ν2 uH3,0L@x,tD

u@x,tD

- ��4 Ν2 uH1,0L@x,tD3

u@x,tD3 + ��4 Ν2 uH1,0L@x,tD uH2,0L@x,tD

u@x,tD2

y

{


In[4]:= transformedTerms = ExpandBterms ��. :

D@u@x, tD, 8x, 2<D ® ��1

Ν D@u@x, tD, tD,

D@u@x, tD, 8x, 3<D ® ��1

Ν D@u@x, tD, 8x, 1<, 8t, 1<D

>F;MatrixForm@transformedTermsD

Out[5]//MatrixForm=

i

k



u@x,tD2 - ��2 Ν uH1,1L@x,tD

u@x,tD

- ��6 Ν uH0,1L@x,tD uH1,0L@x,tD

u@x,tD2 + ��4 Ν2 uH1,0L@x,tD3

u@x,tD3 + ��2 Ν uH1,1L@x,tD

u@x,tD


u@x,tD2 - ��4 Ν2 uH1,0L@x,tD3

u@x,tD3

y

{


In[6]:= residual = Total@transformedTermsD

Out[6]= 0

Verifying the complex exponential series solution u of the equation ut = Νuxx on the periodic domain [0,L)

Because the equation is linear, we may verify each term independently.



In[1]:= u@x_, t_, n_D := u`@nD ExpB-

ikjjjj ��2 n Π

L

y{zzzz2

Ν tF ExpBäikjjjj ��2 n Π

L

y{zzzz xF;

2 Hopf−Cole.nb

In[2]:= terms = Expand@8D@u@x, t, nD, tD,-Ν D@u@x, t, nD, 8x, 2<D<D;


i

k

jjjjjjjjjjjjjjjj

- ��4 ã

��2 ä n Π x

L- ��4 n2 Π2 t Ν

L2 n2 Π2 Ν u`@nD

L2

��4 ã

��2 ä n Π x

L- ��4 n2 Π2 t Ν

L2 n2 Π2 Ν u`@nD

L2

y

{

zzzzzzzzzzzzzzzz

In[4]:= residual = Simplify@8Total@termsD,Hu@x, t, nD �. x ® 0L - Hu@x, t, nD �. x ® LL,HD@u@x, t, nD, xD �. x ® 0L - HD@u@x, t, nD, xD �. x ® LL<, n Î IntegersD

Out[4]= 80, 0, 0<

Verifying that complex exponential series solution u of ut = Νuxx transformed by the Hopf−Cole variant w = -2 Νu-1

ux satisfies the homogeneous Burgers’ equation wt - Νwxx + wwx = 0 on the periodic domain [0,L) with constant Ν



We verify for only a small number of terms because of the nonlinear nature of the equation.

In[1]:= u@x_, t_D :=ân=-3

3u`@nD ExpB-

ikjjjj ��2 n Π

L

y{zzzz2


L

y{zzzz xF;

In[2]:= w@x_, t_D := ��-2 Ν


Hopf−Cole.nb 3



A very large output was generated. Here is a sample of it:

i

k

jjjjjjjjjjjjjjjjjjjjjj

��432 ä ã

- ��12 ä Π x

L- ��72 Π2 t Ν

L2 Π3 Ν2 u`@-3D2

L3ikjjjjjjjã

- ��6 ä �1� �1� x

L- ��

�1�

�1� u`@-3D+ã�1� �1�+�3�+�1�+ã

��1�

L-�1�

u`@3Dy

{zzzzzzz2

+ ��480 ä �3� u

`@-3D u`@-2DL3 H�1�L2 + ��

�1�

�1�+ �31� + ��

128 �5�

L3 H�1�L + ��

L3ikjjjjjjjjjã

�1�

�1�

Show Less Show More Show Full Output Set Size Limit...

In[5]:= residual = FullSimplify@8Total@termsD,Hw@x, tD �. x ® 0L - Hw@x, tD �. x ® LL,HD@w@x, tD, xD �. x ® 0L - HD@w@x, tD, xD �. x ® LL<, 8n Î Integers, t > 0, L > 0, Ν > 0<D;

MatrixForm@residualDOut[6]//MatrixForm=

i

kjjjjjjjj0

0

0

y

{zzzzzzzz

The closed form of the solution:

In[7]:= FullSimplify@w@x, tDD

Out[7]=ikjjj4 ä Π Ν

ikjjj3 u`@-3D + 2 ã

��2 Π Hä L x+10 Π t ΝL

L2 u`@-2D +

ã��4 Π Hä L x+8 Π t ΝL

L2 u`@-1D - ã


L2 u`@1D - 2 ã


L2 u`@2D - 3 ã

��12 ä Π x

L u`@3Dy{

zzzy{zzz �

ikjjjL ikjjju`@-3D + ã


L2 u`@-2D + ã


L2 u`@-1D + ã


L2 u`@0D +

ã��8 Π Hä L x+4 Π t ΝL

L2 u`@1D + ã


L2 u`@2D + ã

��12 ä Π x

L u`@3Dy{

zzzy{zzz

A particular Burgers’ equation solution for solver verification



4 Hopf−Cole.nb

In[1]:= L = 2 Π;Ν = 1;tinitial = 0; tfinal = 1; tstep = Htfinal - tinitialL�10;times = Range@tinitial, tfinal, tstepD;Nsamples = 1024;

u`@n_D :=

Ø≤≤≤≤≤≤≤≤≤≤≤≤≤∞

±

≤≤≤≤≤≤≤≤≤≤≤≤≤

u`@-nD n < 0

4 n � 0

2 n � 1

1 n � 2

0 n � 3

0 n > 3

In[7]:= u@x_, t_D :=ân=-3

3u`@nD ExpB-

ikjjjj ��2 n Π

L

y{zzzz2


L

y{zzzz xF;

In[8]:= w@x_, t_D = ��-2 Ν


In physical space, the closed form of the solution is:

In[9]:= FullSimplify@w@x, tDD

Out[9]= - ��

4 ä I-1 + ã2 ä xM I1 + ã3 t+ä x + ã2 ä xM1 + 2 ã3 t+ä x + 4 ã4 t+2 ä x + 2 ã3 t+3 ä x + ã4 ä x

If, in addition, we require the input arguments to be purely real, the closed form of the solution is:

In[10]:= FullSimplify@ComplexExpand@w@x, tDDD

Out[10]= ��

4 Iã3 t + 2 Cos@xDM Sin@xD2 ã4 t + 2 ã3 t Cos@xD + Cos@2 xD

In physical space, the initial condition is:

In[11]:= FullSimplify@w@x, 0DD

Out[11]= ��

4 HSin@xD + Sin@2 xDL2 + 2 Cos@xD + Cos@2 xD

We verify a zero residual for this solution:


residual = FullSimplify@8Total@termsD,Hw@x, tD �. x ® 0L - Hw@x, tD �. x ® LL,HD@w@x, tD, xD �. x ® 0L - HD@w@x, tD, xD �. x ® LL<, 8n Î Integers<D;

MatrixForm@residualDOut[14]//MatrixForm=

i

kjjjjjjjj0

0

0

y

{zzzzzzzz

In physical space the solution looks like:

Hopf−Cole.nb 5

In[15]:= Plot3D@w@x, tD, 8x, 0, L<, 8t, tinitial, tfinal<, 8PlotRange ® 8Full, Full, Full<,AxesLabel ® 8"x", "t", "w@x,tD"<,Ticks ® 880, Π, 2 Π<, Automatic, Automatic<<D

Out[15]=

We sample the solution to allow comparison with spectral methods:

In[16]:= sampleW@npoints_, times_D := TableB

ChopBNBTableBwB ��2 Π j

npoints, tF, 8j, 0, npoints - 1<FFF

, 8t, times<F;physicalData = sampleW@Nsamples, timesD;ListPlot3D@physicalData, 8PlotRange ® 8Full, Full, Full<,DataRange ® 880, L<, 80, tfinal<<,AxesLabel ® 8"j", "t", "w@xj,tD"<<D

Out[18]=

6 Hopf−Cole.nb

Transform each timestep into spectral space , using the signal processing convention that we normalize the forward transform by1/N:

In[19]:= spectralData = Chop@Map@Fourier@ð, FourierParameters ® 81, -1<D &, physicalDataDD;

In[20]:= waveNumberPairs@xs_D := Module@8n2floor = Floor@Length@xsD�2D,n2ceiling = Ceiling@Length@xsD�2D,<,H*Map from DFT coefficient array to one of 88wavenumber,coefficient<< *LSort@MapThread@List, 8

RotateLeft@Range@-n2floor, n2ceiling - 1D, n2floorD, xs<DD

D;spectralPairs = Map@waveNumberPairs, spectralDataD;spectralAbsPairs = Map@waveNumberPairs, Abs@spectralDataDD;spectralArgPairs = Map@waveNumberPairs, Arg@spectralDataDD;

� Spectrum detail for initial time:

In[24]:= ListLogPlotATooltip@spectralAbsPairs@@1DDD, 9PlotRange ® 8Full, Full<, AxesLabel ® 9kn, Abs@w`D=, Filling ® Axis

=E

Out[24]=

Hopf−Cole.nb 7

In[25]:= sortPairsByMagnitude@xs_D := Sort@xs, Abs@Last@ð1DD > Abs@Last@ð2DD &D;Take@sortPairsByMagnitude@spectralPairs@@1DDD, 16D �� TableForm

Out[26]//TableForm=

1 -1052.94 ä

-1 1052.94 ä

2 -875.924 ä

-2 875.924 ä

3 814.673 ä

-3 -814.673 ä

5 -222.369 ä

-5 222.369 ä

6 154.384 ä

-6 -154.384 ä

4 -115.767 ä

-4 115.767 ä

8 -52.1693 ä

-8 52.1693 ä

9 27.663 ä

-9 -27.663 ä

� Spectrum detail for final time:

In[27]:= ListLogPlotATooltip@spectralAbsPairs@@-1DDD, 9PlotRange ® 8Full, Full<, AxesLabel ® 9kn, Abs@w`D=, Filling ® Axis

=E

Out[27]=

8 Hopf−Cole.nb

In[28]:= sortPairsByMagnitude@xs_D := Sort@xs, Abs@Last@ð1DD > Abs@Last@ð2DD &D;Take@sortPairsByMagnitude@spectralPairs@@-1DDD, 16D �� TableForm

Out[29]//TableForm=

1 -388.425 ä

-1 388.425 ä

2 54.2387 ä

-2 -54.2387 ä

3 -8.44437 ä

-3 8.44437 ä

4 1.34427 ä

-4 -1.34427 ä

5 -0.214898 ä

-5 0.214898 ä

6 0.0343808 ä

-6 -0.0343808 ä

7 -0.00550127 ä

-7 0.00550127 ä

8 0.000880282 ä

-8 -0.000880282 ä

� Spectra details for all sampled times:

Obtain {n, t, value} triples for each sampled quantity :

In[30]:= spectralTriples = Flatten@MapThread@Insert@ð2, ð1, Table@8j, 2<, 8j, Length@ð2D<DD &,8times, spectralPairs<D, 1D;

spectralTriples = Select@spectralTriples, Abs@ð@@3DDD > 0 &D;spectralAbsTriples = Map@8ð@@1DD, ð@@2DD, Abs@ð@@3DDD< &, spectralTriplesD;spectralArgTriples = Map@8ð@@1DD, ð@@2DD, Arg@ð@@3DDD< &, spectralTriplesD;

In[34]:= ListPlot3DAspectralAbsTriples, 9PlotRange ® Full, AxesLabel ® 9kn, t, Abs@w`D==E

Out[34]=

Hopf−Cole.nb 9

a particular hopf-cole solution to burgers' equation

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