Implicit method

Numerical Methods in Geophysics:Implicit Methods

What is an implicit scheme?

Explicit vs. implicit scheme for Newtonian Cooling

Crank-Nicholson Scheme (mixed explicit-implicit)

Explicit vs. implicit for the diffusion equation

Relaxation Methods

Numerical Methods in Geophysics Implicit Methods

What is an implicit method?

Let us recall the ODE:

),( tTfdtdT

=

Before we used a forward difference scheme, what happensif we use a backward difference scheme?

),()(1jj

jj tTfdtOdtTT

=+− −

),(dt1 jjj-j tTfTT +≈⇒



or

Is this scheme convergent?

11 )1( −− +≈

τdtTT jj

jj

dtTT −+≈ )1(0 τ

Does it tend to the exact solution as dt->0? YES, it does (exercise)

Is this scheme stable, i.e. does T decay monotonically? This requires

11

10 <+

<

τdt



11

10 <+

<

τdt

This scheme is always stable! This is calledunconditional stability

... which doesn’t mean it’s accurate!Let’s see how it compares to the explicit method...



0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5

Time(s)

Tem

pera

ture

dt=1.41; tau=0.7

Explicit unstable - implicit stable - both inaccurate

red-analyticblue-explicit

green-implicit



Explicit stable - implicit stable - both inaccurate

0 2 4 6 8-0.5

0

0.5

1

Time(s)

Tem

pera

ture

dt=1; tau=0.7red-analyticblue-explicit

green-implicit



Explicit stable - implicit stable - both inaccurate

0 2 4 6 80

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1

Time(s )

Tem

pera

ture

dt=0.5; tau=0.7red-analyticblue-explicit

green-implicit


0 2 4 6 80

0.1

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0.9

1

Time(s )

Tem

pera

ture

dt=0.1; tau=0.7


Explicit stable - implicit stable - both accurate


green-implicit

It doesn’t look like we gained much from unconditional stability!


Mixed implicit-explicit schemes

We start again with ...

),( tTfdtdT

=

Let us interpolate the right-hand side to j+1/2 so that both sides are defined at the same location in time ...

2),(),( 111 jjjjjj tTftTf

dtTT +

≈− +++

Let us examine the accuracy of such a scheme using our usual tool, the Taylor series.



... we learned that ...

)(62

33

32

2

21 tO

dtTdt

dtTdt

dtdT

tTT

jjj

jj ∆+⎟⎟⎠

⎞⎜⎜⎝

⎛∆+⎟⎟

⎠

⎞⎜⎜⎝

⎛∆+⎟

⎠⎞

⎜⎝⎛=

∆

−+

... also the interpolation can be written as ...

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡∆+⎟⎟

⎠

⎞⎜⎜⎝

⎛∆+⎟

⎠⎞

⎜⎝⎛∆+=+ + )(

22

21

21 3

2

22

1 tOdt

fdtdtdftfff

jjjjj

),( tTfdtdT

=dt

tTdfdt

Td ),(2

2

=since =>



... it turns out that ...this mixed scheme is accurate to second order!

The previous schemes (explicit and implicit) were all first order schemes.

Now our cooling experiment becomes:

)(21

11

jjjj TT

dtTT

+−≈−

++

τ

)2

1()2

1(1 ττdtTdtT jj −≈+⇒ +

leading to the extrapolation scheme



⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−≈⇒ +

τ

τ

21

21

1 dt

dt

TT jj

How stable is this scheme?The solution decays if ...

1

21

21

1 <⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−<−

τ

τdt

dt



1

21

21

1 <⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

−<−

τ

τdt

dt

This scheme is always stable for positive dt and τ!If dt>2 τ, the solution decreases monotonically!

Let us now look at the Matlab code and thencompare it to the other approaches.



t0=1.tau=.7;dt=.1;dt=input(' Give dt : ');

nt=round(10/dt);

T=t0;Ta(1)=1;Ti(1)=1;Tm(1)=1;

for i=1:nt,t(i)=i*dt;T(i+1)=T(i)-dt/tau*T(i); % explicit forwardTa(i+1)=exp(-dt*i/tau); % analytic solutionTi(i+1)=T(i)*(1+dt/tau)^(-1); % implicit Tm(i+1)=(1-dt/(2*tau))/(1+dt/(2*tau))*Tm(i); % mixedend

plot(t,T(1:nt),'b-',t,Ta(1:nt),'r-',t,Ti(1:nt),'g-',t,Tm(1:nt),'k-')xlabel('Time(s)')ylabel('Temperature')



0 2 4 6 8 10-1

-0.8

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-0.2

0

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1

Time(s)

Tem

pera

ture

dt=1.4; tau=0.7red-analyticblue-explicit

green-implicitblack-mixed



0 2 4 6 8 1-0.5

0

0.5

1

Time(s )

Tem

pera

ture

dt=1; tau=0.7 red-analyticblue-explicit




0 2 4 6 80

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Time(s)

Tem

pera

ture

dt=0.5; tau=0.7 red-analyticblue-explicit



0 2 4 6 8 100

0.1

0.2

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0.9

1

Time(s)

Tem

pera

ture

dt=0.1; tau=0.7


1 1.2 1.4 1.6 1.8 20.05

0.1

0.15

0.2

Time(s )

Tem

pera

ture

dt=0.1; tau=0.7



The mixed scheme is a clear winner!


The Diffusion equation

2

2

xCk

tC

∂∂

=∂∂

k diffusivity

The diffusion equation has many applications in geophysics, e.g. temperature diffusion in the Earth, mixing problems, etc.

A centered time - centered space scheme leads to a unconditionally unstable scheme!

Let’s try a forward time-centered space scheme ...



)2( 111 n

jnj

nj

nj

nj CCCsCC −++ +−+=

where

2dxdtks =

how stable is this scheme? We use the following Ansatz

dtieT ω= ikdxeX −=after going into the equation with

nmkmdxndtinm TXeC == − )(ω



... which leads to ...

0)21( 11 =+−+− −sXssXT

and

sT 41−≤

so the stability criterion is

)2/(21 2 kdxdts ≤⇒≤

This stability scheme is impractical as the required time step must be very small to achieve stability.


The Diffusion equation (matrix form)

any FD algorithm can also be written in matrix form, e.g.

)2( 111 n

jnj

nj

nj

nj CCCsCC −++ +−+=

is equivalent to

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+

+

+

nJ

nj

n

nJ

nj

n

C

C

C

s

C

C

C 1

1

1

11

121


The Diffusion equation (matrix form)

... this can be written using operators ...

nnn cLcc +=+1

where L is the tridiagonal scaled Laplacian operator, if the boundary values are zero (blank parts of matrix

contain zeros)



... let’s try an implicit scheme using the interpolation ...

2)()()

2,( tCdttCdttxC ++

≈+

and

( )nj

nj

nj

nj

nj

nj

nj

nj CCCCCC

dxk

dtCC

1111

1112

1

222 −+

+−

+++

+

+−++−=+

... so again we have defined both sides at the same location ... half a time step in the future ...



... after rearranging ...

nj

nj

nj

nj

nj

nj CsCsCsCsCsCs 11

11

111 )2/()1()2/()2/()1()2/( −+

+−

+++ +−+=−++−

... again this is an implicit scheme, we rewrite this in matrix form ...

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−

+

+

+

nJ

nj

n

nJ

nj

n

C

C

C

sss

C

C

C

sss

1

1

1

11

)2/()1()2/()2/()1()2/(

... or using operators ...

nn cVcU =+1



... and the solution to this system of equations is ...

nn cVUc 11 −+ =

... we have to perform a tridiagonal matrix inversion to solve this system.Stability analysis using the Z-transform yields

))(2/)1(())(2/)1(( 1111 −− ++−=+−+ XXssTXXssT

)1/()1( ββ +−=⇒T)cos1( θβ −= s

... where we used ...

2/)(coscos 11 −+=∆= XXxkθ



)cos1( θβ −= s)1/()1( ββ +−=⇒T

... T describes the time-dependent behaviour of the numerical solution, as before we find ...

1|)1/()1(||| ≤+−= ββT

... which means the solution is unconditionally stable ...

... this scheme implies that the FD solution at each grid point is affected by all other points. Physically this could be

interpreted as an infinite interaction speed in the discrete world of the implicit scheme!


The Relaxation Method

Let us consider a space-dependent problem, the Poisson’s equation :

Fzx −=Φ∂+∂ )( 22

applying a centered FD approximation yields ...

( ) Fdx jijijijijiji −=Φ+Φ−Φ+Φ+Φ−Φ +−+− ,1,,11,,1,2 221

rearranging ...

44

2,1,11,1,

,Fdxjijijiji

ji +⎟⎟⎠

⎞⎜⎜⎝

⎛ Φ+Φ+Φ+Φ=Φ +−+−

... so the value at (i,j) is the average of the surrounding values plus a (scaled) source term ...


The Relaxation Method

... the solution can be found by an iterative procedure ...

44

2,1,11,1,1

,Fdxji

mji

mji

mji

mm

ji +⎟⎟⎠

⎞⎜⎜⎝

⎛ Φ+Φ+Φ+Φ=Φ +−+−+

... where m is the iteration index. One can start from an initial guess (e.g. zero) and change the solution until it doesn’t

change anymore within some tolerance e.g.

ε<Φ−Φ + || 1 mij

mij

If there is a stationary state to a diffusion problem, it could be calculated with the time-dependent problem, or with the relaxation

method, assuming dC/dt=0. What is more efficient? (Exercise)


Implicit Methods - Summary

Certain FD approximations to time-dependent partial differential equations lead to implicit solutions. That means to

propagate (extrapolate) the numerical solution in time, a linear system of equations has to be solved.

The solution to this system usually requires the use of matrix inversion techniques.

The advantage of some implicit schemes is that they are unconditionally stable, which however does not mean they

are very accurate.

It is possible to formulate mixed explicit-implicit schemes (e.g. Crank-Nickolson or trapezoidal schemes) , which are

more accurate than the equivalent explicit or implicit schemes.


Implicit method

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