Hello, having discussed both explicit and
implicit methods for a parabolic PDE. Also
discussed about the local truncation error
and then how to estimate it for a particular
approximation, we must discuss more of these
approximations. So, what are the
properties any approximation any method should
have. What kind of a main features?
We have to pay attention. So, the main features
are consistency, stability and
convergence of particular approximation.
So, when I say consistency stability and convergence
which comes first. So, there could
be a debate, but in general when we talk about
any numerical method people
immediately talk about a convergence, whether
your method converges or not. So, may
be this is more intuitive, so immediate concern
for us. So, let us start with a convergence.
So, what do you mean by convergence, for example,
we come across several incidence
where the convergence has to be talked about.
So, when do you say something
converges.
.
.So, for example, in this context we have
approximated the corresponding PDE by a
particular finite difference method. So, then
we are solving, the solution obtained should
converge to the original solution. That is
the exact solution of the equation. So, let
us put
down formally and then proceed further to
talk about consistency and stability.
So, the title I put consistency stability
and convergence. So, as I mentioned I would
like
to talk about convergence first. So, as I
mentioned briefly a one step, we need a formal
definition that is why I would like to put
down here. One step finite difference scheme
approximating a PDE is a convergent scheme
if the solution of the 
finite difference
scheme say call it u i j converges to u of
x t, which is any solution of the PDE as delta
x
delta t goes to 0 so, this is a formal definition.
So, one step finite difference scheme approximating
a PDE is a convergent scheme if the
solution of the finite difference scheme u
i j converges to the exact solution which
is any
solution of the PDE as these parameters goes
to 0. So, as I mentioned once we talk about
the convergence, there are two more you can
see stability and consistency. So, even
consistency if you see consistency for example,
when you talk about stability of a
method, see suppose somebody is little perturbed
it is ok, but somebody is more
perturbed then definitely we say the person
is not stable or the system is not stable.
.
So, that means it is a general usage and even
in this context as well it is same. So, in
the
sense if you give small perturbations and
then if the small perturbations remain small
.then there is no issue, however if the small
perturbations grows up, grows up, grows up
and then blows up. So, then definitely the
particular system is not stable. So, let us
see
the formal definition of stability.
Stability, the error caused by a small perturbation
in the numerical method remains
bonded. Now, for a specific method this could
happen independent of any conditions.
However for a particular method, this could
happen subject to some conditions. So, the
remarks, this could happen 
unconditionally in the entire domain of a
domain of
definition or 
conditionally within a range. So, will when
we come to a specific method
we talk about this.
Then, the next consistency 
given a PDE L u equals to f 
approximated by say L i j u
equals to f, then the 
finite difference scheme l i j u equals to
f is consistent with the PDE.
If L psi L i j psi goes to 0 as delta x delta
t goes to 0 for psi smooth enough. So, that
mean, let us have a understanding given a
PDE L u equal to f approximated by this, then
the finite difference scheme l i j u equal
to f is consistent with the PDE if l psi minus
l i j
psi goes to 0 as this parameter goes to 0
for psi smooth enough.
.
So now what could go wrong, what could go
wrong? So, that means a scheme
approximating a PDE may be stable, but 
has a solution that converges to the solution
of a
different PDE different equation. So, this
is inconsistent that means a scheme
.approximating a PDE may be stable, but the
solution converges to some other some other
PDE as the mesh length goes to 0. So, this
is inconsistency.
.
So, let us consider an example. So, let us
say our operator is, first order for the time
being then L psi is, then L i j say 
forward space forward time. So, accordingly
we get L i
j to be. So, we get psi j plus 1 j by delta
t plus constant times forward space . So,
this is
our L i j, right now in order to compute the
error we have to expand in Taylor series.
So,
let us expand psi i j plus 1. So, this is
evaluated at t squared again evaluated, at
this point
plus. So, similarly 
this we have done number of times. So, now
if we substitute then L i j
psi is L i j psi becomes we can just straight
away substitute.
..
For example, we can see this sitting here.
So, this get cancel. So, first term here delta
t
also cancels. So, first term is dou psi dou
t and here this term cancel with this delta
x
cancels with this. So, first term is a times,
this is just an algebra. So, if you do that
we do
that we get you can verify some series of
notation you could have written simply one
we
go therefore, L psi minus L i j psi equals
to. So, this 
will be which goes to 0, therefore,
scheme is consistent. So, the scheme is consistent
now we should talk more about
consistency, stability, convergence, what
is a interplay among these three properties.
.
.So, there is an interesting theorem in numerical
analysis which talks about this. However
there are some restrictions in order to apply
this theorem. So let us see which ensures
a
what. So, the theorem is Lax Richtmyer Equivalence
Theorem. So, this also called
fundamental theorem of numerical analysis.
So, before putting the statement the net gist
of this consistency plus stability both sides
I am putting consistency, stability ensures
convergence, convergence ensures consistency
and stability, this is the gist of the this
theorem. However as I mentioned there are
some restrictions.
So, let us put down formally if a linear finite
difference scheme is consistent with a well
defined linear IVP that is a initial value
problem. Then 
stability guarantees convergence
as mesh length goes to 0. So, this says if
a linear finite difference scheme is consistent
with a well defined linear IVP, then stability
guarantees convergence as mesh length
goes to 0. So, method that is consistent stable
convergences guaranteed but, in general it
is both ways however the main restriction
is the remark is linear.
So, this is quite important because in general
people look for non-linear because several
of the challenging problems involve non-linear
stuff. However this is restricted to linear
problems. Moreover consistency stability ensuring
convergence is trivial in some sense
however the theorem assures the converse convergence
implies consistency and stability
as well.
.
.So, let us proceed to a little complicated
example the second order. So, that we get
some
idea. Let this be approximated 
from delta t I am using k. So, you may be
thinking every
time the same example so for a change we have
gone to a different example you can see.
So, this looks little heavy let us see how
we proceed. So, the story is the same, we
have
to expand in Taylor series and collect the
coefficients etcetera.
So, let us consider term by terms. So, this
is one term and this and this one term and
this
is another term for me. So, u i j plus 1,
this if you consider u plus k dou u dou t.
So, I am
dropping the evaluation at x i t j, it is
understood 
minus u. So, this become u, u get
cancelled, you get 2 k right then this k square
by 2 with a negative sign this get
cancelled. So, the next term by 3, here actually.
So, there is this should be 3 so then the
next term.
.
So, if you consider this is just a labor alternate
signs. So, ultimately we get this is j plus
1, all should be t there apologies 
this is 2 u and all this k. So, then we are
left with 2 more
terms this and this. So, that also we can
compute 
so many times we can guess. So, there
will be u and u that cancels and here with
a plus h and minus sign and here with a minus
h so it will be 2 times. So, we get there
is a this is a plus sign then u u that will
be 2 u
then plus h minus h they get cancelled, plus
h square term becomes twice. So, h square
by 2 is twice then cubic terms cancels.
..
Now, using all this t i j become 
minus 1 over h square. So, I am giving only
few terms.
So, this is 
an important exercise so even though there
is some labor involved better to do
this. So, this clubbing we can see this get
cancelled. So, h square cancels you get dou
u
by dou t minus this term is what I put. Then
the next term I am going to put is this one,
then the next term is this, then I am putting
this, then we have this term.
Now, case one see even though we clubbed these
terms because this is our actual
equation, but unfortunately there is one more
term with a first derivative first derivative
with respect to t right. So, we have to analyze
carefully suppose k is lambda h. So,
lambda is our grid parameter say k is lambda
h. So, then t i j reduces to 
plus k is lambda
h. So, you have lambda by h plus 2 lambda
square 2 lambda square k is lambda h. So,
you have you have one term. So, let me check
k is lambda h. So, you have this 
then
etcetera. Now, as h goes to 0. So, this is
trouble maker because as h goes to 0 this
is not
getting observed in this therefore, inconsistent
therefore it is inconsistent.
..
Suppose case 2 k is lambda h square then t
i j, this goes to 0 as h goes to 0. So, therefore
consistent therefore, in this case the method
is consistent it is agreeing with the original
PDE. So, this test is very much important
if you come across any finite difference
scheme approximating a given PDE, we do this
test for consistency. Now, having done
consistency check let us see whether we can
discuss about stability of a method.
.
Stability and convergence, so convergence
via differential equation for the error, so
this
is called direct method. So, consider so I
am putting a bar just to make sure that this
is
.exact then initial condition see u bar is
given, then boundary condition u bar some
at is
given u bar b t is given. So, in this case
a equals to 0 b equals to 1. So, this is for
all t
now consider finite difference 
approximating star by u i. So, now we would
like to talk
about convergence right. So, what would happen
at each grid point there will be some
error.
.
So, at mesh points, this is a discretized
and say this is exact, they differ by this
is the
error then following this answers. Now, consider
our scheme 
substitute this in this
scheme we get. So, left hand side u i j plus
1 minus u i j, so you can convert this into
equivalent form. So, that will be u i j plus
1 equal to lambda u i minus 1 j plus 1 minus
2
lambda, this is our standard form. So, then
let us substitute these answers. So, then
what
will happen we get here u i j plus 1 minus.
So, we retain error to the left hand side
we get lambda, then plus the exact solution
we
get this. Now, our aim is to get an equation
for the error and then estimate the error
and
see whether we can make it bounded whether
conditionally or unconditionally. So, that
we talk about the convergence and stability.
..
So, now using Taylor series which is a only
first term I am writing. So, this I will put
it
as a error term x i plus some theta 1 h. So,
this is the error term this is evaluated at
x i t j.
So, then this is Taylor's remainder term then
say some theta 3 k. So, h is increment for
a
space and k i is increment for time. So, where
this is standard Taylor's remainder
conditions, so then using this in our error
equation 
plus k times, so where from these 2
we get this. So, this is 
double star is a difference 
equation for e i j. So, this is the
difference equation for e i j.
.
.Now, let e j is max of e i j. So, that is
at j th level and m is max 
of then, if you carefully
observe. So, we have 1 minus 2 lambda therefore,
for lambda less than or equals to half
coefficients of e i j in double star are positive
or 0. This is an important remark therefore
so for this we need positive or 0 because
we need an estimate. So, therefore for this
range
coefficients positive or 0 therefore, mod
of this is less than or equals to lambda mod
e i
minus 1 j plus 1 minus 2 lambda j plus lambda
plus k m, how we are getting carefully
observe. So, let me repeat we have considered
the answers this is a approximated with
exact differ by an error.
At each grid point we play with the index,
then this is our difference scheme. Then you
substitute these answers in the difference
scheme we get this. Then we expand in Taylor
series and substituted back. So, we get this
now this is a difference equation for e i
j.
Now, let at j th level the maximum error because
at each grid point the error is different.
So, the maximum is E j and correspondingly
you call this. Now, we are estimating the
bound. So, this is we called already m therefore,
we get this 
E j. So, this is 
right now we
obtained.
.
This is less than or equals to E j plus KM
and this 
which is true for every i hence true for
max. Therefore so, max is E j plus 1 are you
getting. So, this is true for every I therefore,
true for max. So, which is nothing but capital
E j plus one, so this is E j plus 
K M, now
you iterate. So, this if we do here 2 times
we get 3 times so, on if you do. Now, what
will
.be this any idea E 0 error at t equal to
0. So, there is no error and j k is t, since
this then
this is t M. So, further when h goes to 0,
K is lambda h square goes to 0, M goes to
at i j.
This also goes to 0 therefore, M goes to 0
therefore, E j plus 1 goes to 0.
That means any E j, as E j this implies u
goes to u bar as h goes to 0 when lambda is
less
than or equals to half and t finite. So, this
is our net conclusion u is agreeing with the
exact only when lambda is less than or equal
to half. So, the remark is lambda greater
than half the error blows up you can verify.
So, in this case, this is conditionally that
means the error is not growing, therefore
it is a stable only in this range. So, this
is called
conditional stability.
Suppose, you do not get any condition and
then we say it is unconditionally stable right.
So, we estimate the bounds the domain in which
the method is stable by using this error
answers. So, that means we formulate the corresponding
differential equation
corresponding to a given finite difference
scheme. Then we find estimates and we find
the domain within which the method is stable.
So, we discuss more, there are other
methods to talk about stability. So, we discuss
in the coming lectures until then.
Thank you.
.
