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The example of the linear wave equation has shown that straightforward application of
finite difference approximation to a given differential equation, does not necessarily
give us proper results. We have seen, in fact, three different ways of discretizing the equation,
and we have seen that in all cases, the solution has been not so exemplary. The solution that
we got from the by applying the final differences approximations, and then doing the corresponding
computation has not given us the correct solution or anything resembling the correct solution,
except for one particular combinational parameters, where, they, the courant number was equal
to 1. So, this should give us, this should tell us that they something more that we need
to do in terms of what kind of approximations that we make before we can start solving them.
So, what it a, what it amounts to saying is that all though we can write down approximations,
final difference approximation of given accuracy for a given derivative, not all approximations
will give as proper solutions. There some which are better than some others. How can
we say which is better and which not good, because in the general case, when we try to
solve Navier-Stokes equations using this particular way, then we do not know the solution, and
if we knew the solutions, we would not be doing the CFD solution at all.
So, if we do not know the solution and if your numerical method is likely to give rise
to errors, then how can we accept such a solution method? Because how can we distinguish the
errors from the real solution when we applied t to a practical case, where we do not know
the solution and where we depend on the accuracy of the method to provide the correct solution?
So, we need to have some method, some way of saying a formal way of saying that if we
do this kind of approximations and if we follow this kind of procedure, then we will get the
satisfactory solution. So, that kind of assurance necessary and that kind of assurance is what
we going to discuss in this class. So, we are looking at the general case of
the scalar transport equation with a time dependent term, advection terms, diffusion
term, source term, and we are looking at a specific idealization of that in form of the
wave equation. And when we look at that, there seems to no reasons why our approximation
should give a wrong answer, except for the fact that we have a truncation error; except
for the fact that we are approximation the derivatives with finite difference approximation
of first order or second order, and in the process of doing this approximation, we are
neglecting certain terms and is it because of the combination of this approximation that
exactly causing is. So, this one question and the other question
is that we have to ask is that is there a combination which will give us a proper solution?
So, we have to look at what kind of combination and is it all question of accuracy or is there
something more to this and is there any guarantee? For example, that we are solving the right
equation is a solution - the computed solutions - not good because we are not solving the
right equation or is it not good because we are solving it only approximately, because
any numerical computation, any computer based solution with finite precision of accuracy
is always going to be approximate. So, is it because of those kind of consideration?
What is a guaranty that computed solutions will approach or will be equal to the exact
solution of the governing equation? So, these are the question that we have in mind and
these are the question that will try to address through formal procedure of the analysis of
the discretized equation. So, now, we can begin to understand, begin
to see some points that we have to consider. Firstly, we are solving the partial differential
equation in an approximate way, in the approximate way by writing a approximate form, of the,
of the derivative and we solving it in a discrete way, we are not looking a solution which is
continuous; we are looking at only discrete points and we are doing method of solutions
which has finite precision. So, we can see that the computed solution
is different from the exact solution in three different ways - one is at instead of looking
at phi of x t, we are looking at phi of x i t n. So, it is at a discrete point, and
second thing is instead of getting a exact solution at x i t n, we only getting an approximate
solution which satisfies only the discretized form or the approximate form of the governing
equation. On top of that, we are getting a solution which is not even an exact solution
of the approximate formal equation, because we are not doing the exact arithmetic, we
have only finite precision arithmetic. So, there are these kinds of errors, this
kind of discrepancies may be playing have a good solution, because of which, we are
getting a wrong answer. So, let us therefore, try to answer question related to this. So,
this kind of analysis, this analysis of the condition of a computed solution approaching
the exact solution partial differential equation is put up in three different stages using
three conditions known as the consistency condition and the stability condition and
the convergence condition. When we say consistency condition, we are
talking about is the discretized equation a very good replica of the actual partial
differential equation. So, that is, can we claim that the discretized equation would
approach the exact equation if we make a discretization very small? So, we are saying that we are
not looking at phi of x y, we are looking at phi of x i t n. So, that is at discrete
points. So, if we had the capacity to make delta x
very small delta and delta to very small, so that we are approaching almost a continuous
variation of x i and t n. So, in that case, we can claim that our discretized equation,
the approximate form of the equation that we are solving. Can we say that this approximate
form will approach the partial differential equation? So, this is known as the consistency
equation, consistency between the partial differential equation or the governing equation
and the discretized equation.
So, we will put this as consistency where we are looking at the assurance that the discretized
equation, where is the same as partial differential equation in the limit as delta x and delta
t tending to 0. The next question is about the solution method. If provided, we have
this consistency condition satisfied; provided we can show that we are solving the correct
equation which has the property of going to very small delta t and approaching the exact
solution for very small delta t and delta x.
Now, this point is important because we saw in the examples, especially with the FTFS
method and the FTCS method which is central in space methods. No matter how small that
delta t was; we were still getting a wrong answer. So, even if those methods are such
that even if we make delta t and delta x very small in the computed solution, even then
the computed solution will not give us a proper solution, which compares favorably with the
exact solution. So, it is not that if in all conditions if we can make delta t delta x
and ah other discretization very small, we cannot guarantee that the solution would be
automatically approach this. So, that is why this query about whether or
not are the discretized equation would approach the exact equation for very small delta x
and delta t is a justified query. So, we will have to considerate that and that is why is
consistency condition is. Now, once we have satisfied ourselves that we are solving an
equation which is consistent, then the second question comes are we solving it properly?
Are we solving it properly in the sense that does the discretization, does the discretized
equation have a property of amplifying errors. Again when we go back to the example of FTFS
and CT and FTCS and even FTBS, we saw that there are, it, the solutions seems to be getting
out of control, out of bounds. The expected variation of phi was between 0 and 1, and
the computed variation of phi was something like plus or minus 400 within a few time steps,
within 5 time steps in case of FTS method for certain conditions.
So, it seems to be amplifying errors, whereas, the true solution denotes something which
is nicely being uniformly being carried forward in the x direction. So, the true solution
of the governing equation is not going to does not have this feature of amplifying errors,
but the discretized form, at least in some discretization seems to be having this nature
of amplifying errors. So, this brings us to the second condition
stability. So, does the discretized equation have the property of stability, whereby errors
that introduced from whatever reason; from finite precision arithmetic or from rounding
of errors or from boundary connections or from truncation error, errors related to discretization,
all this kind of errors or even simple typographically mistakes, those kind of things. Such errors
do they have the property of getting amplified or is a schemes stable in such a way that
those errors will be suppressed and attenuated, and finally, as time progresses we get to
stable solutions.
So, the second question then therefore is stability. So far what we have been looking
at is whether the discretized equation is very good approximation of the partial differential
equation, and we are looking at whether the discretized equation, the computed solution
of the discretized equation has a property of amplifying or attenuating the errors. We
have not said about the comparison between the computed solution and the exact solution.
So, we are talking only about the method of solutions.
So far in the consistency and stability conditions, but now is the time for us to look at the
equivalence between the computed solution and the exact solution of the governing equation
at those discrete points at which we are looking at it. This condition where we are saying
that the computed solution at the discrete points would match with the exact solution
at the same discrete points is known as the convergences conditions, convergence.
So, if you have scheme which is convergent, then we can say that the computed solution
would approach the exact solution at x i t n. So, the convergence criteria is about the computed solution and the exact
solution. Now, how is it different from stability? Stability is about, is only about the discretized
equation. So, the computed solution of the discretized equation – DE: discretized equation - approaches the exact solution of
the discretized solution. So, this stability condition refers to it poses the query on
whether or not the computed solution with all those errors and amplification features;
whether or not it would be an exact solution of this. And convergence is talking only about
the computed solutions and the exact solution of the partial differential equation.
Consistency is not about solution, it is only about whether the discretized equation approaches
the partial differential equation, the governing equation. So, there are three different conditions;
we are addressing three different issues. Ultimately, we want the convergence; ultimately,
we want the computed solution to match with the exact solution of the partial differential
equation at the points x i and t n, but in order to get here, we have to pass these to
hurdles. We to make sure that we are getting this computed solution in the right way by
ensuring that we are solving the right kind of equation and the equation that we are solving
has a right kind of stabilization or attenuation properties so that errors do not amp get,
get, amplified and spoil the solution. So, this condition, this triple criterion
here will, if we can satisfy this condition, - the consistency condition and the stability
condition and convergence condition - then in such a case we can say we can have confidence
in our computed solutions. We can have confidence that the computed solution will approach the
exact solution of the partial differential equation at those grid points we have computing.
It does not mean the computed solution is error free, because of any computation, we
must have some delta x and some delta t; so, that means that there is some discretization
errors; there is an error of approximation that is there, but with this, if we satisfy
the consistency condition and stability condition, then we know that this error is contained
bounded; it does not get amplified and it does not gives us any problem.
So, in that sense, if you are able to satisfy the convergences condition while simultaneously
satisfying this. Then we can say that the computed solution at space point i and time
point n will approach the exact solution which I am denoting by this, at that particular
point subject to plus discretization error, or we can call this as truncation error, which
depends only on delta t and delta x and these are in our control. Once we have computed
solution which is not really sensitive to delta t and delta x in terms of delivering
a proper solution through the stability, then this is in our control and we can reduce delta
t and delta x to as low as possible as we want and then we can make our computed solution
approach the exact solution. So, and we can minimize the error as to as
low as what we desire as it is required for the solution. So, in, in this, under this
condition, we can say that a grid independent solution. So, what we mean by grid independent
solution? A grid a solution which does not depend on either delta x or delta t so which
does not vary with successive reduction of delta x and delta t.
So, let us say that we square dart and we have a 20 by 20 grid; so, that means that
same length, we have 20 intervals in that so that gives you a certain delta x. You make
it 40 by 40, you are reducing delta x by half; you make it 80 by 80, 160 by 160, 1000 by
1000, and what you will see is that as u make the grid finer and finer, obviously, the that
computation cost increases because the matrix a that leads to the differential, equation,
algebraic equations that becomes bigger and bigger, but if we did not have any contents
on, on, the computing resources, then you can make it as high as possible and does you
keep on increasing the grid size or the number of grid points, and as you keep on decreasing
the delta x and delta t, then the parameter that is of interest to you. For example, you
are looking at the pressure gradient, pressure drop in, in, that particular that for given
flow rate, so, in a such a case, you will find that after some number of grid points,
the computed pressure grading does not really change much to you reduce the spacing by another
half; it does not give you any significant variation in the parameter of interest.
So, at that point, you can say that my computed solution which is of interest to me, which
I want to get from my c f d calculation is no longer sensitive to the delta x and delta
t. So, that is what I call as a grid independent solution. So, and grid independent obviously
means not only delta x, but also delta t if you are looking at time dependent problem.
And we can say that grid independent solution is almost equal to the exact solution of the
governing equation. So, the satisfaction of these things is as powerful as this; that
if we are able to generate grid independent solution, then we can rest assured that it
almost as good as the exact solution of the governing equation.
So, this is the kind of power, that is, there in these three conditions that if you are
able to demonstrate the consistency, stability and convergence of a numerical solution procedure,
then we can claim that the grid independent solution which we obtain, from the computed,
from the procedure is, is, the exact solution of the governing equation, but the caveat
is that if we want have this, this is the desired condition this the desired guarantee
that we would like to have. In order to do this, we need to prove these three; these
three are not related in a way.
And these three for the case of linear problem, for a linear partial differential equation,
then there is a theorem called the equivalence theorem of Lax. Lax is obviously a mathematician,
and it is not the laxity on the part of Lax, he is actually made a very nice theorem, which
is, for a linear well posed mathematical problem of in the form partial differential equation,
the equivalence theorem says that consistency and stability are sufficient, sufficient,
to guarantee convergence. This in a way bit of layman speak, but the point is that if
we are able to show the consistency of a discretization and the stability, stability, of the computational
form computed solution of this, then we can rest assured that we will have convergence
condition is satisfied and convergence is guaranteed, and under those condition, we
can say that the difference between the computed solution and the exact solution has a discretization
error, which is, which goes to 0 in the limiting case as this thing goes to 0 and in the limiting
case as the nth ending infinity. So, in that senses, the two will approach.
So, this is a very useful thing, but like all mathematical theorems, it is also not
of a great practical value from the fact that this is limited only to a linear equation.
So, when we have linear partial differential equation, then this theorem works. For a non-linear
problem which is what we normally have in our c f d, then this theorem does not work;
obviously, it is only for the linear thing,but in such a case, we have to do a local linearization;
we can convert the non-linear problem into a linear problem, and then, for that linearized
problem, we can check for consistency using this.
So, in that senses, it is, it is not something we can brush aside as being useless, but at
the same time, it, it, tells us that for a under linearized condition, if you are able
to check for consistency and stability, then we can get the guarantee of convergence, and
while this theorem has those kind of limitations. Experience has shown that if you apply standard
methods for the analysis, consistency and stability and if you follow these as practical
guidelines for determining the parameters that we are going to employee like delta t
and delta x, then in most of the cases, we do get a good solution, but it is not guaranteed
and we also have to worry about other reasons for getting a non convert solution and solve
it, but in general, this is a good point and it is a it is an indication of why and how
problems may arise. So, let us, let us take it for granted that
this is a useful thing for us, useful pointer in terms of what kind of approximation that
we have to, that we are allow to make for derivatives so as to get a proper solution,
and let us investigate this further, and let us investigate how we can prove the consistency
and stability of our discretization, because now the problem reduces to how we can demonstrate
consistency. If there is method of which we can demonstrate the consistency of a method
and, if we can, if they is procedure by which we can demonstrate the stability, then we
can apply these methods to each discretization that we propose and see whether the discretization
satisfies these conditions. If it does, then we can proceed with the solution procedure;
otherwise, if it does not satisfy, in such a case we have look for alternate ways of
discretization.
So, now, let us consider how we can do consistency. We have said consistency is a condition which,
which, tries to look at whether or not the discretizated equation will approach partial
difference equation under the condition of delta x and delta t tending to 0 and any other.
If you have a three dimension equation, then delta y tending 0 and delta z tending to 0
and so on. And so, we are looking at, we have to see under what conditions the discretized equation tends to the partial
differential equation. So, when you pose a question like this, then
we can see, we can start by examining what is the different between the discretized equation
and partial differential equations. We know that the, the, two differ only by the fact
that the discretized equation is in approximation of the partial derivatives of the derivatives
in appearing in the equation and each derivative has approximations which is given by the Taylor
series approximation. So, if you go back to the Taylor series expansion
and then look at error that as arisen in making particular approximation, then we can say
that the difference between the discretized differential equation and partial differential
equations is the truncation error, that is, resulting from the approximation of each derivative
that appears in the partial differential equation. So, if you have five derivatives, then each
derivatives is approximated in a particular way using a finite different approximation
and that each derivatives will have its own truncation error, and the combination of the
all these truncation errors is the difference between the discretized equation partial differential
equation. So, if we can show that the truncation error
goes to 0 as delta t delta x delta y delta z tend to 0, then we can claim that the discretized
equation will approach partial differential equations. So, we can say that the truncation
error we, we, are saying that truncation error, we are calling it truncation error, because
in writing a finite difference approximation, we are truncating the Taylor series which
has infinite number of terms to first three terms or first four terms and so on. So, rest
of the terms is neglected; so, the neglected part is a truncation error.
So, the truncation error of the discretized equation between the discretized equation
and partial differential equation which we are calling as say DE minus PDE; this should
tend to 0 as delta x tends to 0 and delta d tends to 0 in that case.
So, the consistency condition does not say that the two must be equal, it only says that
it has the property that the difference will tends to 0 in the limit as delta x tends to
0 delta t tends to 0. We are not claiming that there is a, there will not be any difference.
We are only saying that solution method as the capability of reducing the error to as
low as is required as is possible. So, it only that the difference will tend to 0 as
delta x is made to go to 0 and delta t is made to go to 0. So, then in order to verify
the consistency, we have look at the truncation error.
So, let us do that for the simple case, the method is very general. We have considered
dou phi by dou t plus u dou phi i dou x equal to 0, and in the, let us consider case of
FTFS, that is, forward in time and forward in space. Then why would be consider this?
Because this has given as an immediately non sequential answer for whatever values of delta
t and whatever values of Courant number that we have taken, and not only that, it has really
blown up very fast as compared to FTBS. So, we can take this as in example and see
whether the FTFS approximation for this particular equation has the, will satisfied the conditional
consistency. So, when we write the FTFS approximation,
we are writing this as phi i and plus 1 minus phi i n by delta t plus u is being taken as
constant. In the example, we have taken it as a 1 meter per second phi i plus 1 n minus
phi i n by delta x equal to 0. Now, this is an approximation, first order approximation
and that results from the expansion of phi i n plus 1 as phi i n plus delta t phi t at
i and n - where phi subscript t is actually dou phi dou t. So, delta t square by factorial
2 phi t t indicating the second derivative delta the delta t t cubed by factorial 3 phi
t t t i and n so on. So, when we make use of the first three terms,
first two terms in this, then we get approximation for phi t which is dou phi by dou t as as
this particular thing. So, we can say that the truncation error resulting from this approximation
is this whole thing, and similarly, the truncation error resulting from this approximation is
the same very similar to this except that instead delta x, we have delta t; we have
delta x, and instead of derivative with respect to time, we have derivative with respect to
space.
So, we can say that the truncation error from the FTFS scheme for this will be given by
delta t square by factorial 2 dou square phi dou t square at i n plus delta t cubed by
factorial 3 coming from, we, if we take the first two terms in the time derivatives plus
u delta x square by factorial 2 plus u delta x cubed by factorial 3 dou x cubed i n plus
so on. So, this is the difference between the discretized equation and the partial differential
equation with either plus or minus depending on whether we do DE minus PDE or PDE minus
DE, but it varies like this, and in this, u is constant. And we are looking what happens
to the truncation error as in the limit as delta x tending to 0 and delta y tending to
0.
So, we can see that, we can look at each term here delta t goes to 0 as this goes to 0;
this thing goes to 0; this thing goes to 0 and these also goes to 0, go to 0 so that
in the limiting case of delta x tending to 0 and delta t tending to 0, the truncation
error goes to 0, and we can see that if we make delta t very small, then this term which
is constant, see for a given functional variation of a phi with respective t and x, then the
derivative - the second derivative - at a particular point with respect to time and
third derivative and and, and the space derivatives these have a fixed value for a given phi of
x t. So, these will not change within the exact case with changes delta x and delta
t. So, once these are fixed, then this thing
can made as small as possible as delta t and delta x tends to 0 and they can be made go
to 0 for very very small values within machine error and so on. So, we can say that the limit
of the truncation error in these cases is equal to 0; so, that means that the FTFS scheme
satisfies the consistency condition. So, we can see that FTFS scheme has a truncation
error which reduces to 0, which goes to in the limiting case delta x and delta t tending
to 0, and we can also show similarly that FTCS scheme and FTBS scheme all the three
schemes that we have employed for this equation they, they go to, they satisfy the consistency
condition. So, at least from these examples, we can confidently
say that the approximations that we have put here are consistent and we can also confidently
say that consistency condition alone does not guarantee us a proper solutions, because
with even though the satisfies the consistency condition, we saw that the computed solution
seemed to be not so good. So, we have to look at the second aspect of a stability to see
whether it satisfies the stability condition, but when look at the this consistency condition
like this and we see that it almost looks like every schemes we consider must obviously
satisfy the consistency condition. So, is it that consistency condition superfluous
by the very fact that way using the Taylor series expansion for writing the approximations
and using the same analysis for looking at consistency is it superfluous or other cases
were the equations are not consistent.
The most famous of this those inconsistent cases is probably the Dufort Frankel scheme
discretization for a different problem, for a problem which is which can be written as dou u by dou t
equal to nu dou square u and dou x square. So, this is for written for different cases,
but in this case u here represents the velocity and t is obviously time and nu can kinematic
velocity. One can see this as a fluid mechanics problem and this is also known as the Stokes
first problem which arises, for example, that when we consider the case of infinite expanse of fluid, that
is, a liquid and we have an infinitely long plate thin plat which is submerged in this
fluid and its horizontal at a particular depth, and at time equal to 0, you suddenly said
this infinitely long wide thin plat in to motion in the horizontal direction, that is,
in its length direction at uniform velocity of a capital U.
So, because the fluid that we are considering is a real fluid with real viscosity and because
of the no slip condition, the fluid which is adjacent to the plate will start moving.
So, we will find that because of the plate movement in this direction, the fluid which
is above it also starts moving, above it also starts moving and above it also starts moving.
So, the variation of the development to the velocity profile with respect to time is encapsulated
in this equation is given by this equation and this is the known as Stokes first problem.
So, this is not any artificial problem, this is a problem which is quite a fluid mechanic
problem, and not only that, we can see that, this is, this is also in a way a subset of
the generic scalar equation. If you at substitute u for phi here like this, then it becomes
like the accumulation diffusion problem. So, the transient diffusion problem and it is
also a transient fluid conduction problem which is given by the same equation. There
are number of cases which are also given by, by this and it also simple equation.
Now, we can write many kinds of finite difference approximations for this and we can for example,
write u i n plus 1 minus u i n by delta t equal to new times u i plus 1 n minus two
u i n plus u i minus n by delta x square. So, this, this is forward in time and center
in space. We know that this is and this is explicit.
So, our first intuitive way of writing a final deferential approximation is like this, because
this is self starting method as part initial condition we know u i n. So, we can compute
the u i n plus 1, and this is first order accurate but we are ok with that first order
accurate given that this is a self staring problem and we can start implementing it straight
away and it is quiet easy have a second order approximation for this without too much of
complexity. So, this is first order in time and second
order in a space, and it gives us a nice and easy expression in terms of an explicit method
for the calculation of u i n plus 1. So, the solutions seems to be nice and smooth in terms
of implementing, but when we actually try it and having seen the difficulty with another
simple linear wave equation that we have considered, we should be suspicious as to whether or not
to get a proper solution, and in fact, one will show later on that the solution that
will get from this FTCS explicit equation for this is only conditionally stable; in
the sense that only for certain, certain, range of values of delta t and delta x can
we hope to get a good solution, and in other cases, we will not get a good solution.
So, in that sense, this is a only we can, which only conditionally stable and the stability condition will be that nu
delta t by delta x square must be less than or equal to half. So, for a given new which
is the kinematic viscosity and delta x for a given grid, delta t must be less than the
value given by this. If it is more than that, then we will get a proper solution; so, that
means that we are limited in how fast we can go forward in time because the time slip is
limited by this. So, it would be nice to have something which
is unconditionally stable, something which will allow us to choose any values of delta
t and delta x and it will also be nice to have not a first order accurate, but is second
order accurate approximation. So, with that, if you word to say that let me not take a
first order accurate thing and I will make it second order accurate.
Then I can, for example, I can write this as u i n plus 1 minus u i n minus 1 by 2 delta
t it will make this as central in time and central in space explicit, still explicit.
So, this is second order accurate in time and second order accurate in space.
So, that is much better than FTCS scheme because that is only first order accurate in time,
but if you do this, if try this, then you will find that it is no longer conditionally
stable. In fact, it is unconditionally stable; that means that no matter how small delta
t and delta x that you choose, you will always get ah an unstable E solution and we have
actually got something like that in the case of FTFS scheme for the linear wave equation.
So, we have seen that kind of thing. So, a straight forward a naive approach to improving
the accuracy, of, of a scheme which seems to be mathematically ok. It has actually is
spoiled the conditional stable condition and is made it into unconditionally unstable condition.
So, if you want to have more higher accuracy and so on, if you want second order accurate
thing, then this is obviously not correct. So, this is where Dufort and Frankel have
stepped in and they have said that let us not go for a fully, if you look at the strokes
first problem, FTFS scheme which is only first order accurate is conditionally stable; central
in time and central in space is second order accurate, but it has become unconditionally
unstable. Therefore, if you want increase the accuracy of time, then we are having to
compromise on the overall solution itself. This is where Dufort and Frankel have stepped
in 1953 and proposed a small modification to the way this CTC scheme is implemented.
We have a central in time, central in space as given by u i of n plus 1 u i of n minus
1 2 delta t, which makes it central in time and central in space, which is given like
this.
And this is unconditionally unstable; so, we cannot hope to get a reasonable solution
with this. So, what Dufort and Frankel have suggested in 1953 was to replace this u i
n as u i n plus 1 plus u i n minus 1 by 2. So, it is taken as the average of the previous
time step in the coming time step, and then, if you now substitute this into this, then
this two and this two will cancel out and we will have u i n plus 1 minus u i n minus
1 by 2 delta t equal to alpha nu by delta x square times u i plus 1 n minus u i in plus
1 minus u i n minus 1 plus u i minus 1 n. So, this is another way of calculating this
u i n plus 1, and this is a second order of approximation, it being a central order approximation.
So, the overall scheme has not compromised on the second order accuracy of either delta
t or delta x. So, the overall scheme is delta t square and delta x square.
So, we have retain the second order accurate nature of this particular discretization,
and not only that, if you look at the terms here, u i n plus 1 is appearing here; otherwise,
it is n minus 1 which is known; u i plus 1 n which is known; u i is value that we have
actually seeking. So, in a way this is nothing new and you have
n minus 1 here and n here. So, in that sense, this can be evaluated explicitly. So, not
only has this retained the character of second order approximation here, this is also an
explicit method which is therefore easy to solve, which is easy to compute and the difference
that this makes as compared to the standard CTS schemes is, whereas, this is unconditionally
unstable. This is unconditionally stable; that means that you are free to choose whatever
value of delta t and delta x that we want to get and it will not amplify errors, which
is very rare for an explicit method. So, in that sense, this is unconditionally
stable and, we are, we have gotten rid of the conditional stable restriction with the
FTFC scheme and we have completely overturned this unconditionally unstable thing while
retaining the same ease computation in the form of explicitness and the same accuracy
of computation in this. So, in that sense, this simple modification that is made here
is a brilliant modification and it has gives us a lot of advantages. Therefore, it is always
nice to explore what approximations we can make in order to improve the solution.
But it does not stop there; the story does not stop there. If we were to look at the
truncation error of the Dufort Frankel error, we will see that it is of this particular
form, the third derivative times delta t square plus nu delta x square by 12 u x x x x, that
is a fourth derivative minus nu delta t square by delta x square times u t t all derivatives
are evaluated it a x plus nu delta x 4 by 360 times u x x x x x x, that is a sixth derivative
plus nu delta t square t 4 by delta x square by 12 delta x square fourth derivative of
time and so on like this. The leading terms in the truncation error or second order thing
here and delta x square here and this is delta t by delta x square.
So, we can say that order of accuracy of the Dufort Frankel scheme delta x square and delta
t square and delta x by delta t delta t by delta x whole square. Will this goes to 0?
As a delta t and delta x go to 0 is not necessary that as delta x and delta t tend to 0, then
these two terms goes to 0, but this term did not go to 0. For example, they can be fixed
ratio in which these two are varying, while E both of them are going to 0, this need not
goes to 0, and therefore, this thing will not go to 0.
So, whereas, this goes to 0 and this also written as delta t square by delta x square
like this will also go to 0. So, for example, if you say that delta t by delta x is a constant
b, then we can write the truncation error of Dufort Frankel scheme as minus delta t
square by 6 derivative plus nu delta x square by 12 fourth derivative minus nu b square
second derivative of time plus nu delta x 4 sixth derivative with respect to x plus
nu delta t square b square tends fourth derivative with respect time.
So, in this as delta t tends to 0 and delta x tends to 0, this will be 0; this will be
0; even this will be 0, because even though b is constant, this is 0 and this is 0 and
so on. So, the rest of terms will also go to 0, but we are left with is this particular
term. So, from that point of view, the Dufort Frankel scheme does not have a truncation
error which goes to 0, but which goes to nu tends and b square tends u delta t u second
derivative of u so that the Dufort Frankel scheme is an approximation not of this, but
this equation dou u by dou t plus nu b square dou square u by dou t square equal to new
dou square u square by d x square.
So, the Dufort Frankel scheme is a proper approximation of this, this, particular equation
probably this equation, and therefore, it approaches this partial difference equation
and not this partial difference equation. So, in that sense, it is inconsistent. So,
the Dufort Frankel approximation with all the desirable features in terms getting as
second order accurate solutions in an easy way and without any restriction in delta t
and delta x is actually inconsistent, but what we can see also is that it is consistent
when you are looking at time dependent solution. If we are looking for a steady state solution, this term will go
to 0 anyway. So, the study state solution obtained by,
by, this method will be consistent solution. So, in that sense, if you are looking at a
study state solution from this transient solution which is very commonly done in CFD solutions,
because that is probably one of the ways overcoming the inherent nonlinearity contained in the
Navier-Stokes equations. So, in such a case, the final solution is not arising from an
inconsistent discretization, but the way through that steady state solution is definitely going
through an inconsistent approximation of the governing equation. So, whatever error that
arises from, from, this term which depends on what the value of b is with respect to
and what the value of second derivative with respect to these other terms.
So, depending on the magnitude of this as opposed to the magnitude of these two at the
particular time and space will actually determine the accuracy of the time dependent solution.
So, even though it has all desirable features, the solution that is the transient solutions
that we get from this need not necessarily be conversing towards the exact solution of
this partial differential equation. So, that is what we mean by lack of consistency.
So, it is not that every scheme that we do is consistent and it is also that arbitrary
approximation that we make. For example, say that u i n is a average of a the two things
here can lead to inconsistent solutions, inconsistence formulation, and we have to be aware of that;
we have to do consistency of our governing discretization, discretized equation. In order
to see that in the limiting case, the assurance that we are actually solving an equation which
resembles closely the partial deferential equation is very important.