For a numerical approach to any practical problems which are framed by Partial Differential Equations, we convert the PDE into any algebric equations with different schemes (implicit or explicit) like FTCS (Forward in time, Central in Space), LAX-Wendroff, etc.
What is consistency, stability and convergence? And how these are tested and defined?
While numerically solving a p.d.e. we replace each derivative term by a Taylor series. Rearranging the terms gives us the equivalent difference formulation + the higher order terms called truncation error terms. The truncation error terms are to be neglected and we actually solve the equivalent difference equation instead of the p.d.e. The way we discretize the p.d.e. (i.e. whether we approximate the derivatives using forward, backward or central differences etc.) gives rise to different schemes like FTCS, Lax Wendroff etc. Three important characteristics a scheme must possess are following:
CONSISTENCY: A finite difference approximation is considered consistent if by reducing the mesh and time step size, the truncation error terms could be made to approach zero. In that case the solution to the difference equation would approach the true solution to the p.d.e.
STABILITY: A finite difference approximation is stable if the errors (truncation, round-off etc) decay as the computation proceeds from one marching step to the next. Stability of a finite difference approximation is assessed using Von-Neumann stability analysis.
CONVERGENCE: means that the solution to the finite difference approximation approaches the true solution of the p.d.e. we the mesh is refined. off course the finite difference formulation must have the same set of ICs and BCs.
The important Lax Equivalence Theorem says that a finite difference approximation for a properly posed p.d.e. satisfying consistency ( meaning truncation error --> zero when time step size and mesh size goes to zero) and stability (meaning error goes on diminishing from time step to time step possesses necessary and sufficient conditions for convergence.
You may consult a book on Numerical analysis by Ryabenkii, V. and Tsynkov, S.
Title of the book is A theoretical introduction to Numerical Analysis
Consistency deals with the extent to which the finite difference equations approximate the partial differential equation.
The stable numerical scheme is one for which errors from any source (round off, truncation, mistakes) are not permitted to grow in the sequence of numerical procedures as the calculation proceeds from one marching step to next.
As far as I know, consistency, stability, and convergence are purely mathematical, and they need not guarantee that round-off errors (also called arithmetic errors) are controlled (to zero), Of course they are connected to Algorithmic issues (as in stability of computation)
Since you are from Mechanical Engg. Dept. - you may follow the following book:
"Computational Fluid Mechanics and Heat Transfer: Series in Computational and Physical Processes in Mechanics and Thermal Sciences" by John C. Tannehill, Dale A. Anderson and Richard H. Pletcher. Taylor and Francis Publishers
Consistency: When the equation has at-least one valid solution.
Stability: The condition of a mathematical problem relates to its sensitivity to changes in its input values. We say that a computation is numerically unstable if the uncertainty of the input values is grossly magnified by the numerical method.
Stability means that errors at any stage of the computation are not amplified but are attenuated as the computation progresses.
Convergence: Methods are said to be convergent because they move closer to the truth as the computation progresses. Often, the rapid convergence may require more refined initial guesses and more complex programming than a method with slower convergence.
Depending on your problem and the algorithm you use, it can have varying degree of consistency, stability and convergence.
Some examples:
Bisection method has linear rate of convergence. It means, with every iteration, your solution improves by one decimal places. But, to a greater extent, bisection method becomes inconsistent for nonlinear equation whose roots are complex! It cannot yield a complex number as the solution! It is a very stable method for equations whose at-least one root is real!
Newton-Raphson method has quadratic rate of convergence. It means, with every iteration, your solution improves by two decimal places, that is faster rate of convergence! Newton Raphson method is consistent for nonlinear equations whose roots are complex, if programmed accordingly. But, results obtained from Newton-Raphson method may oscillate about the local maximum or minimum without converging on a root. It becomes unstable!
One might need to try a different guess or adaptive size methods to circumvent stability-related issues.
Hope this helps!
I can refer you to the following book
Finite difference methods by
Mitchell and Griffiths 1980j
Consistency: This is usually the easy stuff. A numerical approximation is consistent with the PDE if the exact solution to the PDE satisfies the algebraic equation obtained after discretization, at least up to first order in the discretization parameters. A finite-element approximation is consistent if the exact solution satisfies the discrete variational form. To check consistency for finite-difference approximations, you plug in the exact solution, evaluated at the grid points, into your numerical scheme, use Taylor expansions, and check whether the stuff you are left with vanishes as the discretization parameters go to zero. Surprisingly enough, there are examples of numerical schemes that are not consistent at each point in the domain but still are convergent!
Stability: This is perhaps the most crucial property that a scheme must have, and the property that is usually hardest to verify. A numerical approximation is stable if the numerical solution can be bounded from above by the data that is supplied to the problem with a constant that does not depend on the discretization parameters. Stability is often shown by using norms of the solution (“the energy method”). Stability for finite-difference approximations of time dependent problems is often checked by analyzing the growth rate of an initial condition in terms of a wave (von Neumann analysis). As opposed to the energy method, the von Neumann analysis only yields necessary conditions for stability.
Convergence: This is the property you really want in the end. It means that the numerical solution converges to the exact solution as the discretization parameters tend to zero. Convergence is often shown by referring to consistency and stability, since usually consistency + stability implies convergence.
While numerically solving a p.d.e. we replace each derivative term by a Taylor series. Rearranging the terms gives us the equivalent difference formulation + the higher order terms called truncation error terms. The truncation error terms are to be neglected and we actually solve the equivalent difference equation instead of the p.d.e. The way we discretize the p.d.e. (i.e. whether we approximate the derivatives using forward, backward or central differences etc.) gives rise to different schemes like FTCS, Lax Wendroff etc. Three important characteristics a scheme must possess are following:
CONSISTENCY: A finite difference approximation is considered consistent if by reducing the mesh and time step size, the truncation error terms could be made to approach zero. In that case the solution to the difference equation would approach the true solution to the p.d.e.
STABILITY: A finite difference approximation is stable if the errors (truncation, round-off etc) decay as the computation proceeds from one marching step to the next. Stability of a finite difference approximation is assessed using Von-Neumann stability analysis.
CONVERGENCE: means that the solution to the finite difference approximation approaches the true solution of the p.d.e. we the mesh is refined. off course the finite difference formulation must have the same set of ICs and BCs.
The important Lax Equivalence Theorem says that a finite difference approximation for a properly posed p.d.e. satisfying consistency ( meaning truncation error --> zero when time step size and mesh size goes to zero) and stability (meaning error goes on diminishing from time step to time step possesses necessary and sufficient conditions for convergence.
See slide 13 in the lecture linked below.
http://bluebox.ippt.pan.pl/~ljank/docs/PNO-jankowski_lecture-B2.pdf
In general, when you want to solve an operator equation (e.g. a PDE or a 2-pt boundary value problem) Fu = g, you approximate the problem using a finite difference method, Gv = g, where the discrete operator G depends on a discretization parameter h.
For simplicity, the explanation below is carried out for linear operators F and G, but the principle carries over to nonlinear problems as well.
Consistency means that, if you insert the exact solution u into the discrete problem, you get Gu = g - r, where the residual goes to zero as h goes to zero. More precisely, if ||r|| = O(h^p), then the order of consistency is p.
When you solve the discrete problem, you get v = inv(G)*g. Likewise, solving the original problem formally, we have u = inv(F)*g. Therefore the error is
e := v-u = ( inv(G) - inv(F) )*g = inv(G) * ( I - G*inv(F) ) * g = inv(G)*( g - Gu ) = inv(G) * r.
Hence e = inv(G) * r, and taking norms, we get ||e||
For stability
1. If the stability polynomial satisfies the root condition, then the method is stable, in the sense that for h sufficiently small it will deliver accurate results over a closed
interval [0,T].
2. If the stability polynomial satisfies the strong root condition, then the method is elatively stable, meaning that, for h sufficiently small, the parasitic solution components will go to zero as n — oo
3. A method that is stable, but not relatively stable, is called weakly stable
This was also my question for a long time. I tried to find the answer and explain it in section 5.2 of my dissertation, which you can download from the link below.
http://elib.uni-stuttgart.de/opus/frontdoor.php?source_opus=10329
Please have a look on the following:
Abbott, M.B., and Basco, D.R., 1989, "Computational Fluid Dynamics, An Introduction for Engineers", Longman Scientific & Technical, Essex, England.
3.5. Consistency and convergence
3.7. Stability and accuracy
Consistency:
Consistency deals with the extent to which the FDEs approximate the PDEs. The difference between the PDE and the finite difference approximation is defined as the T.E of the difference representation. A finite-difference representation of a PDE is said to be consistent if one can show that the difference between the PDE and its difference representation vanishes as the mesh is refined, i.e.:
limmesh==>0 (PDE-FDE) = Limmesh==>0(T.E) = 0
Stability:
Numerical stability is a concept applicable in strict sense only to marching problem. A stable numerical scheme is one for which error from any source (round-off, truncation, mistake) are not permitted to grow in the sequence of numerical procedure as the calculation proceeds from one marching step to the next.
Convergence:
Generally, one can find that a consistent, stable scheme is convergent. Convergence here means that the solution to the finite-difference equation approaches the true solution to the PDE having the same initial and boundary condition as the mesh is refined.
You can find more details in the following books:
"Tannehill, J., Anderson, D. & Pletcher, R. (1997). Computational fluid mechanics and heat transfer."
"Hoffman, K., & Chiang, S. (2000-4th ed.). Computational fluid dynamics, Vol 1."
Good luck.
To get a solution to a PDE using finite difference method and obtain a finite difference scheme, you must have a consistent, stable and convergent scheme to guarantee a better numerical solution.
You can guarantee the CONSISTENCY of the finite difference scheme by reducing the mesh step size in your grid and consequently the truncation error goes to zero and this is a consistent discretization as the idea of consistency lies on step size of the mesh.
The STABILITY of the finite difference scheme lies on guaranteeing the decaying of the truncation error when you obtain a solution at a mesh point and move to the next point and so on ... it is possible to still the same from one point to te next but it is not allowed to increase and still increase.
Numerical scheme converts differential equations(DE) to difference or discretized equations (numerical Scheme, NS) and then solve these discretized equations to obtain the solution to a given PDE.
1. Consistency : as mesh size and the time step tends to zero, the NS would become DE. The Truncation Error (TE) vanishes as the time step and mesh approaches to zero.
TE is resulting form truncating the Taylor's series ( it i occur when you convert DE to NS) or
TE can be understood as the residual of the discretized equation for values of exact solution to NS.
That mean TE=NS-DE, if NS tends to become DE as s mesh size and the time step tends to zero then numerical scheme is consistent. that is possible only when TE goes zero.
2. Stability : It deals with errors occur in the solution of NS called numerical error such as round off error. Scheme is stable if numerical errors are bounded with time.
It says that computed solution of numerical scheme would become exact solution of numerical scheme. stability condition is requirement NS and nothing to do with DE.
See the consistency and the stability properties do not refer to same errors. (Consistency-TE and Stability numerical errors)
3. Convergence: This is between numerical solution of DE to the exact solution of the DE (check stability, it refers to computed and exact solution of NS). Scheme is convergence if the the numerical solution tend to the exact solution of DE when time and space steps tend to zero.
In simple word;
DE---->NS---->solution
consistency -- no errors in DE--->NS
Stability -- no errors in solution of NS
convergence -- no errors in solution of DE
that means if the scheme is consistent ans stable then its convergent
Equivalence Theorem of Lax states that for a consistent discretization scheme, stability is the necessary and sufficient condition for convergence.
Dear scientist
Is there any relation between these tree definitions (1-consistency,2- stability and 3-convergence) and existence of steady state solution for a PDE?
I mean can we prove that a PDE is theoretically (physically ) stable (convergence to a steady state solution) based on these definitions?
Ali Namadchian:
Yes there is, it is called "Lax Equivalence Theorem" developed by Peter Lax of Courant Institute, it says:
Iff (two sided if) a numerical solution of PDE be consistent and stable it is converging.
If is hardly proved for FDM and FVM but it is known to be valid for BEM and FEM as well. Its analogous equivalent exists for Lagrangian numerical methods like SPH
Hope it helps,
Kaveh
Dear Kaveh Zamani
Thanks very much for your answer, you mentioned that:
Iff (two sided if) a numerical solution of PDE be consistent and stable it is converging.
I think here convergence as it is mentioned in other answers means:
Convergence: A numerical method is said to be convergent if the solution of the discretized equations tends to the exact solution of the differential equation as the grid spacing tends to zero.
but my question is different, because I want to know the convergence of the exact solution of PDE to a stationary solution (the situation where as t-> infinity the solution converges to a time independent solution) .
https://www.farmingdale.edu › faculty › sheldon-gordon › RecentArticles
Stability means the results don't go very high value. Not stable solutions happen with explicit method when r is more than 0.5.
Convergence means there is no high fluctuation. Also, it means the results close to real or experimental results.
If the numerical method dose not amplify noises in initial conditions or ROE (stability), and the truncation error also vanishes when the spatial-temporal step sizes become null (consistency), the numerical results of the PDE will be identical to the real solutions (convergence). This statement is the Lax's equivalence theorem.
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