Remark: foils with

1
Computer Fluid Dynamics E181107
2181106
CFD3
Solvers,Approximation, Stability, Boundedness of
Numerical schemes
Remark foils with black background could be
skipped, they are aimed to the more advanced
courses
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2013
2
SOLVERS
CFD3

• FEM, BEM, FVM, FD transfer PDE into system of
  algebraic equations for Tj (nodal pressures,
  velocities, temperature, concentrations) solved
  by
• Finite methods (Gauss, SVD, LU decomposition,
  frontal methods) N3 operations are required
  suitable for smaller systems.
• Iterative methods (GS, multigrid, GMRES,
  conjugated gradients). Prevails at CFD
  calculations, characterized by number cells
  (nodes) of several millions and parallel
  processing (external as well as internal
  aerodynamics of cars requires up to 108 finite
  volumes, solved in clusters of e.g. 512 and more
  processors)

Iterative methods are not so sensitive to
round-off errors (thats why they can be applied
for such huge systems)
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
3
Mathematical Requirements
CFD3

• Three Mathematical requirements
• consistency (discretized equation for
  must be identical with PDE)
• 
  order of accuracy (m-with respect time,
  n-with respect to spatial
  approximation)
• stability (attenuation of round-off errors or
  glitches of initial conditions)
• convergency. Lax theorem consistent and stable
  numerical scheme converges to exact solution (but
  it holds only for linear systems)

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
4
Physical Requirements
CFD3

• Three Physical requirements
• Conservativeness. Balance of mass should hold
  exactly at an element level and globally.
  Fulfilled by FVM (Finite Volume Method). Not
  exactly satisfied fy FEM.
• Boundedness. Solution should not exhibit local
  min/max in the absence of internal sources (of
  mass, momentum or heat). Solution should be
  bounded by boundary values. Min/max principle.
• Transportivness. Numerical scheme should reflect
  directionality of information transfer
  (convection along streamlines)

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
5
Numerical method-analysis
CFD3
Few examples, how to analyze order of accuracy
and stability of suggested numerical schemes (FD
methods)
Benton
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2013
6
Order of accuracy
CFD3
Taylor expansion
Approximation of first derivative
Higher Order Terms
Accurate for 2nd order polynomials T1,x,x2

Accurate for 1nd order polynomials T1,x

Approximation of second derivative
Accurate for 3rd order polynomials T1,x,x2,x3

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
7
Order of accuracy
CFD3

• Therefore finite differences substituting
  derivatives at node xi are
• First order
• Second order
• Third order

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
8
Stability example (explicit method)
CFD3
Unsteady heat transfer (Fourier equation
parabolic)
T-temperature, a-temperature diffusivity
Finite difference method EXPLICIT (explicit
means that unknown temperatures at a new time
level can be expressed explicitly, without
necessity to solve a system of algebraic
equations).
What is the order of accuracy?
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
9
Stability example (explicit method)
CFD3
Residual of this PDE is therefore
Scheme is consistent, linear with respect time,
cubic with respect space.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
10
Stability example (explicit method)
CFD3
Rewrite the explicit formula to the following
(explicit) form
Unknown temperature at a new time level
Known temperatures at old time
level

• Rules
• Sum of coefficients must be the same on the left
  and on the right side (1AA1-2A). Why? A
  constant solution must be fulfilled exactly!
• All coefficients must be positive for bounded
  solution. Why?
• Scarborough criterion (sum of absolute values of
  off-diagonal elements ? diagonal element,
  criterion for convergency of iterative methods)

Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
11
Stability example (explicit method)
CFD3
So why all coefficients must be positive for
bounded solution? Resulting value T is calculated
as a weighted average of values (sum of weighting
coefficients must be 1). Let us assume only two
values for simplicity and T1ltT2. The solution is
bounded if T1ltTltT2. Let us assume, that the
result is not bounded and TltT1. Then For
positive value (1-A)gt0 it follows that T1gtT2 and
this is contradiction.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
12
Stability example of unbounded solution
CFD3
Let us consider what would happen if A1
(negative value 1-2A)
Evolution of initial condition in node
Initial condition is 0 in all nodes and only in
one node is 1.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
13
Stability example (explicit method)
CFD3
Stability condition can be expressed as a
restriction of time step
Interpretation in terms of penetration theory.
Effective velocity of a thermal pulse
Effective domain of dependency
?x-distance of penetrated thermal pulse at a time
?t
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
14
Stability (hyperbolic/parabolic eqs.)
CFD3
Stability criterion CFL (Courant Friedrichs Levi)
for hyperbolic equations was presented in the
previous lecture as
where c is the speed of sound or a transport
velocity. This CFL criterion represents a linear
restriction of the time step with the spatial
step and seems to be quite different than the
stability criterion for the diffusion driven
phenomena. However, these criteria are almost the
same, taking into account the penetration depth
theory
this is qualitatively the same result (up to a
scale constant)
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
15
Stability example of wrong scheme
CFD3
Richardsons scheme for the solution of previous
equation
operates at 3 time levels, n-1,n,n1 and has
higher orders of accuracy
However, the scheme is practically useless. WHY?
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
16
Stability example of wrong scheme
CFD3
Because this coefficient is always negative
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
17
Stability how to improve Richardson?
CFD3
Richardsons scheme
duFort Frankel scheme
and this solution will be bounded for Alt1/2.
Order of accuracy remains high. However Consisten
cy with Fourier equation is assured only if
.
otherwise the hyperbolic equation of heat
transfer would be solved
where
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
18
Stability hyperbolic equations
CFD3
Leap frog (conditionally stable - von Neumann
analysis)
Always unstable (negative coefficient at Tj1)
.
Lax Fridrichs (conditionally stable)
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2013
19
Stability Neumann
CFD3
More precise (and more complicated) is the
stability analysis suggested by von Neumann. It
is based upon spectral decomposition of solution,
i.e. at a time level n the spatial profile is
substituted by Fourier expansion
This Fourier component is substituted into
differential equation and amplification factor G
is evaluated. Numerical scheme is stable, as soon
as the magnitude of identified amplification
factor decreases. Gain factor G is generally a
complex number. Real part is a measure of dumping
error (artificial viscosity) and imaginary part
is a measure of phase error (dispersion error).
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PDE stability analysis (von Neuman)
CFD3
A general Fourier term of solution of a linear
partial differential equation
km?m/L is wave number (discrete frequency).
Arbitrary initial condition T(0,x) can be
expressed by Fourier expansion and evolution of
individual Fourier terms can be analysed
(exp(at)gt1 indicate growth - instability,
exp(at)lt1 stability).
Why is Euler formula unstable?
Courant number CFL criterion (Courant-Fridrichs-Le
vi)
Gain G absolute value greater than 1 for any
wave number