Lecture # 03 CFD Techniques and Properties of Numerial Solut
Transcript of Lecture # 03 CFD Techniques and Properties of Numerial Solut
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Time-stepping techniques
Unsteady flows are parabolic in time use time-stepping methods to
advance transient solutions step-by-step or to compute stationary solutions
time
space
zone of influence
dependencedomain of
future
resent
past
Initial-boundary value problem u = u(x, t)
ut + Lu = f in (0, T) time-dependent PDE
Bu = 0 on (0, T) boundary conditions
u = u0 in at t = 0 initial condition
Time discretization 0 = t0
< t1
< t2
< . . . < tM = T u0
u0 in
Consider a short time interval (tn, tn+1) , where tn+1 = tn + t
Given un u(tn) use it as initial condition to compute un+1 u(tn+1)
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Space-time discretization
Space discretization: finite differences / finite volumes / finite elements
Unknowns: ui(t) time-dependent nodal values / cell mean values
Time discretization: (i) before or (ii) after the discretization in space
The space and time variables are essentially decoupled and can be discretized
independently to obtain a sequence of (nonlinear) algebraic systems
A(un+1, un)un+1 = b(un) n = 0, 1, . . . , M 1
Method of lines (MOL) L Lh yields an ODE system for ui(t)
duhdt + Lhuh = fh on (tn, tn+1) semi-discretized equations
FEM approximation uh(x, t) =Nj=1
uj(t)j(x), uni u(xi, t
n)
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Properties of time-stepping schemes
Time discretization tn = nt, t = TM M =Tt
Accumulation of truncation errors n = 0, . . . , M 1
loc = O(t)p
glob = M loc = O(t)
p1
Remark. The order of a time-stepping method (i.e., the asymptotic rate at
which the error is reduced as t 0) is not the sole indicator of accuracy
The optimal choice of the time-stepping scheme depends on its purpose:
to obtain a time-accurate discretization of a highly dynamic flow problem
(evolution details are essential and must be captured) or
to march the numerical solution to a steady state starting with some
reasonable initial guess (intermediate results are immaterial)
The computational cost of explicit and implicit schemes differs considerably
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Explicit vs. implicit time discretization
Pros and cons of explicit schemes
easy to implement and parallelize, low cost per time step
a good starting point for the development of CFD software
small time steps are required for stability reasons, especially
if the velocity and/or mesh size are varying strongly
extremely inefficient for solution of stationary problems unless
local time-stepping i. e. t = t(x
) is employed
Pros and cons of implicit schemes
stable over a wide range of time steps, sometimes unconditionally
constitute excellent iterative solvers for steady-state problems
difficult to implement and parallelize, high cost per time step
insufficiently accurate for truly transient problems at large t
convergence of linear solvers deteriorates/fails as t increases
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Predictor-corrector and multipoint methods
Objective: to combine the simplicity of explicit schemes and robustness of
implicit ones in the framework of a fractional-step algorithm, e.g.,
1. Predictor un+1 = un + f(tn, un)t forward Euler
2. Corrector un+1 = un + 12
[f(tn, un) + f(tn+1, un+1)]t Crank-Nicolson
or un+1 = un + f(tn+1, un+1)t backward Euler
Remark. Stability still leaves a lot to be desired, additional correction steps
usually do not pay off since iterations may diverge if t is too large
Order barrier: two-level methods are at most second-order accurate, so
extra points are needed to construct higher-order integration schemes
Adams methods tn+1, . . . , tnm, m = 0, 1, . . .
Runge-Kutta methods tn+ [tn, tn+1], [0, 1]
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Adams methods
Derivation: polynomial fitting
Truncation error: glob = O(t)p
for polynomials of degree p
1 whichinterpolate function values at p points
Adams-Bashforth methods (explicit)
p = 1 un+1 = un + tf(tn, un) forward Euler
p = 2 un+1 = un + t2
[3f(tn, un) f(tn1, un1)]
p = 3 un+1 = un + t12
[23f(tn, un) 16f(tn1, un1) + 5f(tn2, un2)]
Adams-Moulton methods (implicit)
p = 1 un+1 = un + tf(tn+1, un+1) backward Euler
p = 2 un+1 = un + t2
[f(tn+1, un+1) + f(tn, un)] Crank-Nicolson
p = 3 un+1 = un + t12
[5f(tn+1, un+1) + 8f(tn, un) f(tn1, un1)]
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Adams methods
Predictor-corrector algorithm
1. Compute un+1 using an Adams-Bashforth method of order p 1
2. Compute un+1 using an Adams-Moulton method of order p with
predicted value f(tn+1, un+1) instead of f(tn+1, un+1)
Pros and cons of Adams methods
methods of any order are easy to derive and implement
only one function evaluation per time step is performed
error estimators for ODEs can be used to adapt the order
other methods are needed to start/restart the calculation
time step is difficult to change (coefficients are different)
tend to be unstable and produce nonphysical oscillations
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Runge-Kutta methods
Multipredictor-multicorrector algorithms of order p
p = 2 un+1/2 = un + t2
f(tn, un) forward Euler / predictor
un+1 = un + tf(tn+1/2, un+1/2) midpoint rule / corrector
p = 4 un+1/2 = un + t2
f(tn, un) forward Euler / predictor
un+1/2
= un
+t2 f(t
n+1/2
, un+1/2
) backward Euler / corrector
un+1 = un + tf(tn+1/2, un+1/2) midpoint rule / predictor
un+1 = un + t6
[f(tn, un) + 2f(tn+1/2, un+1/2) Simpson rule
+ 2f(tn+1/2, un+1/2) + f(tn+1, un+1)] corrector
Remark. There exist embedded Runge-Kutta methods which perform extra
steps in order to estimate the error and adjust t in an adaptive fashion
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General comments
Pros and cons of Runge-Kutta methods
self-starting, easy to operate with variable time steps
more stable and accurate than Adams methods of the same order
high order approximations are rather difficult to derive; p function
evaluations per time step are required for a pth order method
more expensive than Adams methods of comparable order
Adaptive time-stepping strategy t t t t t t t t
makes it possible to achieve the desired accuracy at a relatively low cost
Explicit methods: use the largest time step satisfying the stability condition
Implicit methods: estimate the error and adjust the time step if necessary
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Lax-Wendroff time-stepping
Consider a time-dependent PDE ut
+ Lu = 0 in (0, T)
1. Discretize it in time by means of the Taylor series expansion
un+1 = un + t
u
t
n+
(t)2
2
2u
t2
n+O(t)3
2. Transform time derivatives into space derivatives using the PDE
u
t= Lu,
2u
t2=
t
u
t
=
t(Lu) = L
u
t= L2u
3. Substitute the resulting expressions into the Taylor series
un+1 = un tLun + (t)2
2L2un +O(t)3
4. Perform space discretization using finite differences/volumes/elements
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Lax-Wendroff scheme for pure convection
Example. Pure convection equation ut
+ v ux
= 0 (1D case)
Time derivatives L = v x
u
t= v u
x,
2u
t2= v2
2u
x2
Semi-discrete scheme un+1 = un vtu
x
n+ (vt)
2
2
2u
x2
n+O(t)3
Central difference approximation in space
ux
i = u
i+1ui1
2x +O(x)2,2
ux2
i
= ui+12ui+ui1(x)2 +O(x)2
Fully discrete scheme (second order in space and time)
un+1i
uni
t + v
uni+1 u
ni1
2x =
v2t
2
uni+1 2u
ni
+ uni1
(x)2 +O
[(t)
2
, (x)
2
]
Remark. LW/CDS is equivalent to FE/CDS stabilized by numerical dissipation
due to the second-order term in the Taylor series (no adjustable parameter)
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Properties of numerical methods
The following criteria are crucial to the performance of a numerical algorithm:
1. Consistency The discretization of a PDE should become exact as the
mesh size tends to zero (truncation error should vanish)
2. Stability Numerical errors which are generated during the solution
of discretized equations should not be magnified
3. Convergence The numerical solution should approach the exact solution ofthe PDE and converge to it as the mesh size tends to zero
4. Conservation Underlying conservation laws should be respected at the
discrete level (artificial sources/sinks are to be avoided)
5. Boundedness Quantities like densities, temperatures, concentrations etc.
should remain nonnegative and free of spurious wiggles
These properties must be verified for each (component of the) numerical scheme
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Consistency
Relationship: discretized equation differential equation
Truncation errors should vanish as the mesh size and time step tend to zero
Example. Pure convection equation ut
+ v ux
= 0 discretized by
CDS in space, FE in time:un+1i u
ni
t+v
uni+1 uni1
2x= O[(t)q, (x)p]
Taylor series expansions: un+1
i = un
i + tutni +
(t)2
22ut2
ni + . . .
uni1 = uni x
ux
ni
+ (x)2
2
2ux2
ni
(x)3
6
3ux3
ni
+ . . .
Hence,un+1
iun
i
t + vuni+1u
n
i1
2x ut
+ v ux
n
i+ = 0 where
= t
2
2u
t2
ni
v(x)2
6
3u
x3
ni
+O[(t)2, (x)4]
residual of the difference scheme for the exact nodal values umj = u(jx, mt)
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Stability
Relationship:numerical solution ofdiscretized equations
exact solution of
discretized equations
Definition 1 Numerical errors (roundoff due to final precision of computers)should not be allowed to grow unboundedly
Definition 2 The numerical solution itself should remain uniformly bounded
Stability analysis: can only be performed for a very limited range problems
Matrix method: Aun+1 = Bun un+1 = Cun, where C = A1B
is assumed to be a linear operator. In practice un = un + en so that
un+1 = Cun for the numerical solution un of the discretized equations
un+1 = Cun for the exact solution un of the discretized equations
en+1 = Cen for the roundoff error en incurred in the solution process
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Convergence
Relationship:numerical solution ofdiscretized equations
exact solution of thedifferential equation
Definition: A numerical scheme is said to be convergent if it produces theexact solution of the underlying PDE in the limit h 0 a n d t 0
Lax equivalence theorem: stability + consistency = convergence
For practical purposes, convergence can be investigated numerically by
comparing the results computed on a series of successively refined grids
The rate of convergence is governed by the leading truncation error of
the discretization scheme and can also be estimated numerically:
u = uh + e(u)hp
+ . . . = u2h + e(u)(2h)p
+ . . . = u4h + e(u)(4h)p
+ . . .
u2h uh e(u)hp(1 2p)
u4h u2h e(u)hp(1 2p)2p
p log
u4hu2hu2huh
log2
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Conservation
Physical principles should apply at the discrete level: if mass, momentum
and energy are conserved, they can only be distributed improperly
Integral form of a generic conservation law
t
V
u dV +
S
f n dS =
V
qdV, f = vu du
accumulation influx source/sink flux function
Caution: nonconservative discretizations may produce reasonably looking
results which are totally wrong (e.g. shocks moving with a wrong speed)
even nonconservative schemes can be consistent and stable
correct solutions are recovered in the limit of very fine grids
Problem: it is usually unclear whether or not the mesh is sufficiently fine
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Boundedness
Convection-dominated / hyperbolic PDEs P e 1, Re 1
spurious undershoots and overshoots occur in the vicinity of steep gradients
quantities like densities, temperatures and concentrations become negative
the method may become unstable or converge to a wrong weak solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
low-order
high-order
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
Idea: make sure that important properties of the exact solution (monotonicity,
positivity, nonincreasing total variation) are inherited by the numerical one
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Analysis of numerical dissipation and dispersion
Modified equation method: the exact solution of the discretized equations
satisfies a PDE which is generally different from the one to be solved
Original PDE Modified equation Aun+1 = Bun
u
t+ Lu = 0
u
t+ Lu =
p=1
2p2pu
x2p+
p=1
2p+12p+1u
x2p+1
Motivation: PDEs are difficult or impossible to solve analytically but their
qualitative behavior is easier to predict than that of discretized equations
Expand all nodal values in the difference scheme in a double Taylor series
about a single point (xi, tn
) of the space-time mesh to obtain a PDE Express high-order time derivatives as well as mixed derivatives in terms
of space derivatives using this PDE to transform it into the desired form
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