Lecture # 03 CFD Techniques and Properties of Numerial Solut

8/9/2019 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)

Lecture # 03 CFD Techniques and Properties of Numerical Solution

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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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Lecture # 03 CFD Techniques and Properties of Numerial Solut

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