analysis of numerical schemes

Analysis of Numerical Schemes

Dr. Sreenivas Jayanti

Department of Chemical Engineering

IIT-Madras

Analysis of Numerical Schemes

• Two parts

• Part I: Given a discretization scheme, how to tell that it

will lead to a satisfactory solution?

• Part II: How to solve together the set of continuity,

momentum, energy and other equations?

Need for Analysis

• Discretization of governing equations using FD or FE or FV

techniques gives a set of discretized equations involving variable

values at neighbouring points

• These discretized equations are the ones that are actually solved to

determine the variable values at grid nodes

• Discretization is not unique and for the same PDE, different discretized

equations can be obtained, e.g. discretization can be done to different

orders of accuracy

• Are all discretizations equivalent? Or are some superior? How can we

ensure that the numerical solution is correct?

A Simple Case Study

• Consider the linear convection equation

• Consider three discretization schemes:

0

x

ua

t

u

FTBS : )( 1

1 n

i

n

i

n

i

n

i uuuu

FTCS: 2/)( 11

1 n

i

n

i

n

i

n

i uuuu

FTFS : )( 1

1 n

i

n

i

n

i

n

i uuuu

at/x

Courant no

The True Solution

• Take a = 1 m/s; initial conditions : u(x,0) = f(x)

where f(x) = 1 for 0.1 x 0.2 and f(x) = 0 otherwise

• True solution: initial pulse gets convected in x-direction at 1 m/s

u(x,t) = u(x-at)

t

u

FTBS Solution for 1 at t = 0.1 s

Fig I.1a: FTBS scheme for various Courant numbers at t=0.1 s

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.15 0.2 0.25 0.3 0.35 0.4

x (m)

f(x, 0.1

) 1.00

0.5

0.25

• Peak location predicted correctly, but the square pulse has been “rounded”

• Decreasing amplitude with decreasing

FTBS Solution for 1 at t = 0.1 s

Fig I.1b. FTBS scheme with various Courant numbers at t=0.1 s

-4

-2

0

2

4

0.05 0.1 0.15 0.2 0.25 0.3 0.35

x (m)

f(x, 0.1

) 1.000

1.111

1.125

• Both shape and amplitude predicted poorly for > 1

FTCS Solution for 1 at t = 0.1 s

• Discretization second-order accurate in space but ...

Fig I.2a FTCS scheme with various Courant numbers at t=0.1s

-10

-5

0

5

10

0.15 0.2 0.25 0.3 0.35

x (m)

f(x, t0

)

0.5

0.25

1

FTFS Solution for 1 at t = 0.05 s

• Solution wildly fluctuating for all Courant numbers

Fig I.2b FTFS scheme with various Courant numbers at t=0.05 s.

-400

-200

0

200

400

0.1 0.15 0.2 0.25 0.3

x (m)

f(x

, t0

) 0.5

0.25

1

Need for Analysis Identified

• Different solutions ranging from exact to thoroughly unsatisfactory

obtained for this simple, linear equation

• Behaviour of the solution appears to depend on choice of t and x

• Need a formal analysis to determine the choice of parameters to yield

an accurate solution

• Three conditions for a good solution:

– Consistency : discretized equation pde

– Stability : exact solution of disc eqn computed solution

– Convergence: computed solution exact solution of pde

• Equivalence theorem of Lax (1954):

– For a well-posed linear initial value problem with a consistent

discretization, stability is necessary and sufficient for convergence

• Convergence assures us that grid independent solution = exact solution

of the governing equation

Properties of Numerical Schemes

• Accuracy : error due to discretization

– leading term in the truncation error, e.g, FTCS ~ O (t, x2 )

– non-uniform grids may reduce accuracy by an order

– cannot be eliminated but can be made as small as one wishes

– higher order schemes reduce error only on sufficiently fine grids

– higher accuracy does not mean a better solution; in fact, second

and higher order schemes usually lead to unphysical oscillations

– for the general case, order of accuracy can be obtained by

successively refining grid and expressing error as ~ tp, xq

– in FV method, overall accuracy depends on the approximation of

derivatives and on the accuracy of surface and volume integrals


• Conservation property:

– conservation law upheld at global as well as at discrete level

– desirable to eliminate spurious source terms

– can be ensured by consistent evaluation of face fluxes so that the

total flux leaving a surface of a control volume is equal to the

total flux entering through the same surface into the

neighbouring cells which have the face as a common boundary

– consistent and inconsistent evaluation of diffusive flux at face I:

I

quadratic interpolation

instead of linear


• Boundedness:

– solution remains within physical bounds, e.g.,

• turbulent kinetic energy is always 0

• 0 mass fraction 0

– for steady flows without source terms, interior values should be

bound between minimum and maximum occurring on the

boundaries, e.g.

• temperature inside a conducting rod cannot be colder than T at

the cold end and it cannot be hotter than T at the hot end, unless

there is heat source/ sink inside the rod

– boundedness can be ensured by having the same sign for all the

coefficients in a discretized equation


• Consistency:

– ensures that the discretized equation tends to partial differential

equation as x and t tend to zero

– a consistent scheme means that we are solving the correct equation

in the limit of fine grid spacing

– consistency can be verified by formal Taylor series expansion

e.g., FTFS scheme for the linear convection equation:

n

iuxt

n

i

n

i

n

i

n

i aux

uua

t

uu)(1

1

),(0)(2

)(2

22 xtuax

ut

ixxnitt


• Stability:

– Error between computed solution and exact solution of the

discretized equation should not be amplified as we march forward

in time

– stability guarantees that the scheme produces a bounded solution if

the exact solution itself is bounded

• Consider D = exact solution of the discretized eqn

N = computed solution using the num scheme

= error = D - N

in = (x,t ) = (ix, nt) = Di

n - Nin

• Then stability requires that

| in+1 / i

n | 1

Stability Analysis

• Stability of a linear equation with constant coefficients is well-

understood and lends itself to simple analysis if the effect of boundary

conditions can be neglected. Under these conditions, von Neumann

stability analysis can be used

• When the effect of boundary conditions has to be taken into account, a

matrix method of stability analysis can be used although this is difficult

to implement in practice as it requires the evaluation of eigenvalues of

large matrices

• For the general non-linear or non-constant coefficient problem, local

stability analysis can be performed on a linearized set of equations

• Only the von Neumann stability analysis is described here. For other

methods, see Hirsch (1988)

Stability Analysis: Formulation

• How to evaluate in and i

n+1 to evalute stability?

• Obtain error evolution equation for the discretization scheme

• Consider FTBS scheme for linear convection equation:

(uin+1 - ui

n ) / t + u0 (uin - ui-1

n ) / x = 0 (1)

• Din = exact soln of (1)

(Din+1 - Di

n ) / t + u0 (Din - Di-1

n ) / x = 0 (2)

• Nin = computed soln of (1) to machine accuracy, we have

(Nin+1 - Ni

n ) / t + u0 (Nin - Ni-1

n ) / x = 0 (3)

Stability Analysis: Formulation

• in = Di

n - Nin Ni

n = Din - i

n

• Substitute in (3) to get

[(Din+1+ i

n+1)-(Din+ i

n)]/ t + u0[(Din +i

n )-(Di-1n +i-1

n )]/x=0

• Rearranging

[(Din+1 -Di

n)]/ t + u0[(Din - Di-1

n )/x

+ [( in+1- i

n)]/ t + u0[(in - i-1

n )]/x = 0

• Since Din satisfies eqn (1) exactly, error eqn is of the same form:

( in+1 - i

n ) / t + u0 ( in - i-1

n ) / x = 0 (4)

• Investigate behaviour of (4) to determine stability

von Neumann Stability Analysis

• In general, in = f(x,t)

• Express (x,t) as a Fourier series (valid for periodic boundaries):

• (x,t) = m( bm ejkmx) j = (-1)

– km = wave no = m/L, m = 0, 1, 2, …., M M = L/x

– bm = amplitude of each wave component

• Since error equation is linear, investigate behaviour of each component

and get overall solution by superposition

• We seek a solution of the form m(x,t) = bm ejkmx = eat ejkmx

• Write min = eant ejkmix and substitute in error eqn (4) to get

von Neumann Stability Analysis

• [eat -1] + [1- e-jkmix ] = 0 = at / x

• min+1 / mi

n = G = amplification factor = eat

• Thus, G = 1- + e-j = km x

• For | G | 1, 0 1

• Scheme stable if 0 = at / x 1

• Courant-Friedrichs-Lewy (CFL) condition

• FTBS scheme conditionally stable

Re

Im

G

FTCS Scheme

• FTCS scheme for linear convection equation:

(uin+1 - ui

n ) / t + u0 (ui+1n - ui-1

n ) / 2 x = 0

• Error equation

(in+1 - i

n ) / t + u0 (i+1n - i-1

n ) / 2x = 0

• Amplification factor

G = 1 - j sin

• Stability: | G|2 = 1+ 2 sin2 > 1

FTCS scheme unconditionally unstable

• Similarly FTFS scheme can be shown to be unconditionally unstable

• Stability behaviour reflected in the Case Study

FTCS Implicit

• FTCS-Implicit scheme for linear convection equation:

(uin+1 - ui

n ) / t + u0 (ui+1n+1 - ui-1

n+1 ) / 2 x = 0

• Error equation

(in+1 - i

n+1 ) / t + u0 (i+1n+1 - i-1

n+1 ) / 2x = 0


G - 1 +/2 (ej - e-j ) = 0 or G = 1/ [1 + j sin ]

• Stability: | G|2 = 1/ [1+ 2 sin2] < 1

FTCS Implicit scheme unconditionally stable

• Gain of stability in implicit schemes countered by need to solve

equations simultaneously

Generic 1-d Scalar Transport Equation

• No source terms, const property and constant given velocity:

/t + u/x = / 2/x2 (1)

• FTCS Explicit scheme

(in+1 - i

n ) / t + u (i+1n - i-1

n ) / 2 x = / (i+1n -2 i

n+i-1n ) / x2

• Put = t /( x2) and = u t / x and rearrange to get

in+1 = (+/2) i-1

n + (1-2) in + (-/2) i+1

n


G = 1+ 2 [cos(kmx) -1] -j sin (kmx)

• Stability: 1/2 and 2 2

Re

Im

G

Generic 1-d Scalar Transport Equation

• Oscillations are produced with FTCS explicit if 2 Pe 2/ where Pe

= ux/ is the mesh Peclet number

• This loss of boundedness can be cured by

– decreasing grid spacing so that Pe < 2 or by

– using an “upwind” scheme for the convective term

• upwinding for convective term assuming u > 0 yields

(in+1 - i

n ) / t + u (in - i-1

n ) /x = / (i+1n -2 i

n+i-1n ) / x2

• or, in+1 = i-1

n + (1-2- ) in + (+) i+1

n

• Stable, oscillation-free solution if (2 + ) < 1

• Stability analysis becomes more complicated in 2- and 3-d, non-

constant coefficients, non-uniform grids etc.

Part II: Solution of Navier-Stokes Equations

• Equations governing fluid flow are coupled, e.g., we have to solve the

continuity and the three momentum equations together to get a solution

for incompressible, isothermal flows. For highly compressible flows,

the energy equation also has to be solved.

• For compressible flows, a natural coupling exists between the

continuity and momentum equations (2-d case):

• U/ t + E/ x + F/ y = 0 (1) - (4)

U = [ u v Et] T Et = e + V2/2

E = [u u2+p-xx uv-xy (Et+p) -uxx - vxy + qx ]T

F = [v uv-xy v2+p-yy (Et+p) -uxy - vyy + qy ]T

Solution for Compressible Flow

• Discretize eqns. (1) to (4) using. e.g., MacCormack (1969) scheme

Predictor step:

Uijn+1 = Uij

n - t/ x (Ei+1,jn - Ei,j

n) - t/ y( Fij+1n - Fij

n)

Corrector step:

Uijn+1 = 1/2 [Uij

n + Uijn+1 - t/ x (Ei,j

n+1 - Ei-1,jn+1)

- t/ y( Fijn+1 - Fij-1

n+1) ] + O(t2, x2)

• Solve (1) for ; (2) for u; (3) for v and (4) for Et

• Calculate u, v, w, e; p = f(, e) ; T = f(, e)

Extension to Incompressible Flow

• MacCormack (1969) explicit scheme has stability limit given by

– t f (t)CFL/(1+2/Re) , f = safety factor ~ 0.9

– (t)CFL= inviscid CFL limit [u/ x + v/ y + a(1/x2 + 1/ y2)0.5 ]-1

– a = speed of sound

– Re = min (u x/, v y/)

• For nearly incompressible flows, speed of sound (t)CFL0

• Even for implicit methods, (t)max ~ 10 (t)CFL

• Special methods necessary for nearly incompressible flows

Methods for Incompressible Flow

• Evaluation of pressure the main problem:

– Continuity for unsteady incompressible flow : U = 0

no natural equation to evaluate pressure

• Special methods for incompressible flow

– artificial compressibility method

– streamfunction-vorticity method

– pressure equation method

– pressure correction method

• Pressure correction method most popular; extension to compressible

flow exists but not discussed today (see Ferziger & Peric, 1999)

Pressure Correction Method

• Consider the staggered grid arrangement

• Discretize x-mom eqn to yield

aeue = nb(anbunb) + b + Ae(pP-pE) (1)

• Discretize y-mom eqn to yield

anvn = nb (anbvnb) + b + An(pP-pN) (2)

• These can be solved if pressure is known

P E

S

P

N

W P E

S

W

u-cell v-cell

p-cell N

N

E


• Guess pressure = p* and solve (1) and (2) for u* and v*:

aeue* = nb (anbunb*) + b + Ae(pP*-pE*) (3)

anvn* = nb (anbvnb*) + b + An(pP*-pN*) (4)

• u* and v* from (3) and (4) do not satisfy continuity as p* is guessed

correct guessed presssure: p = p* + p’

u-velocity correction u = u* + u’

v-velocity correction v = v* + v’

• How to get u’ and v’ for a given p’?

– Eqn (1) - eqn (3)

– ae(ue - ue*) = nb (anb(unb - unb*) + Ae[(pP-pP*)-(pE-pE*)] or

– aeue’ = nb (anbunb’) + Ae[(pP’- pE’)] Ae(pP’- pE’)

• Discretize x-mom eqn to yield

aeue = nb(anbunb) + b + Ae(pP-pE) (1)

• Discretize y-mom eqn to yield

anvn = nb (anbvnb) + b + An(pP-pN) (2)

• These can be solved if pressure is known


• Thus ue’ Ae(pP’- pE’)/ae and vn’ An(pP’- pN’)/an (5)

• Derive pressure correction equation by discretizing continuity

– (ue - uw) y + (vn-vs) x = 0 (6)

• Substitute ue = ue* + u’ etc and substitute eqn (5) for u’ and rearrange

to get the pressure correction equation

aPpP’ = aEpE’ + aWpW’ + aNpN’ + aSpS’ + b (7)

where b = (ue* - uw*) y + (vn*-vs*) x

aE/W = yAe/w/ae/w aN/S = xAn/s/an/s aP = NB aNB

• SIMPLE : Semi-Implicit Method for Pressure-Linked Equations

SIMPLE Algorithm

1. Guess p*, u* and v*

2. Solve disc. x-mom eqn for u*

3. Solve disc. y-mom eqn for v*

4. Solve pressure correction eqn for p’

5. Calculate velocity corrections u’ and v’

6. Calculate other scalars, e.g., T, k ..

7. Update p, u and v and return to 1 if not converged

• Has proved very popular since first proposed by Patankar & Spalding (1968)

• Number of variants exist to improve convergence rate: SIMPLEC, SIMPLER,

SIMPLEST, PISO

• Adapted for collocated grids

• Adapted for compressible flow

• Used in many commercial CFD codes

analysis of numerical schemes

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