2.29 Numerical Fluid Mechanics Spring 2015 · 2.29 Numerical Fluid Mechanics Spring 2015 –Lecture 16 ... – One Dimensional examples ... RRj j j j j j jj f f f f f f ff

PFJL Lecture 16, 1Numerical Fluid Mechanics2.29

2.29 Numerical Fluid Mechanics

Spring 2015 – Lecture 16

REVIEW Lecture 15:

• Finite Volume Methods

– Integral and conservative forms of the cons. laws

– Introduction

– Approximations needed and basic elements of a FV scheme

• Grid generation ⇒ Time-Marching

• FV grids: Cell centered (Nodes or CV-faces) vs. Cell vertex; Structured vs. Unstructured

• Approximation of surface integrals (leading to symbolic formulas)

• Approximation of volume integrals (leading to symbolic formulas)

• Summary: Steps to step-up a FV scheme

– One Dimensional examples

• Generic equation:

• Linear Convection (Sommerfeld eqn): convective fluxes

– 2nd order in space

1/ 2

1/ 21/ 2 1/ 2 ( , )j

j

xjj j x

d xf f s x t dx

dt


TODAY (Lecture 16):

FINITE VOLUME METHODS

• Summary: Steps to step-up a FV scheme

• Examples: One Dimensional examples

– Generic equations

– Linear Convection (Sommerfeld eqn): convective fluxes

• 2nd order in space, 4th order in space, links to CDS

– Unsteady Diffusion equation: diffusive fluxes

• Two approaches for 2nd order in space, links to CDS

• Approximation of surface integrals and volume integrals revisited

• Interpolations and differentiations

– Upwind interpolation (UDS)

– Linear Interpolation (CDS)

– Quadratic Upwind interpolation (QUICK)

– Higher order (interpolation) schemes


References and Reading Assignments

• Chapter 29.4 on “The control-volume approach for elliptic equations” of “Chapra and Canale, Numerical Methods for Engineers, 2014/2010/2006.”

• Chapter 4 on “Finite Volume Methods” of “J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics. Springer, NY, 3rd edition, 2002”

• Chapter 5 on “Finite Volume Methods” of “H. Lomax, T. H. Pulliam, D.W. Zingg, Fundamentals of Computational Fluid Dynamics (Scientific Computation). Springer, 2003”

• Chapter 5.6 on “Finite-Volume Methods” of T. Cebeci, J. P. Shao, F. Kafyeke and E. Laurendeau, Computational Fluid Dynamics for Engineers. Springer, 2005.


One-Dimensional Example I

Linear Convection (Sommerfeld) Eqn, Cont’d

• The resultant linear algebraic system is circulant tri-diagonal (for

periodic BCs)

• This is as the 2nd order CDS!, except that it is written in terms of

cell averaged values instead of values at FD nodes/points

– It is also 2nd order in space

– Has same properties as classic CDS for

• Non-dissipative (check Fourier analysis or eigenvalues of BP which are

imaginary), but can provide oscillatory errors

• Stability (recall tables for FD schemes, linear convection eqn.) of time-marching

– If centered in time, centered in space, explicit: stable with CFL condition:

– If implicit in time: unconditionally stable for all

( 1,0,1) 02 P

d cdt x

Φ B Φ

1c tx

,t x

( , ) ( , ) 0x t c x tt x


One-Dimensional Example IILinear Convection (Sommerfeld) Eqn: 4th order approx.

• 1D exact integral equation still

• Use 4th order accurate surface/volume integrals

– Replace piecewise-constant approx. to (x) with piece-wise quadratic

approx (ξ= x – xj ): (note defined over more than 1 cell)

– Satisfy (average) constraints, i.e. choose a, b, c so that:

– This gives:

– Next, we need to evaluate the values of (x) at the boundaries so as to

compute the advective fluxes at these boundaries:

1/ 2 1/ 2 0j

j j

d xf f

dt

2( ) a b c

'P s

/ 2 / 2 3 / 2

1 13 / 2 / 2 / 2

1 1 1( ) , ( ) , ( )x x x

j j jx x xd d d

x x x

1 1 1 1 1 12

2 26, ,

2 2 24j j j j j j j ja b c

x x

1/ 2 1/ 2 1/ 2 1/ 2, , ,L R L Rj j j jf f f f

j-1/2 j+1/2∆ x

j-2 j-1 j j+1 j+2

L R L R

x

21

Image by MIT OpenCourseWare.


One-Dimensional Example IILinear Convection (Sommerfeld) Eqn: 4th order approx.

• Since f = c compute at edges:

• Resolve flux discontinuity again, use average values

• Done with “integrals” we can substitute in 1D conv. eqn:

• For periodic domains:

1 2 1 11/ 2 1/ 2

1 1 2 11/ 2 1/ 2

2 5 2 5, ,

6 65 2 5 2

,6 6

j j j j j jL Lj j

j j j j j jR Rj j

1/ 2 1/ 2 1/ 2 1/ 21/ 2

2 1 11/ 2

ˆ2 2

7 7ˆ12

L R L Rj j j j

j

j j j jj

f f c cf

f c

1/ 2 1/ 2 1/ 2 1/ 21/ 2

1 1 21/ 2

ˆ2 2

7 7ˆ12

L R L Rj j j j

j

j j j jj

f f c cf

f c

2 1 1 21/ 2 1/ 2 1/ 2 1/ 2

8 8ˆ ˆ 012

j j j j j j jj j j j

d x d x df f f f x c

dt dt dt

( 1, 8,0,8,1) 012 P

d cdt x

Φ B Φ

j-1/2 j+1/2∆ x

j-2 j-1 j j+1 j+2

L R L R

x

21



Centered

Differences

(from Lecture 10)


One-Dimensional Example III

2nd order approx. of diffusion equation:

• 1D exact integral equation same form!

but with:

• Approximation of surface (flux) integral: Approach 1

– Direct: we know that to second-order (from CDS and from )

– Substitute into integral equation:

– In the matrix form, with Dirichlet BCs:

• Semi-discrete FV scheme is as CDS in space,

but in terms of cell-averaged data

1/ 2 1/ 2 0j

j j

d xf f

dt

2

2

( , ) ( , )x t x tt x

fx

1 1 121/ 2 1/ 2 1/ 2

1/ 2

ˆ ˆ( ) and j j j j j jj j j

j

f O x f fx x x x

2( )j j O x

1 11/ 2 1/ 2

2ˆ ˆ 0j j j j jj j

d x df f x

dt dt x

2 (1, 2,1) ( )ddt x

Φ B Φ bc

j-1/2 j+1/2∆ x

j-2 j-1 j j+1 j+2

L R L R

x

21



One-Dimensional Example III

2nd order approx. of diffusion equation:

• Approximation of surface (flux) integral: Approach 2

– Use a piece-wise quadratic approx.:

• Note that a, b, c remain as before, they are set by the volume average constraints

• Since a, b are “symmetric”:

• There are no flux discontinuities in this case

– Substitute into integral equation:

– In the matrix form, with Dirichlet BCs:

• Semi-discrete FV scheme is as CDS in space,

but in terms of cell-averaged data

2

2

( , ) ( , )x t x tt x

1 21/ 2 1/ 2

1/ 2

1 21/ 2 1/ 2

1/ 2

( )

( )

j jR Lj j

j

j jR Lj j

j

f f O xx x

f f O xx x

1 11/ 2 1/ 2

2ˆ ˆ 0j j j j jj j

d x df f x

dt dt x

2( ) 2a b c a bx

2 (1, 2,1) ( )ddt x

Φ B Φ bc

Expressing fluxes at the surface based on cell-averaged (nodal)

values: Summary of Two Approaches and Boundary Conditions

• Set-up of surface/volume integrals: 2 approaches (do things in opposite order)

1. (i) Evaluate integrals using classic rules (symbolic evaluation); (ii) Then, to obtain the unknown symbolic values, interpolate based on cell-averaged (nodal) values

Similar for other integrals:

2. (i) Select shape of solution within CV (piecewise approximation); (ii) impose volume constraints to express coefficients in terms of nodal values; and (iii) then integrate. (this approach was used in the examples).

Similar for higher dimensions:

• Boundary conditions:

– Directly imposed for convective fluxes

– One-sided differences for diffusive fluxes

2.29 Numerical Fluid Mechanics PFJL Lecture 16, 10

( ) ( ) ( )( ) ( )( ) ( )

( ' )

( )

i i

i Pi

P

Pe

a a

aa P

e PV

e S

i x xx xii x

F s

iii F f dA

( ) ( ) ( )i i

( ) ( ) ( )i i

( ) ( ) ( )( ) ( ) ( )a a( ) ( ) ( )i ia ai i

( ) ( ) ( )i i

( ) ( ) ( )a a( ) ( ) ( )i i

( ) ( ) ( )( ) ( ) ( )i x x( ) ( ) ( )( ) ( ) ( )a a( ) ( ) ( )i x x( ) ( ) ( )a a( ) ( ) ( )

( ' )e P( ' )e P( ' )( ' )F s( ' )( ' )e P( ' )F s( ' )e P( ' )F s( ' )F s( ' )e PF se P( ' )e P( ' )F s( ' )e P( ' )( ' )( ' )e Pe P( ' )e P( ' )( ' )e P( ' )( ' )F s( ' )( ' )F s( ' )( ' )e P( ' )F s( ' )e P( ' )( ' )e P( ' )F s( ' )e P( ' )

1( , , )V V

S s dV dV etcV

( ) ( )( ' )

( ) ( ' ) ( ' )e

e e eSe P

e P P

i F f dA FF s

ii s s

( )e e( )e e( )( )( )e ee e( )e e( )( )e e( )

( ' )e P( ' )e P( ' )( ' )F s( ' )( ' )e P( ' )F s( ' )e P( ' )F s( ' )F s( ' )e PF se P( ' )e P( ' )F s( ' )e P( ' )( ' )( ' )e Pe P( ' )e P( ' )( ' )e P( ' )( ' )F s( ' )( ' )F s( ' )( ' )e P( ' )F s( ' )e P( ' )( ' )e P( ' )F s( ' )e P( ' )

( ) ( ' ) ( ' )e P P( ) ( ' ) ( ' )P( ) ( ' ) ( ' )( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )( ) ( ' ) ( ' )P( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )e P e P( ) ( ' ) ( ' )e P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )e P( ) ( ' ) ( ' )( ) ( ' ) ( ' )s s( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )( ) ( ' ) ( ' )P( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )P( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )P( ) ( ' ) ( ' )( ) ( ' ) ( ' ) ( ) ( ' ) ( ' ) ( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )( ) ( ' ) ( ' )e P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )e P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )e P( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )e P( ) ( ' ) ( ' )( ) ( ' ) ( ' )s s( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )s s( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )s s( ) ( ' ) ( ' ) ( ) ( ' ) ( ' )s s( ) ( ' ) ( ' )

( , ) ( , );

( , ) ;i

i

a

a P P P

x y x y etc

x y etc

( , ) ( , );iax y x y etc( , ) ( , );x y x y etc( , ) ( , );i

x y x y etci

( , ) ( , );i

( , ) ( , );x y x y etc( , ) ( , );i

( , ) ( , );ax y x y etca( , ) ( , );a( , ) ( , );x y x y etc( , ) ( , );a( , ) ( , );

Approach 1: Evaluate integrals symbolically, then

interpolate based on neighboring cell-averages

• Surface/Volume integrals: Approach 1

(i) Evaluate integrals based on classic rules (symbolic evaluation)

(ii) Then, to obtain the unknown symbolic values, interpolate based on

neighboring cell-averaged (nodal) values

• If we utilize this approach 1

– Symbolic evaluation:

• To evaluate total surface fluxes (convective + diffusive),

values of and its gradient normal to the cell face at one or more locations on

that face are needed. They have to be expressed as a function of nodal values

• Similar for volume integrals

– Next is interpolation:

• Express the ’s as a function of nodal values. Numerous possibilities. We

already saw some of the most common, provided again next.

2.29 Numerical Fluid Mechanics PFJL Lecture 16, 11

. ( . ) .S S S

F n dA v n dA q n dA

To evaluate total surface fluxes (convective + diffusive),

F n dA v n dA q n dA F n dA v n dA q n dA. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .F n dA v n dA q n dAF n dA v n dA q n dA F n dA v n dA q n dAF n dA v n dA q n dAF n dA v n dA q n dA. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .To evaluate total surface fluxes (convective + diffusive),

F n dA v n dA q n dA. ( . ) .F n dA v n dA q n dA. ( . ) . F n dA v n dA q n dA F n dA v n dA q n dA. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) . . ( . ) .F n dA v n dA q n dA. ( . ) .. ( . ) .F n dA v n dA q n dA. ( . ) .


Approx. of Surface/Volume Integrals:

Classic symbolic formulas

• Surface Integrals

– 2D problems (1D surface integrals)

• Midpoint rule (2nd order):

• Trapezoid rule (2nd order):

• Simpson’s rule (4th order):

– 3D problems (2D surface integrals)

• Midpoint rule (2nd order):

• Higher order more complicated to implement in 3D

• Volume Integrals:

– 2D/3D problems, Midpoint rule (2nd order):

– 2D, bi-quadratic (4th order, Cartesian):

2( ) ( )2e

ne see eS

f fF f dA S O y

4( 4 ) ( )6e

ne e see eS

f f fF f dA S O y

ee S

F f dA

2 2( , )e

e e eSF f dA S f O y z

P P PVS s dV s V s V

16 4 4 4 436P P s n w e se sw ne nwx yS s s s s s s s s s

1,V V

S s dV dVV

(summary from Lecture 15)

2( )e

e e e e e e eSF f dA f S f S O y f S

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e

Notation used for a Cartesian 2D and 3D grid.Image by MIT OpenCourseWare.

Interpolations and Differentiations

(to obtain fluxes “F ” as a function of cell-average values)e

• Upwind Interpolation (UDS) for convective fluxes

– Approximates by its value at the node upstream of e“e”. This is equivalent to using backward or forward-

difference approx for a first derivative (depends on

direction of flow) => Upwind Differencing

– This approximation never yields oscillatory solutions (boundedness criterion), but it is numerically diffusive:

• Taylor expansion about xP:

• UDS retains only first term: 1st order scheme in space

• Leading truncation error is “diffusive”, it has the form of a diffusive flux

• The numerical diffusion is (has 2 components when flow is oblique to the grid)2.29 Numerical Fluid Mechanics PFJL Lecture 16, 13

Scheme,

which is also called Donor-cell.

if . 0

if . 0P e

eE e

v n

v n

if . 0 if . 0 if . 0 v n if . 0 if . 0 if . 0

if . 0 if . 0

if . 0 if . 0 e if . 0 v n if . 0

if . 0 v n if . 0

if . 0 if . 0

if . 0 if . 0

2 2

22

( )( )2

e Pe P e P

P P

x xx x Rx x

ˆ. . . ...e e e P xe e eP

f v n f v n v n xx

UDS retains only first term: 1 order scheme in space

ˆ ˆ f v n f v n . f v n f v n . f v n f v n . f v n f v n . e P f v n f v n e P

f v n f v n f v n f v n e P f v n f v n e P e P f v n f v n e P

f v n f v n f v n f v n . f v n f v n . . f v n f v n . e P f v n f v n e P e P f v n f v n e P v n x v n x v n x v n x . v n x . . v n x .

.e

v n x

Leading truncation error is “diffusive”, it has the form of a diffusive

The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid)v n xThe numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid)v n xThe numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid).The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid)v n xThe numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid).The numerical diffusion is (has 2 components when flow is oblique to the grid) The numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid)v n xThe numerical diffusion is (has 2 components when flow is oblique to the grid)The numerical diffusion is (has 2 components when flow is oblique to the grid)v n xThe numerical diffusion is (has 2 components when flow is oblique to the grid)

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e




(to obtain fluxes “Fe” as a function of cell-average values)

• Linear Interpolation (CDS) for convective fluxes

– Approximates e (value at face center) by its linear

interpolation between two nearest nodes:

• e is the interpolation factor

– This approx. is 2nd order accurate (for convective fluxes):

• Use Taylor exp. of E about xP to eliminate 1st derivative in Taylor exp. of e (previous slide)

• Truncation error is proportional to square of grid spacing, on uniform/non-uniform grids.

• As all approximations of order higher than one, this scheme can provide oscillatory

solutions

• Corresponds to central differences, hence its CDS name (gives avg. if uniform grid spacing)

(1 ) where e Pe E e P e e

E P

x xx x

2 2 22

22 2

2 2 2

2 22 2

( ) ( )( )2 2

( ) ( )( )( ) (1 ) '2 2

E P E P E PE P E P

P P E P E PP P

e P e P E ee P e P E e P e

P P P

x x x x Rx x Rx x x x x x x x

x x x x x xx x R Rx x x

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e



– Approximation is 2nd order accurate if e is midway between P and E (e.g. uniform

grid)

– When the grid is non-uniform, the formal accuracy is 1st order, but error reduction

when grid is refined is asymptotically 2nd order



• Linear Interpolation (CDS) for diffusive fluxes

– Linear profile between two nearest nodes leads to simplest approx. of

gradient (diffusive fluxes)

– Taylor expansions of ’s around xe, one obtains:

2 2 2 3 3 3

32 3

( ) ( ) ( ) ( )2 ( ) 6 ( )

e P E e e P E ex

E P E Pe e

x x x x x x x x Rx x x x x x

E P

e E Px x x

(1 )E P

P

E P

x xx x

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e



– Coefficients in terms of nodal coordinates:

– Uniform grids: coefficients of ’s are 3/8 for node D, 6/8 for node U and -1/8 for node UU

– Somewhat more complex scheme than CDS (larger computational molecules by one node in

each direction)

– Approximation is 3nd order accurate on both uniform and non-uniform grids. For uniform grids:

• But, when this interpolation scheme is used with midpoint rule for surface integral, becomes 2nd order



• Quadratic Upwind Interpolation (QUICK), convective fluxes

– Approx. by quadratic profile between two nearest nodes.

– In accord with convection, third point chosen on upstream side:

• i.e. chose W if flow is from P to E, or EE if flow from E to P.

This gives:

where D, U and UU denote the downstream, first upstream and second

upstream, respectively

3 3

33

6 3 1 38 8 8 48e U D UU

U

x Rx

1 2( ) ( )e U D U U UUg g

1 2( ) ( ) ( ) ( );( ) ( ) ( ) ( )

e U e UU e U D e

D U D UU U UU D UU

x x x x x x x xg gx x x x x x x x

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e



= Symmetric formula for e: no need for “upwind” as with 0th or 2nd order polynomials (donor-cell & QUICK)

– With (x), one can insert e= (xe) in symbolic integral formula. For a uniform Cartesian grid:

• Convective Fluxes: (similar formulas used for ϕ values at corners)

• For Diffusive Fluxes (1st derivative):

– This FV approximation often called a 4th-order CDS (linear poly. interpol. was 2nd-order CDS)

– Polynomials of higher-degree or of multi-dimensions can be used, as well as cubic splines (to

ensure continuity of first two derivatives at the boundaries). This increases the cost.

Interpolations and Differentiations(to obtain fluxes “Fe= f (e)” as a function of cell-average values)

• Higher Order Schemes (for convective/diffusive fluxes)

– Interpolations of order higher than 3 make sense if integrals are

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

ealso approximated with higher order formulas

– In 1D problems, if Simpson’s rule (4th order error) is used for the integral, a polynomial interpolation of order 3 can be used:

=> 4 unknowns, hence 4 nodal values (W, P, E and EE) needed

2 30 1 2 3( )x a a x a x a x

21 2 3

27 272 3 for a uniform Cartesian grid: 24

E P W EE

e e

a a x a xx x x

27 27 3 348

P E W EEe

( Note: higher-order,

approach 1 →≈ approach 2 ! )



– Ex. 2: use a parabola, fit the values on either side of the cell face and the derivative on the

upstream side (equivalent to the QUICK scheme, 3rd order)

– Similar schemes are obtained for derivatives (diffusive fluxes), see Ferziger and Peric (2002)

• Other Schemes: more complex and difficult to program

– Large number of approximations used for “convective” fluxes: Linear Upwind Scheme,

Skewed Upwind schemes, Hybrid. Blending schemes to eliminate oscillations at higher order.

Interpolations and Differentiations(to obtain fluxes “Fe= f (e)” as a function of cell-average values)

• Compact Higher Order Schemes

– Polynomial of higher order lead too large computational

molecules => use deferred-correction schemes and/or

compact (Pade’) schemes

– Ex. 1: obtain the coefficients of by

yj+1

xi-1 xi xi+1

yj-1

y

ji

x

yj

NW

WW W

SW S SE

E EE

N NE

∆y

∆x

nw

s

nw neneP

sw se

e

fitting two values and two 1st derivatives at the two nodes on

either side of the cell face. With evaluation at xe:

• 4th order scheme:

• If we use CDS to approximate derivatives, result retains 4th order:

2 30 1 2 3( )x a a x a x a x

3 1 x+ 4 4 4e U D

Ux

4( )2 8

P Ee

P E

x O xx x

4( )2 16

P E P E W EEe O x


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2.29 Numerical Fluid MechanicsSpring 2015

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2.29 Numerical Fluid Mechanics Spring 2015 · 2.29 Numerical Fluid Mechanics Spring 2015 –Lecture 16 ... – One Dimensional examples ... RRj j j j j j jj f f f f f f ff

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