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AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

Outline

1 Conservative Finite Difference Methods in One Dimension

2 Forward, Backward, and Central Time Methods

3 Domain of Dependence and CFL Condition

4 Linear Stability Analysis

5 Formal, Global, and Local Order of Accuracy

6 Upwind Schemes in One Dimension

7 Nonlinear Stability Analysis

8 Multidimensional Extensions

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Note: The material covered in this chapter equally applies to scalar

conservation laws and to the Euler equations, in one and multipledimensions. In order to keep things as simple as possible, it is presentedin most cases for scalar conservation laws: rst in one dimension, then inmultiple dimensions.

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Conservative Finite Difference Methods in One Dimension

Recall that scalar conservation laws are simple scalar models of the

Euler equations that can be written in strong conservation form as u t

+ f (u )

x = 0 (1)

Suppose that a 1D space is divided into grid points x i and cells[x i 1/ 2 , x i +1 / 2], where x i +1 / 2 is called a cell edgeAlso suppose that time is divided into time-intervals [t n , t n+1 ]The conservation form of a nite difference method applied to thenumerical solution of equation (1) is dened as follows

t

u

t

n

i = (f n

i +1 / 2 f n

i

1/ 2) (2)

where the subscript i designates the point x i , the superscript ndesignates the time t n , a hat designates a time-approximation, and

= t

x , t = t n+1 t n , x = x i +1 / 2 x i 1/ 2

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One interpretation of the nite difference approach ( 2) and theconservation form label is the approximation of the following integralform of equation (1)

1 x Z

x i +1 / 2

x i 1 / 2

[u (x , t n +1 ) dx u (x , t n )] dx

= 1 x Z

t n +1

t n[f `u (x i +1 / 2 , t ) f `u (x i 1/ 2 , t )] dt

which clearly describes a conservation law

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Not every nite difference method can be written in conservationform: Those which can are called conservative and their associatedquantities f ni +1 / 2 are called conservative numerical uxes

nite difference methods derived from the conservation form of theEuler equations or scalar conservation laws tend to be conservativenite difference methods derived from other differential forms (for

example, primitive or characteristic forms) of the aforementionedequations tend not to be conservative

Conservative nite differencing implies correct shock and contactlocations

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Like many approximation methods, conservative nite difference

methods can be divided into implicit and explicit methodsin a typical implicit method

u t n

i

=

u t (u ni K 1 , ..., u ni + K 2 ; u n +1i L 1 , ..., u n +1i + L 2 )f

ni +1 / 2 = f (u

ni K 1 +1 , ..., u

ni + K 2 ; u

n +1i L 1 +1 , ..., u

n +1i + L 2 )

= the solution of a system of equations is required at eachtime-stepin a typical explicit method

u

t

n

i

=

u

t (u ni K 1 , ..., u

ni + K 2 )

f ni +1 / 2 = f (u

ni K 1 +1 , ..., u

ni + K 2 )

= only function evaluations are incurred at each time-step(u ni K 1 , ..., u

ni + K 2 ) and ( u

n +1i L 1

, ..., u n +1i + L 2 ) are called the stencil or direct numerical domain of dependence of u n +1i

K 1 + K 2 + 1 and L1 + L2 + 1 are called th e st enc il wi d th s AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
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Like any proper numerical approximation, proper nite difference

approximation becomes perfect in the limit x 0 and t 0an approximate equation is said to be consistent if it equals the trueequations in the limit x 0 and t 0a solution to an approximate equation is said to be convergent if itequals the true solution of the true equation in the limit x 0 and t 0

Hence, a conservative approximation is consistent whenf (u , ..., u ) = f (u )

= in this case, the conservative numerical ux f is said to beconsistent with the physical ux

A conservative numerical method and therefore a conservativenite difference method automatically locates shocks correctly(however, it does not necessarily reproduce the shape of the shockcorrectly)A method that explicitly enforces the Rankine-Hugoniot relation is

called a shock-capturing methodAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

AA214B NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
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Forward, Backward, and Central Time Methods

Forward Time Methods

Forward Time (FT) conservative nite difference methods

correspond to the choices

t

u t

n

i = u n+1i u

ni and f

ni +1 / 2 = f (u

ni K 1 +1 ,..., u

ni + K 2 )

with Forward Space (FS) approximation of the term u

x (x i , t n ), this

leads to the FTFS scheme

u n +1i = u

ni ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 = f (u

ni +1 ) (3)

with Backward Space (BS) approximation of the term u x (x i , t n ), this

leads to the FTBS scheme

u n +1i = u ni ( f ni +1 / 2 f ni 1/ 2), with f

ni +1 / 2 = f (u

ni ) (4)

with Central Space (CS) approximation of the term u x (x i , t n ), this

leads to the FTCS scheme

u n +1i = u

ni ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 =

12

(f (u ni +1 ) + f (u ni )) (5)


AA214B NUMERICAL METHODS FOR COMPRESSIBLE FLOWS
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Backward Time Methods

Backward Time (BT) conservative nite difference methods

correspond to the choices

t

u t

n

i = u n+1i u

ni and f

ni +1 / 2 = f (u

n +1i K 1 +1 ,..., u

n+1i + K 2 )

with Forward Space (FS) approximation of the term u x (x i , t n ), this

leads to the BTFS schemeu n +1i = u

ni ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 = f (u

n +1i +1 ) (6)


leads to the BTBS scheme

u n +1i = u

ni ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 = f (u

n +1i ) (7)


leads to the BTCS scheme

u n +1i = u

ni ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 =

12 f (u n +1i +1 ) + f (u n +1i )

(8)AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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Central Time Methods

Central Time (CT) conservative nite difference methods correspond

to the choices

t

u t

n

i =

12

(u n+1i u n 1i ) and f

ni +1 / 2 = f (u

ni K 1 +1 ,..., u

ni + K 2 )

with Forward Space (FS) approximation of the term u x (x i , t n ), this

leads to the CTFS schemeu n +1i = u

n 1i 2 ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 = f (u

ni +1 ) (9)


leads to the CTBS scheme

u n +1i = u

n 1i 2 ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 = f (u

ni ) (10)


leads to the CTCS scheme

u n +1i = u

n 1i 2 ( f

ni +1 / 2 f

ni 1/ 2), with f

ni +1 / 2 =

12

(f (u ni +1 ) + f (u ni ))

(11)AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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Domain of Dependence and CFL Condition

Numerical and Physical Domains of Dependence

Recall the theory of characteristics: A point in the x t plane is

inuenced only by points in a nite domain of dependence andinuences only points in a nite range of inuence

Hence, the physical domain of dependence and physical range of inuence are bounded on the right and left by the waves with thehighest and lowest speedsIn a well-posed problem, the range of inuence of the initial andboundary conditions should exactly encompass the entire ow in the

x t planeAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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The direct numerical domain of dependence of a nite difference

method is its stencil: For example, if the solution approximated byan implicit nite difference method can be written as

u n+1i = u (u ni K 1 ,..., u

ni + K 2 ; u

n+1i L1 ,..., u

n +1i + L2 )

its direct numerical domain of dependence is the region of the x t plane covered by the points (u ni K 1 ,..., u

ni + K 2 ; u

n+1i L1 ,..., u

n+1i + L2 )

Similarly, if the solution approximated by an explicit nite differencemethod can be written as

u n+1i = u (u ni K 1 ,..., u

ni + K 2 )

its direct numerical domain of dependence is the region of the x t

plane covered by the points (u ni K 1 ,..., u ni + K 2 )The full (or complete ) numerical domain of dependence of a nitedifference method consists of the union of its direct numericaldomain of dependence and the domain covered by the points of thex t plane upon which the numerical values in the direct numerical

domain of dependence depend uponAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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The Courant-Friedrichs-Lewyor (CFL) condition

The full numerical domain of dependence must contain the physical domain of dependence

Any numerical method that violates the CFL condition missesinformation affecting the exact solution and may blow up to innity:For this reason, the CFL condition is necessary but not sufficient for

numerical stabilityAA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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Scalar Conservation Laws

Consider rst the linear advection problem

u t

+ a u x

= 0

u (x , 0) = 1 if x < 00 if x 0

Assume that a > 0: The exact solution is

u (x , t ) = u (x at , 0) = 1 if x at < 00 if x at 0

The FTFS approximation with x = cst is

u n +1i = (1 + a)u ni au

ni +1

u 0i = u (i x , 0) = 1 if i < 00 if i 0

where as before, = t x


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Then

u 1i =1 if i 2

1 + a if i = 10 if i 0

u 2i =1 if i 3(1 + a)(1 a) if i = 2

(1 + a)(1 + a) if i = 10 if i 0

and so forthThe rst two time-steps reveal that FTFS moves the jump in thewrong direction (left rather than right!) and produces spuriousoscillations and overshootsFurthermore, the exact solution yields u (0, t ) = 1, but FTFS yieldsu 10 = 0!


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This is because FTFS violates the CFL condition


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FTCS satises the CFL condition

However, it almost always blow up (as will be seen in a homework):This illustrates the fact that the CFL condition is a necessary butnot sufficient condition for numerical stabilityYou can also check that when applied to the solution of any scalarconservation law, the BTCS method always saties the CFLcondition


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For scalar conservation laws, the CFL condition translates into a

simple inequality restricting the wave speedlinear advection equation and explicit forward-time method withu n +1i = u (u

ni K 1 , ..., u

ni + K 2 )

in the x t plane, the physical domain of dependence is the line of slope 1/ ain the x t plane, the full numerical domain of dependence of u n +1i

is bounded on the left by a line of slope t K 1 x =

K 1 and on the rightby a line of slope t K 2 x =

K 2

hence, the CFL condition is

K 2

a K 1

K 2 a K 1

which requires that waves travel no more than K 1 points to the rightor K 2 points to the left during a single time-stepif K 1 = K 2 = K , the above condition becomes

|a | K

for this reason, a is called the CFL number or the Courant number keep in mind however that in general, a = a(u ) and therefore theCFL number depends in general on the solutions range


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For scalar conservation laws, the CFL condition translates into asimple inequality restricting the wave speed (continue)

linear advection equation and implicit backward-time method withu n +1i = u (u

ni K 1 , ..., u

ni + K 2 ; u

n +1i L 1

, ..., u n +1i + L 2 )if L1 > 0 and L2 = 0, the full numertical domain of dependence of

u n +1i includes everything to the left of x = x i and beneath t = t

n +1

in the x t planeif L1 = 0 and L2 > 0, the full numertical domain of dependence of u n +1i includes everything to the right of x = x i and beneath t = t

n +1

in the x t planeif L1 > 0 and L2 > 0, the full numertical domain of dependence of u n +1i includes everything in the entire x t plane beneath t = t

n +1

conclusion: as long as their stencil includes one point to the left andone to the right, implicit methods avoid CFL restrictions by using theentire computational domain (hence, this includes BTCS but notBTFS or BTBS)


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The Euler Equations

In 1D, the Euler equations have three families of waves that denethe physical domain of dependenceFor each family of waves, a CFL condition of a given numericalmethod can be established as in the case of a scalar conservationlaw: Then, the overall CFL condition is the most restrictive of allestablished CFL conditionsFor example, if K 1 = K 2 = K , A is the Jacobian matrix of theconservative ux vector, and (A) denotes its spectral radius(A) = max( |u a|, |u | , |u + a|) , the CFL condition of an explicit

forward-time method becomes recall (5)

(A) K

(A) is called the CFL number or the Courant number


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The Euler Equations

(A) K

For supersonic ows, all waves travel in the same direction, eitherleft or right the minimum stencil allowed by the CFL conditioncontains either W ni 1 and W ni for right-running supersonic ow, orW ni and W ni +1 for left-running supersonic owFor subsonic ows, waves travel in both directions, and theminimum stencil should always contain W ni 1 , W

ni , and W

ni +1

Hence, a smart or adaptive stencil is needed in general for the Eulerequations!


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Linear Stability Analysis

Unstable solutions exhibit signicant suprious oscillations and/orovershootsUnstable solutions of linear problems exhibit unbounded suprious

oscillations: Their errors grow to innity as t Hence the concept of instability discussed here for the solution of linear problems is that of ubounded growthSince unstable solutions typically oscillate, it makes sense to describethe solution of a linear problem such as a linear advection problem

as a Fourier series (sum of oscillatory trigonometric functions)


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von Neumann Analysis

The Fourier series for the continuous (in space) solution u (x , t n ) on

any spatial domain [a, b ] is

u (x , t n ) = an0 +

m =1

anm cos 2mx ab a

+

m =1

b nm sin 2mx ab a

(12)For the discrete solution u (x i , t n ), the Fourier series is obtained bysampling (12) as follows

u (x i , t n ) = an0 +

m =1

anm cos 2mx i ab a

+

m =1

b nm sin 2mx i ab a

(13)Assume x i +1 x i = x = cst , x 0 = a, and x N = b x i a = i x and b a = N x : This transforms (13) into

u (x i , t n ) = an0 +

m =1

anm cos 2m i N

+ b nm sin 2m i N

(14)


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Recall that samples can only support wavelengths of 2x or longer

(the Nyquist sampling theorem states that samples spaced apart by x perfectly represent functions whose shortest wavelengths are4 x ): Hence (14) is truncated as follows

u (x i , t n ) an0 +N / 2

m =1

anm cos2mi

N + b nm sin

2mi N

An equivalent expression in the complex plane using I as thenotation for the pure imaginary number ( I 2 = 1) is

u (x i , t n ) N / 2

m = N / 2

C nm e 2 Imi

N (15)

From (14), (15), and Eulers formula e I = cos + I sin it followsthat

C n0 = an0 , C

nm =

anm Ib nm2

, C n m = anm + Ib nm

2AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWS

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Hence, each term of the Fourier series can be written as

u ni m = C nm e (2 Imi

N ) = C nm e m Ii (16)

where m = 2 mN and m = N / 2, , N / 2Because of linearity the amplication factor

G m =

C n+1mC nm = G m ()

does not depend on n: However, it depends on (since u n+1i and u ni

are produced by the numerical scheme being analyzed) which itself depends on t Hence, each term of the Fourier series can be expressed as

u ni m = C nmC n 1m

C 2mC 1m

C 1mC 0m

C 0m e I m i = G m G m G m C 0m e

I m i

= G nm C 0m e

I m i

where G nm

= ( G m ())n


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Finally, assume that C 0m = 1 (for example): This leads to

u ni m = G nm ()e

I m i

Conclusionsthe linear approximation is linearly stable if |G m ( )| < 1 for all mit is neutrally linearly stable if |G m ( )| 1 for all m and |G m ( )| = 1for some mit is linearly unstable if |G m ( )| > 1 for some m

Each of the above conclusion can be re-written in terms of = t x Application (in class): apply the von Neumann analysis to determinethe stability of the FTFS scheme for the linear advection equation



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Matrix Method

Shortcomings of the von Neumann stability analysis methodrequires the solution to be periodic ( u ni = u 0i )requires constant spacing x does not account for the boundary conditions

Alternative method: so-called Matrix (eigenvalue analysis) Method

based on the fact that for a linear problem and a linearapproximation method, one has

u n+1 = M ()u n , where u n = [u no , u n1 , , u

nN ]

T

and M is an amplication matrix which depends on theapproximation scheme and on This implies

u n = M n ()u 0 (17)



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Matrix Method

u n = M n ()u 0

Suppose that M is diagonalizable

M () = Q 1 ()Q , = diag (1 (), , N ())

ThenM n () = Q 1n Q (Qu )n = n ()(Qu )0

Conclusionsthe linear approximation is linearly stable if (M ( )) < 1it is neutrally linearly stable if (M ( )) = 1it is linearly unstable if (M ( )) > 1


AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSLi St bilit A l i
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Matrix Method

Advantages of the Matrix Method for (linear) stability analysisdoes not require the solution to be periodic

does not require constant grid spacingincorporates the effects of the boundary conditions

Major shortcoming: the computation of (M ) is not in general atrivial matter

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AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSFormal Global and Local Order of Accuracy

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Formal, Global, and Local Order of Accuracy

Besides formal order of accuracy, one way to measure the order of accuracy is to reduce x and t simultaneously while maintaining = t x constant and xing the initial and boundary conditionsIn this case, a method is said to have global p -th order of accuracy(in space and time) if

e C x x p = C t t p , e i = u (x i , t ni ) u ni

for some constant C x and a related constant C t

Other error measures can be obtained by using the 1-norm, 2-norm,or any vector norm, or if the error is measured pointwise


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Determining analytically the global order of accuracy dened abovecan be challenging: For this reason, it is usually predicted bycomparing two different numerical solutions obtained using the same

numerical method but two different values of x p =

log( e 2 / e 1 )log( x 2 / x 1)

= log( e 2 / e 1 )

log( t 2 / t 1)

where e l i = |u (x i , t n ) u ni | is the absolute error for x l and

t l = x l , l = 1 , 2


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Another way to measure the order of accuracy is to assume that the

solution is perfect at time t n

that is, u n

i = u (x i , t n

) i , which isusually true for n = 0 and measure the local (in time) truncationerror induced by a single time-step

e i = u (x i , t n+1 ) u n+1i

t

Now, let 0 and x 0 while maintaining = t x , and theinitial and boundary conditions xed: Then, a method is said tohave local p -th order of accuracy (in space and time) if

e i C x x p = C t t p , e i = u (x i , t ni ) u ni

for some constant C x and a related constant C t Unlike the global order of accuracy, the local order of accuracy isrelatively easy to determine analyticallyExample (in class): determine analytically the local order of accuracy of the FTFS scheme for the linear advection equation


AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSUpwind Schemes in One Dimension
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Upwind Schemes in One Dimension

In 1D, there are right-running waves and left-running waves: Forright-running waves, right is the downwind direction and left is theupwind directionSimilarly for left-running waves, left is the downwind direction andright is the upwind directionThen every numerical approximation to a scalar conservation lawcan be described as

Centered : if its stencil contains equal numbers of points in bothdirectionsUpwind : if its stencil contains more points in the upwind directionDownwind : if its stencil contains more points in the downwind

directionUpwind and downwind stencils are adjustable or adaptive stencils:Upwind and downwind methods test for wind direction and then,based on the outcome of the tests, select either a right- or aleft-biased stencil


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p


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p


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Upwinding ensures shock avoidance if the shock reverses the wind,

whereas central differencing does not


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Upwinding does not ensure shock avoidance if the shock does notreverse the wind

downwinding on the right above avoids the shock but violates theCFL condition and thus would create larger errors than crossing theshock would


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General remarksupwind methods are popular because of their excellent shockcapturing abilityamong simple FT or BT methods, upwind methods outdo centered

methods: However, higher-order upwind methods often have nospecial advantages over higher-order centered methodsSample techniques for designing methods with upwind and adaptivestencils

ux averaging methodsux splitting methodswave speed splitting methods


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Introduction to Flux Splitting

Flux splitting is dened as

f (u ) = f + (u ) + f (u )df +

du 0,

df

du 0

Hence, f + (u ) is associated with a right-running wave and f (u ) is

associated with a left-running waveUsing ux splitting, the governing conservation law becomes

u t

+ f +

x +

f

x = 0

which is called the ux split formThen, f

+

x can be discretized conservatively using at least one pointto the left, and f

x can be discretized conservatively using at leastone point to the right, thus obtaining conservation and satisfactionof the CFL condition


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Unfortunately, ux splitting cannot describe the true connectionbetween uxes and waves unless all waves run in the same direction

if all waves are right-running, the unique physical ux splitting is

f +

= f and f

= 0if all waves are left-running, the unique physical ux splitting isf = f and f + = 0

This is the case only for scalar conservation laws away from sonicpoints, and for the supersonic regime in the Euler equations


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Assume that f +

x is discretized with a leftward bias so that the

approximation at x = x i is centered or biased towards x = x i 1/ 2

f +

x

f +i 1/ 2 x

for some f +i 1/ 2

Assume that f

x is discretized with a rightward bias so that theapproximation at x = x i is centered or biased towards x = x i +1 / 2

f

x

f i +1 / 2 x

for some f i +1 / 2

Using forward Euler to perform the time-discretization leads to

u n+1i = u ni ( f

+i 1/ 2 + f

i +1 / 2)

which is called the ux split form of the numerical approximation


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A method in ux split form is conservative if and only if

f +i +1 / 2 + f i +1 / 2 = g ni +1 g ni for someg ni (18)

Proof u n+1i = u

ni ( f

+i 1/ 2 + g

ni g

ni + f

i +1 / 2)

compare with the conservative form u n +1i = u ni ( f i +1 / 2 f i 1/ 2)

= f ni +1 / 2 = f

i +1 / 2 + g ni , f

ni 1/ 2 = f

+i 1/ 2 + g

ni

require now thatf ni +1 / 2 = f

n( i +1) 1/ 2 f

i +1 / 2 + g ni = f +i +1 / 2 + g

ni +1

= f +i +1 / 2 + f

i +1 / 2 = g ni +1 g

ni

Since there are no restrictions on g ni , every conservative method hasinnitely many ux split forms that are useful for nonlinear stabilityanalysis


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Example: Design a rst-order upwind method for Burgers equationusing ux splitting then re-write it in conservation formfor Burgers equation, the unique physical ux splitting is

f (u ) = u 2

2 = max(0 , u )

u 2

| {z } f + (u )

+ min(0 , u )u 2

| {z } f (u )


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Example: Design a rst-order upwind method for Burgers equationusing ux splitting then re-write it in conservation form (continue)

a ux split form of Burgers equation is u t

+ 12

x

(max(0 , u )u ) + 12

x

(min(0 , u )u ) = 0

a backward-space approximation of f +

x gives

x

(max(0 , u )u ) max(0 , u n

i )u n

i max(0 , u n

i 1)u n

i 1 x

a forward-space approximation of f

x gives

x

(min(0 , u )u ) min(0, u ni +1 )u

ni +1 min(0 , u

ni )u ni

x combining these with a FT approximation yields

u n +1i = u ni

2

(max(0 , u ni )u ni max(0 , u

ni 1)u

ni 1)

2 (min(0 , u ni +1 )u

ni +1 min(0 , u

ni ))



d l l
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Example: Design a rst-order upwind method for Burgers equationusing ux splitting then re-write it in conservation form (continue)

the reader can check that the rst-order upwind method described inthe previous page can be re-written in conservation form using

g ni =

12

(u ni )2



I d i W S d S li i
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Introduction to Wave Speed Splitting

In contrast with ux splitting, wave speed splitting uses thegoverning equations in non conservation form and tends to yield nonconservative approximationsHence in most cases, ux splitting is preferred over wave speed

splitting ... except when the ux function has the property

f (u ) = df du

u = a(u )u

which means that f (u ) is a homogeneous function of degree 1: This

property makes ux splitting and wave speed splitting essentiallyequivalent



I t d ti t W S d S litti g
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For scalar conservation laws, wave speed splitting can be written asa(u ) = a+ (u ) + a (u )

a+ (u ) 0, a (u ) 0 (19)

Then, the scalar conservation law can be written as

u t

+ a+ u x

+ a u x

= 0

which is called the wave speed split formThen, a

+ u

x can be discretized conservatively using at least one point

to the left, and a u x can be discretized conservatively using at least

one point to the right, thus obtaining satisfaction of the CFLcondition



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Next, consider vector conservation laws such as the Euler equationsSplit the Jacobian matrix as follows

A(u ) = A+ (u ) + A (u )

where the eigenvalues of A+ are non negative and those of A arenon positive

A+ 0, A 0

Recall that A+ and A are obtained by computing and splitting theeigenvalues of AThe wave speed split form of the Euler equations can then bewritten as

u t + A

+ u x + A

u x = 0

Again, A+ u x can then be discretized conservatively using at leastone point to the left, and A u x can be discretized conservativelyusing at least one point to the right, thus obtaining satisfaction of the CFL condition



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If f is a homogeneous function of degree 1, then

f = Au f

= A

u However, the above ux vector splitting may or may not satisfydf +

du 0 and df

du 0



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Assume that a+ u x is discretized with a leftward bias so that the

approximation at x = x i is centered or biased towards x = x i 1/ 2

a+ u x

a+i 1/ 2u ni u ni 1

x

Assume that a u x is discretized with a rightward bias so that the

approximation at x = x i is centered or biased towards x = x i +1 / 2

a u x

ai +1 / 2u ni +1 u ni

x

Using forward Euler to perform the time-discretization leads to

u n+1i = u ni a

i +1 / 2(u ni +1 u

ni ) a

+i 1/ 2(u

ni u

ni 1)

which is called the wave speed split form of the numericalapproximation



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The ux split form and wave speed form are connected via

f i +1 / 2 = a

i +1 / 2(u ni +1 u

ni )

From the above relation and equation ( 18), it follows that

(a+i +1 / 2 + a

i +1 / 2)(u ni +1 u

ni ) = g

ni +1 g

ni for some ux functiong

ni

(20)Hence, the transformation from conservation form to wave speedform and vice versa is

f ni +1 / 2 = ai +1 / 2(u

ni +1 u ni ) + g ni , f ni 1/ 2 = a+i 1/ 2(u

ni u ni 1) + g ni

(21)



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u n+1i = u ni a

i +1 / 2(u ni +1 u ni ) a+i 1/ 2(u ni u ni 1)

The above notation for the wave speed split form is the standardnotation when wave speed splitting is used to derive newapproximation methodsWave speed split form is also often used as a preliminary step innonlinear stability analysis, in which case the standard notation is

u n+1i = u ni + C

+i +1 / 2(u

ni +1 u

ni ) C

i 1/ 2(u ni u

ni 1)

Hence

C +i +1 / 2 = a

i +1 / 2 , C

i 1/ 2 = a+i 1/ 2 C

i +1 / 2 = a+i +1 / 2



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p p g

If a method is derived using wave speed splitting and not justwritten in wave speed split form, the splitting (19) becomes

a(u ) = C + (u ) C (u ), C + (u ) 0, C (u ) 0

The conservation condition ( 20) becomes(C i +1 / 2 C

+i +1 / 2)(u

ni +1 u

ni ) = (g

ni +1 g

ni )

And equations (21) become

f ni +1 / 2 = C +i +1 / 2(u ni +1 u ni )+ g ni , f ni 1/ 2 = C

i 1/ 2(u ni u ni 1)+ g ni (22)


AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSNonlinear Stability Analysis
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Focus is set here on explicit FT difference approximationsRecall that unstable solutions exhibit signicant suprious oscillationsand/or overshootsRecall also that linear stability analysis focuses on these oscillationsand relies on the Fourier series representation of the numericalsolution: It requires only that this solution should not blow up, ormore specically, that each component in its Fourier seriesrepresentation should not increase to innity

because of linearity, this is equivalent to requiring that eachcomponent in the Fourier series should shrink by the same amount orstay constant at each time-step


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Similarly, nonlinear stability analysis focuses on the spuriousoscillations of the numerical solution, but without representing it bya Fourier series

it can require that the overall amount of oscillation remains bounded,which is known as the Total Variation Bounded (TVB) condition

it can also require that the overall amount of oscillation, as measuredby the total variation, either shrinks or remains constant at eachtime-step this is known as the Total Variation Diminishing (TVD)conditionhowever, whereas not blowing up and shrinking are equivalentnotions for linear equations, these are different notions for nonlinearequations: In particular, TVD implies TVB but TVB does notnecessarily imply TVD



Monotonicity Preservation
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The solution of a scalar conservation law on an innite spatialdomain is monotonicity preserving : If the initial condition ismonotone increasing (decreasing), the solution is monotoneincreasing (decreasing) at all timesSuppose that a numerical approximation inherits this monotonicitypreservation property: Then, if the initial condition is monotone, thenumerical solution cannot exhibit a spurious oscillation

Monotonocity preservation was rst suggested by the Russianscientist Godunov in 1959: It is a nonlinear stability condition, butnot a great one for the following reasons:

it does not address the case of nonmonotone solutionsit is a too strong condition:

it does not allow even an insignicant spurious oscillation that does

not threaten numerical stabilityattempting to purge all oscillatory errors, even the small ones, maycause much larger nonoscillatory errors

Godunovs theorem: For linear methods (NOT to be confused withlinear problems), monotonicity preservation leads to rst-order accuracy at best



Total Variation Diminishing
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TVD was rst proposed by the american applied mathematicianAmiram Harten in 1983 as a nonlinear stability conditionThe total variation of the exact solution may be dened as follows

TV (u (, t )) = supi

i = |u (x i +1 , t ) u (x i , t )|

Laney and Caughey (1991):the total variation of a function on an innite domain is a sum of extrema maxima counted positively and minima counted

negatively with the two innite boundaries always treated as extrema and counting each once, and every other extrema counting twice



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Numerical effects that can cause the total variation to increase



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The exact solution of a scalar conservation law is TVD

TV (u (, t 2)) TV (u (, t 1)) , t 2 t 1

What about the numerical solution of a scalar conservation law?The total variation of a numerical approximation at time t n may beequally dened as

TV (u n ) =

i = |u ni +1 u

ni |

it is the sum of extrema maxima counted positively and minimacounted negatively with the two innite boundaries always treatedas extrema and counting each once, and every other extremacounting twice

Now, a numerical approximation inherits the TVD property if

n, TV u n+1 TV (u n )



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Important result: TVD implies monotonicity preservation andtherefore implies nonlinear stabilityProof: Suppose that the initial condition is monotone

the TV of the initial condition is u u if it is monotone

increasing and u u if it is monotone decreasingif the numerical solution remains monotone, TV = cst ; otherwise, itdevelops new maxima and minima causing the TV to increaseif the approximation method is TVD, this cannot happen andtherefore the numerical solution remains monotone

TVD can be a stronger nonlinear stability condition than themonotonicity preserving condition



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Drawback: Clipping phenomenon (illustrated with the linear

advection of a triangle-shaped initial condition)

The TV should increase by x between time-steps but a TVDscheme will not allow this clipping error here this error is O ( x )because it happens at a nonsmooth maximum, but for most smoothextrema it is O ( x 2 )



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Summary of what should be known about TVD

in practice, most attempts at constructing a TVD scheme end upenforcing stronger nonlinearity stability conditions such as thepositivity condition discussed nextTVD implies monotonicity preservation: This is desirable whenmonotonicity preservation is too weak but less desirable whenmonotonicity preservation is too strong given that TVD can be

strongerTVD tends to cause clipping errors at extrema: In theory, clippingdoes not need to occur at every extrema since, for example, thelocal maximum could increase provided that a local maximumdecreased or a local minimum increased or a localmaximum-minimum pair disappeared somewhere else and may be

only moderate when it occurs: However, in practice, most TVDschemes clip all extrema to between rst- and second-order accuracyin theory, TVD may allow large spurious oscillations but in practice itrarely does in any case, it does not allow the unbounded growthtype of instability



Positivity
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Recall that the wave speed split form of a FT scheme is given byu n+1i = u

ni + C

+i +1 / 2 (u

ni +1 u

ni ) C

i 1/ 2(u ni u

ni 1)

C +i +1 / 2 0 and C

i +1 / 2 0 (23)

Suppose that a given FT numerical scheme can be written in wavespeed split form with

C +i +1 / 2 0, C

i +1 / 2 0 and C +i +1 / 2 + C

i +1 / 2 1 i (24)

Condition (24) above is called the positivity condition (also proposed

rst by Harten in 1983)What is the connection between the positivity condition and thenonlinear stability of a scheme?



Positivity
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The answer is: The positivity condition implies TVDExample: FTFS applied to the nonlinear advection equation u t + a(u )

u x = 0 is positive if 1 a

ni +1 / 2 0

Proof:from (22) with f i +1 / 2 = f (u i +1 ) = a i +1 / 2u i +1 andg ni = f (u ni ) = a ni +1 / 2u ni , it follows that FTFS can be written in wavespeed split form for the purpose of nonlinear stability analysis asfollows u n +1i = u

ni + C +i +1 / 2(u

ni +1 u

ni ) C i 1/ 2(u

ni u ni 1) with

C +i +1 / 2 = ani +1 / 2 and C

i 1/ 2 = 0Hence C +i +1 / 2 + C

i +1 / 2 = ani +1 / 2 and therefore the condition ( 24)

becomes in this case 1 ani +1 / 2 0note that also in this case, the positivity condition is equivalent to

the CFL condition


AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSMultidimensional Extensions
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The extension to multiple dimensions of the computational part of the material covered in this chapter may be tedious in some casesbut is straightforward (except perhaps for the characteristic theory)The expressions of the Euler equations in 2D and 3D can be

obtained from Chapter 2 (as particular cases of the expression of theNavier-Stokes equations in 3D)For simplicity, the focus is set here on the 2D Euler equations

W

t +

F x (W )

x +

F y (W )

y = 0


AA214B: NUMERICAL METHODS FOR COMPRESSIBLE FLOWSMultidimensional Extensions
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2D structured grid



Multidimensional Extensions
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W

t +

F x (W )

x +

F y (W )

y = 0

For the above 2D Euler equations, the equivalent of equation (2) ona 2D structured grid is

t

W t

n

i , j = x ( F nx i +1 / 2 , j F

nx i 1 / 2 , j ) y ( F

ny i , j +1 / 2 F

ny i , j 1 / 2 )

where

x = t

x i , y = t y j

x i = x i +1 / 2, j x i 1/ 2, j j , y j = y i , j +1 / 2 y i , j 1/ 2 i

and F x i +1 / 2 , j and F y i , j +1 / 2 are constructed exactly like f i +1 / 2 in 1D



Multidimensional Extensions
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For example, a 2D version of FTCS has the following conservative

numerical uxes

F nx i +1 / 2 , j = 1

2F x (W ni +1 , j ) + F x (W

ni , j )

= 1

2

(v x ) i +1 , j + ( v x ) i , j (v 2x )i +1 , j + ( v 2x ) i , j + p i +1 , j + p i , j

(v x v y ) i +1 , j + ( v x v y )i , j (Ev x ) i +1 , j + ( Ev x ) i , j + ( pv x )i +1 , j + ( pv x ) i , j

F ny i , j +1 / 2 = 1

2F y (W ni , j +1 ) + F y (W

ni , j )

= 12

(v y ) i , j +1 + ( v y ) i , j (v x v y ) i , j +1 + ( v x v y )i , j

(v 2y ) i , j +1 + ( v 2y ) i , j + p i , j +1 + p i , j (Ev y )i , j +1 + ( Ev y ) i , j + ( pv y ) i , j +1 + ( pv y ) i , j

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