Numerical Methods for Hyperbolic Conservation Laws Lecture 2personal.psu.edu/wxs27/NotesNumCons/Lecture2.pdfNumerical Methods for Hyperbolic Conservation Laws Lecture 2 Wen Shen Department

Numerical Methods for Hyperbolic Conservation LawsLecture 2

Wen Shen

Department of Mathematics, Penn State UniversityEmail: wxs27@psu.edu

Oxford, Spring, 2018

Lecture Notes online: http://personal.psu.edu/wxs27/NotesNumCons/

Wen Shen (Penn State) Numerical Methods for Hyperbolic Conservation Laws Lecture 2Oxford, Spring, 2018 1 / 47

Godunov’s method of kth order

General approach, from tn to tn+1: Given {Qni }, compute {Qn+1

1 Construct piecewise polynomial function (of degree k − 1) from cell averages{Qn

i };2 Solve the conservation law exactly on [tn, tn+1];

3 Construct new cell averages {Qn+1i }.

Conservative form:

Qn+1i = Qn

i −∆t

∆x[Fi+ 1

2− Fi− 1

First order Godunov: piecewise constant approximation

qt + f (q)x = 0, f (q) = Aq

Need to solve a Riemann problem at each cell boundary xi .

Observations:• Solution of Riemann problem is constant along cell boundary, i.e., ξ = 0.

• Flux f (q) is constant along ξ = 0;

• Need to find solution of Riemann problem only along ξ = 0.

Upwind method

qt + f (q)x = 0, f (q) = Aq

• If λp ≥ 0 for all p, the Riemann problem solution with (ql , qr ) alongξ = x/t = 0 is ql .

⇒ F(Qi−1,Qi ) = Qi−1 ⇒ Qn+1i = Qn

i −∆t

∆x[f (Qi )− f (Qi−1)]

• If λp ≤ 0 for all p, the Riemann problem solution with (ql , qr ) alongξ = x/t = 0 is qr .

⇒ F(Qi−1,Qi ) = Qi , ⇒ Qn+1i = Qn

i −∆t

∆x[f (Qi+1)− f (Qi )]

CFL condition: ν ≤ 1

In general, denote byQ↓

i− 12

=q↓(Qni−1,Q

the value of q of the solution of Riemann problem along ξ = 0.

Numerical flux:

Fi− 12

∫ tn+1

f (q↓(Qni−1,Q

ni ))dt = f (Q↓

i− 12

Wave propagation form

Qi − Qi−1 =m∑

αp,i− 12rp

Wp,i− 12

Example: m = 3, λ1 < 0 < λ2 < λ3

Observation:• Positive velocity waves enter at xi− 1

2(left)

• Negative velocity waves enter at xi+ 12

(right)

The effect on cell average at tn+1:Consider W2,i− 1

2= α2,i− 1

2r2, with speed λ2 > 0

cell average is changes by

−λ2∆t · W2,i− 1

for W3,i− 12

: −λ3∆t · W3,i− 1

for W1,i+ 12

: −λ1∆t · W1,i+ 1

(note that λ1 < 0, so −λ1 > 0)

Add up:

Qn+1i = Qn

i −∆t

[λ1W1,i+ 1

2+ λ2W2,i− 1

2+ λ3W3,i− 1

Using the notation

(λp)+= max{λp, 0}, (λp)−= min{λp, 0} ∀p

Suppose that Riemann problem solution consists of m waves Wp with speed λp:

Qn+1i = Qn

i −∆t

(λp)+Wp,i− 12

(λp)−Wp,i+ 12

= Qni −

[A+∆Qi− 1

2+A−∆Qi+ 1

i −∆t

∆x[(net-effect of positive waves) + (net-effect of negative waves)]

(definitions coming next...)

qt + Aqx = 0, Wp = αprp with speed λp

Define

Λ+=diag((λ1)+, (λ2)+, · · · , (λm)+), Λ−=diag((λ1)−, (λ2)−, · · · , (λm)−)

andA+=RΛ+R−1, A−=RΛ−R−1

A+ + A− = RΛ+R−1 + RΛ−R−1 = R(Λ− + Λ−)R−1 = RΛR−1 = A

⇒ splitting of A into positive and negative speed propagation

Let ∆Qi− 12

= Qi − Qi−1.

A+∆Qi− 12

= RΛ+[R−1(Qi − Qi−1)

]= RΛ+αi− 1

m∑p=1

(λp)+αp,i− 12rp

Similarly

A−∆Qi− 12

= RΛ−[R−1(Qi − Qi−1)

]= RΛ−αi− 1

m∑p=1

(λp)−αp,i− 12rp

Relate to numerical flux

Riemann solution along x = xi− 12:

Q↓i− 1

= q↓(Qi−1,Qi ) = Qi−1 +∑

p:λp<0

Wp,i− 12

F ni− 1

2= f (Q↓

i− 12

) = AQi−1 +∑

p:λp<0

AWp,i− 12

= AQi−1 +∑

p:λp<0

λpWp,i− 12

= AQi−1 +m∑

(λp)−Wp,i− 12

= f (Qi−1) + A−∆Qi− 12

F ni+ 1

2= AQi +

m∑p=1

(λp)−Wp,i+ 12

Alternatively,

Q↓i− 1

= Qi −∑

p:λp>0

Wp,i− 12

we get

F ni− 1

2= AQi −

∑p:λp>0

(λp)+Wp,i− 12

= f (Qi−1)− A+∆Qi− 12

Qn+1i = Qn

i −∆t

[F ni+ 1

2− F n

i− 12

i −∆t

(λp)−Wp,i+ 12

(λp)+Wp,i− 12

Can be extended to non-linear problems.

CLAWPACK Software

– developed by Randall LeVeque and his group

– written in Fortran

– for both linear and nonlinear problems

– for both scalar equation and systems

– finite volume methods, first order and high resolutions

– both 1D and several space dimensions (2D and 3D)

– public domain

Download at:

http://www.amath.washington.edu/~claw

Read Chapter 5 in the textbook.

Discrete Fourier analysis

– A quick way to check stability.

Recall a naive method for qt + uqx = 0:

Qn+1j − Qn

∆t+ u

Qnj+1 − Qn

2∆x= 0, F n

j− 12

2(Qj−1 + Qj)

Fourier mode:Qn

j = (λ)ne ik(j∆x), λ ∈ C

We haveQn+1

j = λQnj , Qn

j±1 = e±i∆xkQnj

Plug inλ− 1

∆t+ u

e i∆xk − e−i∆xk

2∆x= 0

λ = λ(k) = 1− u∆t

∆x· 1

[e i∆xk − e−i∆xk

]= 1− u

∆xi sin(k∆x)

– wave dispersion relation

Stability requirement: |λ(k)| ≤ 1 for all k

But here:|λ(k)| > 1 for all mesh ratios ∆t/∆x and almost all modes.

⇒ always unstable.

Upwind scheme

qt + uqx = 0 (u > 0), Qn+1j = Qn

j −∆t

∆xu[Qn

j − Qnj−1

]Plug in the Fourier modes and simplify:

λ(k) = 1− u∆t

[1− e−ik∆x

]= 1− ν + ν [cos(k∆x)− i sin(k∆x)]

|λ(k)|2 = (1− ν + ν cos k∆x)2 + ν2 sin2(k∆x)

= (1− ν)2 + 2(1− ν)ν cos k∆x + ν2 cos2(k∆x) + ν2 sin2(k∆x)

= (1− ν)2 + 2(1− ν)ν cos k∆x + ν2

= 1− 2ν(1− ν)(1− cos k∆x)

= 1− 4ν(1− ν) sin2(k∆x/2)

⇒ stability condition: |λ(k)|2 ≤ 1 if 0 ≤ 1− ν ≤ 1. (same as CFL condition)

Lax-Wendroff scheme

• approximate q at the mid-point in time, i.e. tn+ 12

= tn + 12 ∆t

• evaluate flux also at that point

⇒ a second order method in both x and t

numerical flux: F ni− 1

= f (Qn+ 1

i− 12

) where

i− 12

i−1 + Qni )− ∆t/2

[f (Qn

i )− f (Qni−1)

](Lax-F)

stencil CFL condition: |ν| ≤ 1

For scalar case: qt + uqx = 0, let ν = u∆t/∆x

f (Qn+ 1

i− 12

i−1 + Qni )− u

2ν(Qn

i − Qni−1)

Plug in

Qn+1i = Qn

i − ν[f (Q

)− f (Qn+ 1

i− 12

= · · ·

2ν(1 + ν)Qn

i−1 + (1− ν2)Qni −

2ν(1− ν)Qn

Discrete Fourier analysis: dispersion relation:

λ(k) =1

2ν(1 + ν)e−ik∆x + (1− ν2)− 1

2ν(1− ν)e ik∆x

= · · ·= 1− iν sin k∆x − 2ν2 sin2(k∆x/2)

then|λ(k)|2 =

∣∣1− 4ν2(1− ν2) sin4(k∆x/2)∣∣

Stability condition:|λ(k)|2 ≤ 1 if ν2 ≤ 1, i.e., |ν| ≤ 1.

(same as CFL condition)

Accurary (formal)

Taylor series expansion at (x , tn):

q(x , tn+1) = q(x , tn) + ∆t · qt(x , tn) +1

2(∆t)2qtt(x , tn) + · · ·

Usingqt = −Aqx , qtt = −Aqxt = −A(−Aqx)x = A2qxx

q(x , tn+1) = q(x , tn)−∆t · Aqx(x , tn) +1

2(∆t)2A2qxx(x , tn) + · · ·

Compare with Lax-Wendroff:

Qn+1i = Qn

i −∆tAQn

i+1 − Qni−1

2∆x+

2(∆t)2A2 (Qn

i−1 − 2Qni + Qn

(∆x)2

⇒ second order method for smooth solutions

numerical flux for Lax-Wendroff can be written as

F ni− 1

i−1 + Qni )− 1

∆xA2(Qn

i − Qni−1)

(unstable numerical flux) + (diffusion, stabilizing flux)

The Beam-Warming method

For qt + uqx = 0 (with u > 0) or qt + Aqx = 0 (with λi > 0).

Use one-sided (upwind) finite difference of second order:

qx(xi , tn) =1

2∆x[3q(xi , tn)− 4q(xi−1, tn) + q(xi−2, tn)] +O(∆x2)

qxx(xi , tn) =1

∆x2[q(xi , tn)− 2q(xi−1, tn) + q(xi−2, tn)] +O(∆x2)

We have

Qn+1i = Qn

i −∆t

2∆xA[3Qn

i − 4Qni−1 + Qn

2(∆t/∆x)2A2

i − 2Qni−1 + Qn

Numerical flux for Beam-Warming can be written:

F ni− 1

2= AQn

i−1 +1

2A(1− ∆t

∆xA)(Qn

i−1 − Qni−2)

(upwind) + (high order corrections)

Comment: Not advantageous with one sided finite differences.

Discrete Fourier analysis: leave as an exercise ***

Numerical simulations

Example: qt + qx = 0 with periodic boundary condition, and ν = ∆t/∆x = 0.8

Upwind method:

Excessive dissipation, correct phase position, no oscillations

Lax-Wendroff method:

Sharper capturing of the smooth pulse, phase error, oscillation behind jumps.

Beam-Warming method:

Sharper capturing of the smooth pulse, phase error, oscillation before jumps.

summary of simulation results

High order method (LW)

(+) High accuracy in smooth region

(-) Oscillation around singularities – dispersive

(-) Phase error, incorrect wave position

Low order method (Upwind)

(-) Low accuracy even in smooth region

(+) No oscillations around singularities (monotone)

(+) No phase shift, correct wave position

Idea: Combine them!

Combining the two methods

Compare the numerical fluxes:

Upwind : Fi− 12

= A−Qni + A+Qn

LW : Fi− 12

= (A−Qni + A+Qn

i−1) +1

2|A| (I − ∆t

∆x|A|) · (Qn

i − Qni−1)

= (Upwind flux) + (high order correction term)

where|A| = R |Λ|R−1 = A+ − A−, where |Λ| = diag(|λp|)

Limiter:Change the magnitude of the correction term, depending on the solution.•: in smooth region, use the correction term•: around singularities, remove the correction term

This is more complicated for systems.

Solution is superposition of waves of several families.

At a point, some family is smooth, and some can have a jump.

⇒ one can construct limiters based on each family (in the solution of Riemannproblem)

General setting:

Assume that we have two methods, a lower and a higher order, with numericalfluxes FL(Qi−1,Qi ) and FH(Qi−1,Qi ).

Flux Limiter

F ni− 1

2= FL(Qn

i−1,Qni ) + Φn

i− 12

[FH(Qn

i−1,Qni )−FL(Qn

i−1,Qni )]

If Φni− 1

= 0, then F ni− 1

= FL(Qni−1,Q

If Φni− 1

= 1, then F ni− 1

= FH(Qni−1,Q

The REA algorithm with piecewise linear reconstruction

From cell average Qni , construct a piecewise linear (affine) function

qn(x , tn) = Qni + σn

i (x − xi ), for xi− 12≤ x ≤ xi+ 1

2(xi− 1

2+ xi+ 1

2), center of Ci

and σni : slope of qn in cell Ci . (TBD)

Then, in each cell, we have (regardless of σni )

qn(xi , tn) = Qni

Solve the equations exactly on [tn, tn+1], and form new cell averages {Qn+1i }, and

repeat the process.

Consider the scalar equation qt + uqx = 0, u > 0.

Solving it on [tn, tn+1], the solution at tn+1 is

qn(x , tn+1) = qn(x − u∆t, tn).

CFL condition: ν = u∆t/∆x ≤ 1

qn(x , tn+1)

qn(x , tn)

Qni−1

Take cell-average over Ci :

∆x · Qn+1i =

i−1 +∆x

2σni−1) + (Qn

i−1 − (u∆t − ∆x

∆x − u∆t

i −∆x

2σni ) + (Qn

i − (u∆t − ∆x

]simplify

Qn+1i =

i−1 +1

2(∆x − u∆t)σn

]+ (1− u

i −1

2u∆tσn

]= ...

= Qni −

∆x(Qn

i − Qni−1)− u∆t

2∆x(∆x − u∆t)(σn

i − σni−1)

Choices of slopes σni

If we choose σni ≡ 0: ⇒ Godunov’s method (i.e., upwind)

3 other choices:

• Centered slope: σni =

Qni+1 − Qn

2∆x(Fromm’s method)

• Upwind slope: σni =

Qni − Qn

∆x(Beam-Warming)

• Downwind slope: σni =

Qni+1 − Qn

∆x(Lax-Wendroff)

⇒ Formally all are of 2nd order.

Fromm’s method:

Qn+1i = Qn

i −u∆t

i+1 + 3Qni − 5Qn

i−1 + Qni−2

]− u2∆t2

4∆x2

i+1 − Qni − Qn

i−1 + Qni−2

]Wen Shen (Penn State) Numerical Methods for Hyperbolic Conservation Laws Lecture 2Oxford, Spring, 2018 34 / 47

Total variation: definitions

For discrete data {Qi}:TV(Q) =

|Qi − Qi−1|

For function q(x):

TV(q) = sup

N∑j=1

|q(yi )− q(yi−1)|

where the sup is taken over all subdivisions of the real line

−∞ = y0 < y1 < · · · < yN =∞, ∀N

TV (q) = lim supε→0

∫ ∞−∞|q(x)− q(x − ε)| dx

If q(x) is differentiable:

TV (q) =

∫ ∞−∞|q′(x)| dx

TVD methods

Definition: A two-level method is called total variation diminishing (TVD) if

TV(Qn+1) ≤ TV(Qn)

Definition: A two-level method is called monotonicity-preserving if

Qni ≥ Qn

i+1, ∀i implies Qn+1i ≥ Qn+1

i+1 , ∀i

TVD implies monotonicity-preserving!

More observations:• For transport equation, in the exact solution, as t evolves, TV does not change

• The averaging operator does not increasing the TV!

So, if the reconstruction step is TVD, then the method is TVD.

Slope-limiter methods:

min-mod limiter

σni = minmod

i − Qni−1

∆x,Qn

i+1 − Qni

)where

minmod(a, b) =

a, if |a| < |b| and ab > 0

b, if |a| > |b| and ab > 0

0, if ab ≤ 0

Example of minmod limiter:

One can improve upon minmod limiter!

Superbee limiter (introduced by Roe)

σni = maxmod(σ

(1)i , σ

(2)i )

σ(1)i = minmod

i+1 − Qni

∆x, 2

Qni − Qn

σ(2)i = minmod

i+1 − Qni

∆x,Qn

i − Qni−1

)and maxmod selects the argument with larger modulus.

Monotonized central-difference limiter (MC limiter):

(Propose by van Leer)

σni = minmod

i+1 − Qni−1

2∆x, 2

Qni − Qn

∆x, 2

Qni+1 − Qn

Example:

Mostly TVD!

Problem of superbee: tends to steepen the slope even in smooth region.

MC limiter: does not steepen up smooth slopes – a good default choice.

These limiters make the method nonlinear (even though the equation is linear).

Numerical Methods for Hyperbolic Conservation Laws Lecture 2personal.psu.edu/wxs27/NotesNumCons/Lecture2.pdfNumerical Methods for Hyperbolic Conservation Laws Lecture 2 Wen Shen Department

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