Runge-Kutta-Chebyshev Methods...Runge-Kutta-Chebyshev Methods Mirela Dar˘ au˘ Department of...

Transcript of Runge-Kutta-Chebyshev Methods...Runge-Kutta-Chebyshev Methods Mirela Dar˘ au˘ Department of...

IntroductionStability PolynomialsIntegration Formulas

Numerical SimulationsSummary

Runge-Kutta-Chebyshev Methods

Mirela Dărău

Department of Mathematics and Computer ScienceTU/e

CASA Seminar, 26th November 2008

Mirela Dărău Runge-Kutta-Chebyshev Methods


Outline

1 IntroductionStabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation

2 Stability PolynomialsFirst-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials

3 Integration Formulas

4 Numerical SimulationsStability RegionsSimulations



Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation

Motivation

consider

w ′(t) = F (t ,w(t)), t > 0, w(0) = w0, (1)

representing semi-discrete, multi-space PDEparabolic problems→ stiff problems having a symmetricJacobian with spectral radius proportional to h−2, h spatial meshwidthstandard explicit methods: severe stability constraint⇒ highlyinefficientunconditionally stable implicit methods→ costly in higher spacedimension




Motivation

consider

w ′(t) = F (t ,w(t)), t > 0, w(0) = w0, (1)





Motivation

consider

w ′(t) = F (t ,w(t)), t > 0, w(0) = w0, (1)





Motivation

consider

w ′(t) = F (t ,w(t)), t > 0, w(0) = w0, (1)





Explicit Runge-Kutta

REMEMBER:

wn0 = wn,

wnj = wn + τΣj−1k=0αjk F (tn + ckτ,wnk ), j = 1, . . . , s,

wn+1 = wns,

wn being the approximation to the exact solution w at time t = tn andτ = tn+1 − tn the step-size.

Specifying a particular method:s ∈ Z (number of stages)ck and αjkcj = Σ

j−1k=0αjk




Stabilized Runge-Kutta methods

explicit⇒ avoid algebraic system solutionspossess extended real stability interval with a length proportionalto s2, s number of stagesuseful for: modestly stiff, semi-discrete parabolic problems forwhich the implicit system solution is really costly in terms of CPUtime3 families:

RKCROCKDUMKA






RKCROCKDUMKA






RKCROCKDUMKA






RKCROCKDUMKA




Stability Functions. Stability Regions.

consider the general form of a Runge-Kutta method:

wn+1 = wn + τΣsi=1biF (tn + ciτ,wni ),

wni = wn + τΣsj=1αijF (tn + cjτ,wnj ), i = 1, . . . , s.

stability analysis→ considering the complex test equationw ′(t) = λw(t)let z = τλ; by applying the method to the test equation we havewn+1 = R(z)wn, with R the stability function of the methodR(z) = 1 + zbT (I − zA)−1e, where b = (bi ), A = (αij ) ande = (1,1, . . . ,1)T

for explicit methods R(z) is a polynomial of degree ≤ sstability region: S = {z ∈ C : |R(z)| ≤ 1}




Model Problem

Consider the advection-diffusion-reaction equation

ut + aux = εuxx + λu(1− u), 0 < x < 1, t > 0,ux (0, t) = 0, t > 0,

u(1, t) =12

(1 + sin(ωt)), t > 0,

u(x ,0) = v(x), 0 < x < 1,

wherea - advection velocity

ε - diffusion coefficient

λ - source term coefficient

ω - frequency.



First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials

Shifted Chebyshev Polynomials

Theorem

For any explicit, consistent Runge-Kutta method we have βR ≤ 2s2.The optimal stability polynomial is the shifted Chebyshev polynomialof the first kind

Ps(z) = Ts(

1 +zs2

).

Sketch of proof Chebyshev polynomials:Ts(x) = cos(s arccos(x)), x ∈ [−1,1] ORT0(z) = 1, T1(z) = z, Tj (z) = 2zTj−1(z)− Tj−2(z), 2 ≤ j ≤ s,z ∈ C.

⇒ |Ps(x)| ≤ 1 for −2s2 ≤ x ≤ 0.Uniqueness: largest stability

boundary.-50 -40 -30 -20 -10

-1.0

-0.5

0.5

1.0

P2

P3

P4

P5




Shifted Chebyshev Polynomials

The coefficients of Ps are given by

γ0 = γ1 = 1, γi =1− (i − 1)2/s2

i(2i − 1)γi−1 for i = 2, . . . , s.

P2(z) = 1 + z + 18P3(z) = 1 + z + 427 z

2 + 4729 z3

P4(z) = 1 + z + 532 z2 + 1128 z

3 + 18192 z4

P5(z) =1 + z + 425 z

2 + 283125 z3 + 1678125 z

4 + 169765625 z5

-50 -40 -30 -20 -10

-4

-2

2

4P5 undamped

For s large and z → 0, Ps(z) = ez − 13 z2 + O(z3)⇒ leading error

coefficient 1/3.




Second-Order Stability Polynomials

For actual computational practice first-order consistency is oftentoo lowWe look for polynomials of consistency order p ≥ 2, coefficientsso that βR is as large as possibleRiha proved the existence ∀p ≥ 1 and s ≥ pFor p = 2: a suitable approximate polynomial in analytical form

Bs(z) =23

+1

3s2+

(13− 1

3s2

)Ts

(1 +

3zs2 − 1

), βR ≈

23

(s2−1).

this generates about 80% of the optimal interval






Bs(z) =23

+1

3s2+

(13− 1

3s2

)Ts

(1 +

3zs2 − 1

), βR ≈

23

(s2−1).







Bs(z) =23

+1

3s2+

(13− 1

3s2

)Ts

(1 +

3zs2 − 1

), βR ≈

23

(s2−1).







Bs(z) =23

+1

3s2+

(13− 1

3s2

)Ts

(1 +

3zs2 − 1

), βR ≈

23

(s2−1).







Bs(z) =23

+1

3s2+

(13− 1

3s2

)Ts

(1 +

3zs2 − 1

), βR ≈

23

(s2−1).





Consistency Order

Expanding Bs(z) we get:

Bs(z) = 1 + z +z2

2+

−4 + s2

10(−1 + s2)z3 + . . .

⇒ second order consistency!!

For s large and z → 0, Bs(z) =ez − 115 z

3 + O(z4)⇒ leading errorcoefficient 1/15. For the optimalpolynomial the error is ≈ 0.074.

-15 -10 -5

-3

-2

-1

1

2

3B5 undamped




Stability Polynomials

StabilityPolynomials

First orderconsistency

Second orderconsistency

Ps

Bs




Damped Ps

For Ps and Bs the stability intervals contain interior points where|R(z)| = 1 instability⇒ we introduce a little dampingDamped form of Ps:

Ps(z) =Ts(ω0 + ω1z)

Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.

Stability interval: −ω0 ≤ ω0 + ω1z ≤ ω0 ⇒ βR = 2ω0ω1 ;Ps(z) ∈ [−Ts(ω0)−1,Ts(ω0)−1].

Convenient: ω0 = 1 + εs2 , ε smallDamping ≈ 5%Stability boundary ≈ 1.93s2

-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps



Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps



Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps



Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps



Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps



Ts(ω0), ω1 =

Ts(ω0)T ′s(ω0)

, ω0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T ′′s (ω0)

(T ′s(ω0))2(Ts(ω0 + ω1z)− Ts(ω0)), ω1 =

T ′s(ω0)T ′′s (ω0)

.

βR ≈ (ω0+1)T′′s (ω0)

T ′s (ω0)≈ 23 (s

2 − 1)(1− 215ε

).

Damping ≈ 5%Stability boundaryβR ≈ 0.9794βR,undamped

-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T ′′s (ω0)

(T ′s(ω0))2(Ts(ω0 + ω1z)− Ts(ω0)), ω1 =

T ′s(ω0)T ′′s (ω0)

.

βR ≈ (ω0+1)T′′s (ω0)

T ′s (ω0)≈ 23 (s

2 − 1)(1− 215ε

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T ′′s (ω0)

(T ′s(ω0))2(Ts(ω0 + ω1z)− Ts(ω0)), ω1 =

T ′s(ω0)T ′′s (ω0)

.

βR ≈ (ω0+1)T′′s (ω0)

T ′s (ω0)≈ 23 (s

2 − 1)(1− 215ε

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T ′′s (ω0)

(T ′s(ω0))2(Ts(ω0 + ω1z)− Ts(ω0)), ω1 =

T ′s(ω0)T ′′s (ω0)

.

βR ≈ (ω0+1)T′′s (ω0)

T ′s (ω0)≈ 23 (s

2 − 1)(1− 215ε

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping



Method DescriptionAnsatz: Rj (z) = aj + bjTj (ω0 + ω1z), aj = 1− bjTj (ω0), 1 ≤ j ≤ s.Imposing Chebyshev recursion:

R0(z) = 1, R1(z) = 1 + µ̃1z,Rj (z) = (1− µj − νj ) + µjRj−1(z) + νjRj−2(z) + µ̃jRj−1(z)z + γ̃jz,

where j = 2, . . . , s and

µ̃1 = b1ω1, µj =2bjω0bj−1

, νj =−bjbj−2

, µ̃j =2bjω1bj−1

, γ̃j = −aj−1µ̃j .

The RKC integration formulas are then of the form:

wn0 = wn,wn1 = wn + µ̃1τFn0, (2)

wnj = (1− µj − νj )wn + µjwn,j−1 + νjwn,j−2 + µ̃jτFn,j−1 + γ̃jτFn0, j = 2, swn+1 = wns,

Fnk = F (tn + ckτ,wnk ), wn-approximation of the exact solution at tnMirela Dărău Runge-Kutta-Chebyshev Methods


First-Order Formulas

R(z): first-order damped polynomial Ps.we select bj so that

Rj (z) =Tj (ω0 + ω1z)

Tj (ω0), ω1 =

Ts(ω0)T ′s(ω0)

, j = 1, . . . , s.

⇒ bj = 1Tj (ω0) , j = 0, . . . , s.

Observation: Rj (z) = ecj z + O(z2) with

cj =Ts(ω0)T ′s(ω0)

T ′j (ω0)Tj (ω0)

≈ j2

s2(1 ≤ j ≤ s − 1) cs = 1.





Rj (z) =Tj (ω0 + ω1z)

Tj (ω0), ω1 =

Ts(ω0)T ′s(ω0)

, j = 1, . . . , s.

⇒ bj = 1Tj (ω0) , j = 0, . . . , s.



T ′j (ω0)Tj (ω0)

≈ j2

s2(1 ≤ j ≤ s − 1) cs = 1.





Rj (z) =Tj (ω0 + ω1z)

Tj (ω0), ω1 =

Ts(ω0)T ′s(ω0)

, j = 1, . . . , s.

⇒ bj = 1Tj (ω0) , j = 0, . . . , s.



T ′j (ω0)Tj (ω0)

≈ j2

s2(1 ≤ j ≤ s − 1) cs = 1.



Second-Order Formulas

R(z): second-order damped polynomial Bs.we select bj so that

Rj (z) = 1 + bjω1T ′j (ω0)z +12

bjω21T′′j (ω0)z

2 + O(z3)

matchesRj (z) = 1 + cjz +

12

(cjz)2 + O(z3)

⇒ bj =T ′′j (ω0)

(T ′j (ω0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T ′s(ω0)T ′′s (ω0)

T ′′j (ω0)T ′j (ω0)

≈ j2 − 1

s2 − 1(2 ≤ j ≤ s − 1) cs = 1.





Rj (z) = 1 + bjω1T ′j (ω0)z +12


2 + O(z3)


12

(cjz)2 + O(z3)

⇒ bj =T ′′j (ω0)

(T ′j (ω0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T ′s(ω0)T ′′s (ω0)

T ′′j (ω0)T ′j (ω0)

≈ j2 − 1

s2 − 1(2 ≤ j ≤ s − 1) cs = 1.





Rj (z) = 1 + bjω1T ′j (ω0)z +12


2 + O(z3)


12

(cjz)2 + O(z3)

⇒ bj =T ′′j (ω0)

(T ′j (ω0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T ′s(ω0)T ′′s (ω0)

T ′′j (ω0)T ′j (ω0)

≈ j2 − 1

s2 − 1(2 ≤ j ≤ s − 1) cs = 1.



Stability RegionsSimulations

Stability Regions

-2.5 -2.0 -1.5 -1.0 -0.5

-3

-2

-1

1

2

3RKF4

-6 -4 -2

-1.5

-1.0

-0.5

0.5

1.0

1.5P2 with damping

-30 -25 -20 -15 -10 -5

-3

-2

-1

1

2

3P4 with damping

-2.0 -1.5 -1.0 -0.5

-1.5

-1.0

-0.5

0.5

1.0

1.5

B2 with damping

-10 -8 -6 -4 -2

-2

-1

1

2B4 with damping




From Stability to Instability

a = −1, λ = 1, ε = 10−2, ω = 10, Nt = 100, Nx = 70,71,72,73


test4.movMedia File (video/quicktime)






a = −1, λ = 1, ε = 10−2, ω = 10

Nt = 100, stability function: P2, B2, P4, B4; Nx = 105,70,122,115








a = −1, λ = 1, ε = 1, ω = 10

Nt = 200, stability function: P2, B2, P4, B4; Nx = 20,11,40,23








a = −1, λ = 1, ε = 10, ω = 10, Nt = 10000

Stab Function NxP2 45B2 20P4 85B4 50




Changing ω

a = −1, λ = 1, ε = 1, ω = 10,3, Nt = 200, stability function: P2





Summary

stability polynomials with extended stability regionbased on these Runge-Kutta-type numerical methodstested on the advection-diffusion-reaction equation→ stabilityregion grows for different numbers of stages



Summary




Summary



IntroductionStabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation

Stability PolynomialsFirst-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials

Integration FormulasNumerical SimulationsStability RegionsSimulations

Summary

Runge-Kutta-Chebyshev Methods...Runge-Kutta-Chebyshev Methods Mirela Dar˘ au˘ Department of...

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Transcript of Runge-Kutta-Chebyshev Methods...Runge-Kutta-Chebyshev Methods Mirela Dar˘ au˘ Department of...