Runge-Kutta-Chebyshev Methods - TU/e Methods Mirela Dar˘ au ... let z = ˝ ; by applying the method...

Transcript of Runge-Kutta-Chebyshev Methods - TU/e Methods Mirela Dar˘ au ... let z = ˝ ; by applying the method...

IntroductionStability PolynomialsIntegration Formulas

Numerical SimulationsSummary

Runge-Kutta-Chebyshev Methods

Mirela Darau

Department of Mathematics and Computer ScienceTU/e

CASA Seminar, 26th November 2008

Mirela Darau 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 h2, 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 + j1k=0jk F (tn + ck,wnk ), j = 1, . . . , s,

wn+1 = wns,

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

Specifying a particular method:s Z (number of stages)ck and jkcj =

j1k=0jk




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 + si=1biF (tn + ci,wni ),

wni = wn + sj=1ijF (tn + cj,wnj ), i = 1, . . . , s.

stability analysis considering the complex test equationw (t) = w(t)let z = ; 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 = (ij ) 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 = uxx + u(1 u), 0 < x < 1, t > 0,ux (0, t) = 0, t > 0,

u(1, t) =12

(1 + sin(t)), t > 0,

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

wherea - advection velocity

- diffusion coefficient

- source term coefficient

- frequency.



First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials

Shifted Chebyshev Polynomials

Theorem

For any explicit, consistent Runge-Kutta method we have R 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) = 2zTj1(z) Tj2(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

0 = 1 = 1, i =1 (i 1)2/s2

i(2i 1)i1 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 R 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

), R

23

(s21).

this generates about 80% of the optimal interval






Bs(z) =23

+1

3s2+

(13 1

3s2

)Ts

(1 +

3zs2 1

), R

23

(s21).







Bs(z) =23

+1

3s2+

(13 1

3s2

)Ts

(1 +

3zs2 1

), R

23

(s21).







Bs(z) =23

+1

3s2+

(13 1

3s2

)Ts

(1 +

3zs2 1

), R

23

(s21).







Bs(z) =23

+1

3s2+

(13 1

3s2

)Ts

(1 +

3zs2 1

), R

23

(s21).





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(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.

Stability interval: 0 0 + 1z 0 R = 201 ;Ps(z) [Ts(0)1,Ts(0)1].

Convenient: 0 = 1 + s2 , smallDamping 5%Stability boundary 1.93s2

-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps


Ps(z) =Ts(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps


Ps(z) =Ts(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps


Ps(z) =Ts(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps


Ps(z) =Ts(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Ps


Ps(z) =Ts(0 + 1z)

Ts(0), 1 =

Ts(0)T s(0)

, 0 > 1.



-40 -30 -20 -10

-4

-2

2

4P5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T s (0)

(T s(0))2(Ts(0 + 1z) Ts(0)), 1 =

T s(0)T s (0)

.

R (0+1)Ts (0)

T s (0) 23 (s

2 1)(1 215

).

Damping 5%Stability boundaryR 0.9794R,undamped

-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T s (0)

(T s(0))2(Ts(0 + 1z) Ts(0)), 1 =

T s(0)T s (0)

.

R (0+1)Ts (0)

T s (0) 23 (s

2 1)(1 215

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T s (0)

(T s(0))2(Ts(0 + 1z) Ts(0)), 1 =

T s(0)T s (0)

.

R (0+1)Ts (0)

T s (0) 23 (s

2 1)(1 215

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping




Damped Bs

Damped form of Bs:

Bs(z) = 1 +T s (0)

(T s(0))2(Ts(0 + 1z) Ts(0)), 1 =

T s(0)T s (0)

.

R (0+1)Ts (0)

T s (0) 23 (s

2 1)(1 215

).


-15 -10 -5

-3

-2

-1

1

2

3B5 with damping



Method DescriptionAnsatz: Rj (z) = aj + bjTj (0 + 1z), aj = 1 bjTj (0), 1 j s.Imposing Chebyshev recursion:

R0(z) = 1, R1(z) = 1 + 1z,Rj (z) = (1 j j ) + jRj1(z) + jRj2(z) + jRj1(z)z + jz,

where j = 2, . . . , s and

1 = b11, j =2bj0bj1

, j =bjbj2

, j =2bj1bj1

, j = aj1j .

The RKC integration formulas are then of the form:

wn0 = wn,wn1 = wn + 1Fn0, (2)

wnj = (1 j j )wn + jwn,j1 + jwn,j2 + jFn,j1 + jFn0, j = 2, swn+1 = wns,

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


First-Order Formulas

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

Rj (z) =Tj (0 + 1z)

Tj (0), 1 =

Ts(0)T s(0)

, j = 1, . . . , s.

bj = 1Tj (0) , j = 0, . . . , s.

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

cj =Ts(0)T s(0)

T j (0)Tj (0)

j2

s2(1 j s 1) cs = 1.





Rj (z) =Tj (0 + 1z)

Tj (0), 1 =

Ts(0)T s(0)

, j = 1, . . . , s.

bj = 1Tj (0) , j = 0, . . . , s.


cj =Ts(0)T s(0)

T j (0)Tj (0)

j2

s2(1 j s 1) cs = 1.





Rj (z) =Tj (0 + 1z)

Tj (0), 1 =

Ts(0)T s(0)

, j = 1, . . . , s.

bj = 1Tj (0) , j = 0, . . . , s.


cj =Ts(0)T s(0)

T j (0)Tj (0)

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 + bj1T j (0)z +12

bj21Tj (0)z

2 + O(z3)

matchesRj (z) = 1 + cjz +

12

(cjz)2 + O(z3)

bj =T j (0)

(T j (0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T s(0)T s (0)

T j (0)T j (0)

j2 1

s2 1(2 j s 1) cs = 1.





Rj (z) = 1 + bj1T j (0)z +12

bj21Tj (0)z

2 + O(z3)


12

(cjz)2 + O(z3)

bj =T j (0)

(T j (0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T s(0)T s (0)

T j (0)T j (0)

j2 1

s2 1(2 j s 1) cs = 1.





Rj (z) = 1 + bj1T j (0)z +12

bj21Tj (0)z

2 + O(z3)


12

(cjz)2 + O(z3)

bj =T j (0)

(T j (0))2 , j = 2, . . . , s, b0 = b1 = b2


cj =T s(0)T s (0)

T j (0)T j (0)

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, = 1, = 102, = 10, Nt = 100, Nx = 70,71,72,73


test4.movMedia File (video/quicktime)






a = 1, = 1, = 102, = 10

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








a = 1, = 1, = 1, = 10

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








a = 1, = 1, = 10, = 10, Nt = 10000

Stab Function NxP2 45B2 20P4 85B4 50




Changing

a = 1, = 1, = 1, = 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 - TU/e Methods Mirela Dar˘ au ... let z = ˝ ; by applying the method...

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Transcript of Runge-Kutta-Chebyshev Methods - TU/e Methods Mirela Dar˘ au ... let z = ˝ ; by applying the method...