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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
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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}
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
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.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
Stability Polynomials
StabilityPolynomials
First orderconsistency
Second orderconsistency
Ps
Bs
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
First-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Observation: Rj (z) = ecj z + O(z3) with
cj =T ′s(ω0)T ′′s (ω0)
T ′′j (ω0)T ′j (ω0)
≈ j2 − 1
s2 − 1(2 ≤ j ≤ s − 1) cs = 1.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Observation: Rj (z) = ecj z + O(z3) with
cj =T ′s(ω0)T ′′s (ω0)
T ′′j (ω0)T ′j (ω0)
≈ j2 − 1
s2 − 1(2 ≤ j ≤ s − 1) cs = 1.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Observation: Rj (z) = ecj z + O(z3) with
cj =T ′s(ω0)T ′′s (ω0)
T ′′j (ω0)T ′j (ω0)
≈ j2 − 1
s2 − 1(2 ≤ j ≤ s − 1) cs = 1.
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
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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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
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Stability RegionsSimulations
From Stability to Instability
a = −1, λ = 1, ε = 10−2, ω = 10, Nt = 100, Nx = 70,71,72,73
Mirela Dărău Runge-Kutta-Chebyshev Methods
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IntroductionStability PolynomialsIntegration Formulas
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Stability RegionsSimulations
a = −1, λ = 1, ε = 10−2, ω = 10
Nt = 100, stability function: P2, B2, P4, B4; Nx = 105,70,122,115
Mirela Dărău Runge-Kutta-Chebyshev Methods
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IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stability RegionsSimulations
a = −1, λ = 1, ε = 1, ω = 10
Nt = 200, stability function: P2, B2, P4, B4; Nx = 20,11,40,23
Mirela Dărău Runge-Kutta-Chebyshev Methods
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IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stability RegionsSimulations
a = −1, λ = 1, ε = 10, ω = 10, Nt = 10000
Stab Function NxP2 45B2 20P4 85B4 50
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
Stability RegionsSimulations
Changing ω
a = −1, λ = 1, ε = 1, ω = 10,3, Nt = 200, stability function: P2
Mirela Dărău Runge-Kutta-Chebyshev Methods
test1.movMedia File (video/quicktime)
test2.movMedia File (video/quicktime)
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStability PolynomialsIntegration Formulas
Numerical SimulationsSummary
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
Mirela Dărău Runge-Kutta-Chebyshev Methods
IntroductionStabilized Explicit Runge-Kutta MethodsAdvection-Diffusion-Reaction Equation
Stability PolynomialsFirst-Order Stability PolynomialsSecond-Order Stability PolynomialsDamped Stability Polynomials
Integration FormulasNumerical SimulationsStability RegionsSimulations
Summary