implicit two-derivative runge-kutta methods · implicit two-derivative runge-kutta methods angela...
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Two-Derivative Runge-Kutta
Methods
Angela Tsai(joint work with Shixiao Wang and Robert Chan)
Department of Mathematics
The University of Auckland
SciCADE 2011, Toronto, Canada11-15 July
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Outline of Topics
1 Two-Derivative Runge-Kutta (TDRK) Methods
2 TDRK Methods for ODEs
3 TDRK Methods for PDEs
4 Discussion/Conclusion
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Two-derivative Runge-Kutta (TDRK) methods belong to thefamily of multi-derivative Runge-Kutta methods – they areone-step multi-stage methods.
We consider an autonomous ODE system y′(t) = f(y) withinitial condition y0 = y(t0) and known second derivativey′′(t) = f ′(y)f(y) =: g(y).
Numerical Scheme:
Yi = yn + h
s∑
j=1
aijf(Yj) + h2
s∑
j=1
aijg(Yj), i = 1, . . . , s,
yn+1 = yn + h
s∑
i=1
bif(Yi) + h2
s∑
i=1
big(Yi).
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Two-derivative Runge-Kutta (TDRK) methods belong to thefamily of multi-derivative Runge-Kutta methods – they areone-step multi-stage methods.
We consider an autonomous ODE system y′(t) = f(y) withinitial condition y0 = y(t0) and known second derivativey′′(t) = f ′(y)f(y) =: g(y).
Numerical Scheme:
Yi = yn + h
s∑
j=1
aijf(Yj) + h2
s∑
j=1
aijg(Yj), i = 1, . . . , s,
yn+1 = yn + h
s∑
i=1
bif(Yi) + h2
s∑
i=1
big(Yi).
3/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Two-derivative Runge-Kutta (TDRK) methods belong to thefamily of multi-derivative Runge-Kutta methods – they areone-step multi-stage methods.
We consider an autonomous ODE system y′(t) = f(y) withinitial condition y0 = y(t0) and known second derivativey′′(t) = f ′(y)f(y) =: g(y).
Numerical Scheme:
Yi = yn + h
s∑
j=1
aijf(Yj) + h2
s∑
j=1
aijg(Yj), i = 1, . . . , s,
yn+1 = yn + h
s∑
i=1
bif(Yi) + h2
s∑
i=1
big(Yi).
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
In a non-autonomous system, the variable t can be treated asa component of the y vector.
Block Matrix Form:
Y = e ⊗ yn + h(A ⊗ IN )F (Y ) + h2(A ⊗ IN )G(Y ),
yn+1 = yn + h(bT⊗ IN )F (Y ) + h2(bT
⊗ IN )G(Y ),
where e = [1]s×1, A = [aij ]s×s, A = [aij ]s×s, b = [bi]s×1,
b = [bi]s×1, and
Y =
Y1
Y2
...Ys
, F (Y ) =
f(Y1)f(Y2)
...f(Ys)
, G(Y ) =
g(Y1)g(Y2)
...g(Ys)
.
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
In a non-autonomous system, the variable t can be treated asa component of the y vector.
Block Matrix Form:
Y = e ⊗ yn + h(A ⊗ IN )F (Y ) + h2(A ⊗ IN )G(Y ),
yn+1 = yn + h(bT⊗ IN )F (Y ) + h2(bT
⊗ IN )G(Y ),
where e = [1]s×1, A = [aij ]s×s, A = [aij ]s×s, b = [bi]s×1,
b = [bi]s×1, and
Y =
Y1
Y2
...Ys
, F (Y ) =
f(Y1)f(Y2)
...f(Ys)
, G(Y ) =
g(Y1)g(Y2)
...g(Ys)
.
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
(tn, yn)
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
(tn, yn)
Φh(tn, yn)
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
(tn, yn)
Φh(tn, yn)
(tn+1, yn+1)
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Extended Butcher Tableau:
c A A
bT bT
Stability Function: For the standard test problem y′(t) = λy,yn+1 = R(z)yn, where
R(z) = 1 + (zbT + z2 bT )(I − zA − z2A)−1e, with z = hλ.
Symmetry Conditions:
(tn, yn)
Φh(tn, yn)
(tn+1, yn+1)
Φ−h(tn+1, yn+1)
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Symmetry Conditions:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Symmetry Conditions:PAP = ebT
− A
PAP = −ebT + A
Pb = b
P b = −b
where P is the permutation matrix which reverses the stages.
Order Conditions: As for RK methods, we compare the TaylorSeries expansions of the exact and numerical solutions,y(tn + h) and yn+1 respectively, to derive the order conditionsof methods.
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Basic Background
Symmetry Conditions:PAP = ebT
− A
PAP = −ebT + A
Pb = b
P b = −b
where P is the permutation matrix which reverses the stages.
Order Conditions: As for RK methods, we compare the TaylorSeries expansions of the exact and numerical solutions,y(tn + h) and yn+1 respectively, to derive the order conditionsof methods.
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Discussion/Conclusion
Order Conditions
Order conditions assuming C(1):
Order Tree Order Condition
1 bT e = 1
2 bT c + bT e = 1
2
3 bT c2 + 2bT c = 1
3
bT Ac + bT Ae + bT c = 1
6
4 bT c3 + 3bT c2 = 1
4
bT cAc + bT cAe + bT c2 + bT Ac + bT Ae = 1
8
bT Ac2 + 2bT Ac + bT c2 = 1
12
bT A2c + bT AAe + bT Ac + bT Ac + bT Ae = 1
24
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Order Conditions
Order conditions assuming C(1):
Order Tree Order Condition
1 bT e = 1
2 bT c + bT e = 1
2
3 bT c2 + 2bT c = 1
3
bT Ac + bT Ae + bT c = 1
6
4 bT c3 + 3bT c2 = 1
4
bT cAc + bT cAe + bT c2 + bT Ac + bT Ae = 1
8
bT Ac2 + 2bT Ac + bT c2 = 1
12
bT A2c + bT AAe + bT Ac + bT Ac + bT Ae = 1
24
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
Example of Labelling Trees:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
bT
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
bT
c
c
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
bT
c
c
bT
cc2
2
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
bT
c
c
bT
cc2
2
bT
c2
2
8/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
c
c
A
bT
c Ae
bT
c
A
bT
Ae
bT
c
c
bT
cc2
2
bT
c2
2
bT
c
c
8/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
cc2
2
bT
c2
2
bT
c
c
1
2bT c3 + 1
2bT c2 + bT c2 = 1
8
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
cc2
2
bT
c2
2
bT
c
c
1
2bT c3 + 3
2bT c2 = 1
8
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Simplifying Assumptions and Labelling Trees
Stage Order Conditions:
C(q) : Ack−1 + (k − 1)Ack−2 =ck
k, k = 1, . . . , q.
Bushy Tree Conditions:
B(p) : bT ck−1 + (k − 1)bT ck−2 =1
k, k = 1, . . . , p.
Example of Labelling Trees:
bT
cc2
2
bT
c2
2
bT
c
c
1
2bT c3 + 3
2bT c2 = 1
8
bT c3 + 3bT c2 = 1
4
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
9/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
9/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
9/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
9/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
9/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
In our study, we include two special groups of explicit TDRKmethods:
GROUP A:
c Ae1 A
b1 bT⇒
c A
bT
GROUP B:
c A Ae1
bT b1
We also constructed embedded explicit TDRK methods tocompare with some popular embedded explicit RK methods.
Explicit TDRK methods can easily have stage order 2, i.e.they satisfy the C(2) conditions.
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TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
TDRK45b/TDRK5b: p = 5, q = 2
0 0 0 0 01
3
1
180 0 0
1
2
1
80 0 0
4
5−
2
125
42
1250 0
1 5
48
9
280 25
336order 5
1 1
60 1
30 order 4
TDRK5b requires 1f + 3g function evaluations per step, andR(z) = 1 + z + z2
2+ z3
6+ z4
24+ z5
120+ z6
720.
TDRK45b is an embedded method which requires 1f + 4gfunction evaluations per step.
10/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
TDRK45b/TDRK5b: p = 5, q = 2
0 0 0 0 01
3
1
180 0 0
1
2
1
80 0 0
4
5−
2
125
42
1250 0
1 5
48
9
280 25
336order 5
1 1
60 1
30 order 4
TDRK5b requires 1f + 3g function evaluations per step, andR(z) = 1 + z + z2
2+ z3
6+ z4
24+ z5
120+ z6
720.
TDRK45b is an embedded method which requires 1f + 4gfunction evaluations per step.
10/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Explicit TDRK Methods
TDRK45b/TDRK5b: p = 5, q = 2
0 0 0 0 01
3
1
180 0 0
1
2
1
80 0 0
4
5−
2
125
42
1250 0
1 5
48
9
280 25
336order 5
1 1
60 1
30 order 4
TDRK5b requires 1f + 3g function evaluations per step, andR(z) = 1 + z + z2
2+ z3
6+ z4
24+ z5
120+ z6
720.
TDRK45b is an embedded method which requires 1f + 4gfunction evaluations per step.
10/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Implicit TDRK Methods
We have constructed several implicit TDRK methods, forexample, TDRK244sss is a 2-stage, order-4, stage-order-4,semi-implicit, symmetric, and stiffly-accurate method:
0 0 0 0 0
1 1
2
1
2
1
12−
1
12
1
2
1
2
1
12−
1
12
R(z) =12 + 6z + z2
12 − 6z + z2
The implicit TDRK methods we constructed range from order3 to 6, the order-3 and 5 methods are L-stable and theorder-4 and 6 methods are A-stable.
11/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Implicit TDRK Methods
We have constructed several implicit TDRK methods, forexample, TDRK244sss is a 2-stage, order-4, stage-order-4,semi-implicit, symmetric, and stiffly-accurate method:
0 0 0 0 0
1 1
2
1
2
1
12−
1
12
1
2
1
2
1
12−
1
12
R(z) =12 + 6z + z2
12 − 6z + z2
The implicit TDRK methods we constructed range from order3 to 6, the order-3 and 5 methods are L-stable and theorder-4 and 6 methods are A-stable.
11/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Constructing Implicit TDRK Methods
We have constructed several implicit TDRK methods, forexample, TDRK244sss is a 2-stage, order-4, stage-order-4,semi-implicit, symmetric, and stiffly-accurate method:
0 0 0 0 0
1 1
2
1
2
1
12−
1
12
1
2
1
2
1
12−
1
12
R(z) =12 + 6z + z2
12 − 6z + z2
The implicit TDRK methods we constructed range from order3 to 6, the order-3 and 5 methods are L-stable and theorder-4 and 6 methods are A-stable.
11/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Stiff ODE Problems
Prothero-Robinson Problem (PR):
y′(t) = λ(y(t) − φ(t)) + φ′(t),
we show the results for φ(t) = sin(t) and two cases for theimplicit methods,
PR1b: y0 = φ0 and λ = −104. Exact solution is y(t) = φ(t).PR1d: y0 = 1 and λ = −104. Exact solution isy(t) = φ(t) + (y0 − φ0) exp(λt).
Kaps Problem:
y′(t) =
[−y1(1 + y1) + y2
λ(y21 − y2) − 2y2
], y(0) =
[11
], Re(λ) ≫ 1,
with exact solution y(t) = [exp(−t), exp(−2t)]T .
12/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Stiff ODE Problems
Prothero-Robinson Problem (PR):
y′(t) = λ(y(t) − φ(t)) + φ′(t),
we show the results for φ(t) = sin(t) and two cases for theimplicit methods,
PR1b: y0 = φ0 and λ = −104. Exact solution is y(t) = φ(t).PR1d: y0 = 1 and λ = −104. Exact solution isy(t) = φ(t) + (y0 − φ0) exp(λt).
Kaps Problem:
y′(t) =
[−y1(1 + y1) + y2
λ(y21 − y2) − 2y2
], y(0) =
[11
], Re(λ) ≫ 1,
with exact solution y(t) = [exp(−t), exp(−2t)]T .
12/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Stiff ODE Problems
Prothero-Robinson Problem (PR):
y′(t) = λ(y(t) − φ(t)) + φ′(t),
we show the results for φ(t) = sin(t) and two cases for theimplicit methods,
PR1b: y0 = φ0 and λ = −104. Exact solution is y(t) = φ(t).PR1d: y0 = 1 and λ = −104. Exact solution isy(t) = φ(t) + (y0 − φ0) exp(λt).
Kaps Problem:
y′(t) =
[−y1(1 + y1) + y2
λ(y21 − y2) − 2y2
], y(0) =
[11
], Re(λ) ≫ 1,
with exact solution y(t) = [exp(−t), exp(−2t)]T .
12/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Stiff ODE Problems
Prothero-Robinson Problem (PR):
y′(t) = λ(y(t) − φ(t)) + φ′(t),
we show the results for φ(t) = sin(t) and two cases for theimplicit methods,
PR1b: y0 = φ0 and λ = −104. Exact solution is y(t) = φ(t).PR1d: y0 = 1 and λ = −104. Exact solution isy(t) = φ(t) + (y0 − φ0) exp(λt).
Kaps Problem:
y′(t) =
[−y1(1 + y1) + y2
λ(y21 − y2) − 2y2
], y(0) =
[11
], Re(λ) ≫ 1,
with exact solution y(t) = [exp(−t), exp(−2t)]T .
12/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Explicit Methods for PR Problem
−2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2−14
−12
−10
−8
−6
−4
log10
(Work Done)
log 10
|Err
or|
(a) Efficiency Diagram, Prothero−Robinson, λ=−1, order 5, xf=2.8π
F5CK5DP5TDRK5cTDRK5eRK6
−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−16
−14
−12
−10
−8
−6
−4
log10
(Work Done)
log 10
|Err
or|
(b) Efficiency Diagram, Prothero−Robinson, λ=−200, order 5, xf=2.8π
13/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Embedded Explicit Methods for PR Problem
−2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8−14
−12
−10
−8
−6
log10
(Work Done)
log 10
|Err
or|
(a) Efficiency Diagram: Prothero−Robinson, λ=−1, variable stepsize, (4,5) pair
F45DP45CK45TDRK45b
−1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2−16
−14
−12
−10
−8
−6
log10
(Work Done)
log 10
|Err
or|
(b) Efficiency Diagram: Prothero−Robinson, λ=−200, variable stepsize, (4,5) pair
14/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1−16
−14
−12
−10
−8
−6
−4
−2
0
2
log10
(Work Done)
log 10
|y E
rror
|Efficiency Diagram: PR1b, λ=−104 at t=10 (full Newton)
Gauss2Gauss3Radau3Radau5TDRK232ssLTDRK334sLTDRK244sssTDRK344sssTDRK353ssLTDRK366fss
15/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
3 TDRK232ssL
{O(h3/z2) = O(h/λ2)O(h4/z) = O(h3/λ)
{O(h/λ2) small h
O(h3/λ) large h
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
3 TDRK232ssL
{O(h3/z2) = O(h/λ2)O(h4/z) = O(h3/λ)
{O(h/λ2) small h
O(h3/λ) large h
3 TDRK334sL O(h4) O(h4)
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
3 TDRK232ssL
{O(h3/z2) = O(h/λ2)O(h4/z) = O(h3/λ)
{O(h/λ2) small h
O(h3/λ) large h
3 TDRK334sL O(h4) O(h4)
4 TDRK244sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
4 TDRK344sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
3 TDRK232ssL
{O(h3/z2) = O(h/λ2)O(h4/z) = O(h3/λ)
{O(h/λ2) small h
O(h3/λ) large h
3 TDRK334sL O(h4) O(h4)
4 TDRK244sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
4 TDRK344sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
5 TDRK353ssL O(h4/z2) = O(h2/λ2) O(h2/λ2)
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
Order Behaviour – Error for PR problem when h → 0 andz = λh → ∞:
p Method Local Error Global Error
3 TDRK232ssL
{O(h3/z2) = O(h/λ2)O(h4/z) = O(h3/λ)
{O(h/λ2) small h
O(h3/λ) large h
3 TDRK334sL O(h4) O(h4)
4 TDRK244sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
4 TDRK344sss O(h5/z2) = O(h3/λ2) O(h2/λ2)
5 TDRK353ssL O(h4/z2) = O(h2/λ2) O(h2/λ2)
6 TDRK366fss O(h7/z2) = O(h5/λ2) O(h4/λ2)
16/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for PR Problem
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5−16
−14
−12
−10
−8
−6
−4
−2
0
2
log10
(Work Done)
log 10
|y E
rror
|Efficiency Diagram: PR1d, λ=−104 at t=10 (full Newton)
Gauss2Gauss3Radau3Radau5TDRK232ssLTDRK334sLTDRK244sssTDRK344sssTDRK353ssLTDRK366fss
17/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for Kaps Problem
−3.5 −3 −2.5 −2 −1.5 −1−14
−12
−10
−8
−6
−4
−2
0
log10
(Work Done)
log 10
|| y
Err
or ||
2Efficiency Diagram: K3, λ = 106 at t=1
Gauss2Gauss3Radau3Radau5TDRK232ssLTDRK334sLTDRK244sssTDRK344sssTDRK353ssLTDRK366fss
18/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Implicit Methods for HIRES Problem
−1 −0.5 0 0.5 1 1.5−16
−14
−12
−10
−8
−6
−4
log10
(Work Done)
log 10
|| y
Err
or ||
2Efficiency Diagram: HIRES Problem at t=321.8122
Gauss2Gauss3Radau3Radau5TDRK232ssLTDRK334sLTDRK244sssTDRK344sssTDRK353ssLTDRK366fss
19/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
Uj
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
Uj
Uj−1 Uj+1
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Semi-discretization (or Method of Lines) is used toapproximate PDEs by
firstly, discretize the spatial variables of PDEs to get a set ofODEs,and then integrate along the time variable.
However, many popular classical PDE methods are not MOL.Why?Two main disadvantages of MOL:
Stability is restricted by spatial discretization, possibly leadingto unstable methods.Approximation to higher order derivatives depends on thediscretization used and often leads to non-optimal spread-outschemes.
Uj
Uj−1 Uj+1Uj−1 Uj+1Uj−2 Uj Uj+2
20/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Consider the advection/wave equation,
∂u
∂t+ a
∂u
∂x= 0 on the interval (0, 1) with u(0, t) = u(1, t).
By using central differences, we semi-discretize the PDE to anODE system du(t)/dt = Ahu(t) with spatial stepsizeh = 1/N , and then integrate the system by an explicit RKmethod with temporal stepsize δ. It follows that z∗ must stayinside the stability region of the RK method to ensure thetime integration is stable, where z∗ = δλk, for k = 1, . . . , Nand λk are the eigenvalues of Ah.
21/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Consider the advection/wave equation,
∂u
∂t+ a
∂u
∂x= 0 on the interval (0, 1) with u(0, t) = u(1, t).
By using central differences, we semi-discretize the PDE to anODE system du(t)/dt = Ahu(t) with spatial stepsizeh = 1/N , and then integrate the system by an explicit RKmethod with temporal stepsize δ. It follows that z∗ must stayinside the stability region of the RK method to ensure thetime integration is stable, where z∗ = δλk, for k = 1, . . . , Nand λk are the eigenvalues of Ah.
21/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods
Consider the advection/wave equation,
∂u
∂t+ a
∂u
∂x= 0 on the interval (0, 1) with u(0, t) = u(1, t).
By using central differences, we semi-discretize the PDE to anODE system du(t)/dt = Ahu(t) with spatial stepsizeh = 1/N , and then integrate the system by an explicit RKmethod with temporal stepsize δ. It follows that z∗ must stayinside the stability region of the RK method to ensure thetime integration is stable, where z∗ = δλk, for k = 1, . . . , Nand λk are the eigenvalues of Ah.
21/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Classical PDE Methods – Method of Lines
−3 −2 −1 0 1−4
−3
−2
−1
0
1
2
3
4
Re(z)
Im(z
)
Advection Equation: Stability Regions and Eigenvalues, a = 0.1, N = 40
RK1
RK2
RK3
RK4
δ λk, δ=1.0
−3 −2 −1 0 1−4
−3
−2
−1
0
1
2
3
4
Re(z)
Im(z
)
RK1
RK2
RK3
RK4
δ λk, δ=0.6
22/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
We want to develop new discretization methods whichovercome the disadvantages of MOL and unify MOL and otherclassical PDE methods under the same RK/TDRK structure.
The idea is simple: we discretize the temporal variable t first.This means that the spatial discretization can then be chosenin a more flexible way to meet stability and/or computationalrequirements.
Let f(η) be a smooth function of η and we examine
∂u
∂t= f(P(u)), (1)
where P(u) be a linear partial differential operator withconstant coefficients. For examples: P(u) = ∂
∂xu and
P(u) = ∂2
∂x2 u.
23/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
We want to develop new discretization methods whichovercome the disadvantages of MOL and unify MOL and otherclassical PDE methods under the same RK/TDRK structure.
The idea is simple: we discretize the temporal variable t first.This means that the spatial discretization can then be chosenin a more flexible way to meet stability and/or computationalrequirements.
Let f(η) be a smooth function of η and we examine
∂u
∂t= f(P(u)), (1)
where P(u) be a linear partial differential operator withconstant coefficients. For examples: P(u) = ∂
∂xu and
P(u) = ∂2
∂x2 u.
23/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
We want to develop new discretization methods whichovercome the disadvantages of MOL and unify MOL and otherclassical PDE methods under the same RK/TDRK structure.
The idea is simple: we discretize the temporal variable t first.This means that the spatial discretization can then be chosenin a more flexible way to meet stability and/or computationalrequirements.
Let f(η) be a smooth function of η and we examine
∂u
∂t= f(P(u)), (1)
where P(u) be a linear partial differential operator withconstant coefficients. For examples: P(u) = ∂
∂xu and
P(u) = ∂2
∂x2 u.
23/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
A Novel Semi-Discretization Method
Differentiate (1) with respect to t, we get
∂2u
∂t2= fη(P(u))P(f(P(u)))
= fηP(f).
Compare withd2y
dt2= fyf for y′(t) = f(y).
Similarly, we can derive all the higher derivatives and applythe tree theory for ODEs on PDEs.
This enables us to apply ODE methods directly to PDEs.
If we apply the explicit trapezoidal rule to the wave equationwith appropriate compact schemes to approximate the spatialderivatives, we have a method which has order-2 behaviour inboth time and space. In fact, this is the well-knownLax-Wendroff scheme.
24/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
An Application to PDE
One popular method for solving PDE problems, such as theheat equation, is Crank-Nicolson method, which is an order-2method.
Heat Equation:
Ut = Uxx,
with I.C. U(x, 0) = sin(πx) and B.C. U(0, t) = U(1, t) = 0.
Crank-Nicolson method: use 3-point second orderapproximation to Uxx and implicit midpoint or trapezoidalrule to solve the resulting tridiagonal system of ODEs.
25/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
An Application to PDE
One popular method for solving PDE problems, such as theheat equation, is Crank-Nicolson method, which is an order-2method.
Heat Equation:
Ut = Uxx,
with I.C. U(x, 0) = sin(πx) and B.C. U(0, t) = U(1, t) = 0.
Crank-Nicolson method: use 3-point second orderapproximation to Uxx and implicit midpoint or trapezoidalrule to solve the resulting tridiagonal system of ODEs.
25/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
An Application to PDE
One popular method for solving PDE problems, such as theheat equation, is Crank-Nicolson method, which is an order-2method.
Heat Equation:
Ut = Uxx,
with I.C. U(x, 0) = sin(πx) and B.C. U(0, t) = U(1, t) = 0.
Crank-Nicolson method: use 3-point second orderapproximation to Uxx and implicit midpoint or trapezoidalrule to solve the resulting tridiagonal system of ODEs.
25/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6−10
−9
−8
−7
−6
−5
−4
log10
(Work Done)
log 10
|Err
or|
Efficiency Diagram of heat equation at t = 1.0
Crank−NicolsonTDRK244sss
26/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation and Order Reduction
Practically, Crank-Nicolson method performs better thanother higher order methods which suffer order reduction.
Analyze the diffusion equation with non-homogeneousboundary values
∂u
∂t=
∂2u
∂x2on the interval (0, 1),
u(0, t) = p(t), u(1, t) = 0, and u(x, 0) = 0.
After obtaining the semi-discrete system, we can decouple itto a equivalent non-homogenous ODE system which can thenbe written as the Prothero-Robinson equation.
We conduct experiments for p(t) = tα with different α valuesto compare the order behaviour of three methods:Crank-Nicolson, TDRK244sss and Gauss 2-stage methods.
27/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation and Order Reduction
Practically, Crank-Nicolson method performs better thanother higher order methods which suffer order reduction.
Analyze the diffusion equation with non-homogeneousboundary values
∂u
∂t=
∂2u
∂x2on the interval (0, 1),
u(0, t) = p(t), u(1, t) = 0, and u(x, 0) = 0.
After obtaining the semi-discrete system, we can decouple itto a equivalent non-homogenous ODE system which can thenbe written as the Prothero-Robinson equation.
We conduct experiments for p(t) = tα with different α valuesto compare the order behaviour of three methods:Crank-Nicolson, TDRK244sss and Gauss 2-stage methods.
27/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation and Order Reduction
Practically, Crank-Nicolson method performs better thanother higher order methods which suffer order reduction.
Analyze the diffusion equation with non-homogeneousboundary values
∂u
∂t=
∂2u
∂x2on the interval (0, 1),
u(0, t) = p(t), u(1, t) = 0, and u(x, 0) = 0.
After obtaining the semi-discrete system, we can decouple itto a equivalent non-homogenous ODE system which can thenbe written as the Prothero-Robinson equation.
We conduct experiments for p(t) = tα with different α valuesto compare the order behaviour of three methods:Crank-Nicolson, TDRK244sss and Gauss 2-stage methods.
27/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation and Order Reduction
Practically, Crank-Nicolson method performs better thanother higher order methods which suffer order reduction.
Analyze the diffusion equation with non-homogeneousboundary values
∂u
∂t=
∂2u
∂x2on the interval (0, 1),
u(0, t) = p(t), u(1, t) = 0, and u(x, 0) = 0.
After obtaining the semi-discrete system, we can decouple itto a equivalent non-homogenous ODE system which can thenbe written as the Prothero-Robinson equation.
We conduct experiments for p(t) = tα with different α valuesto compare the order behaviour of three methods:Crank-Nicolson, TDRK244sss and Gauss 2-stage methods.
27/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation with u(0, t) = tα
−2 −1 0−6
−5
−4
−3
−2
−1lo
g 10|E
rror
|α = 1.1
Crank−NicolsonGauss 2
−2 −1 0−7
−6
−5
−4
−3
−2Order Plot: α = 1.5
−2 −1 0−8
−7
−6
−5
−4
−3
−2α = 1.9
−2 −1 0−10
−8
−6
−4
−2
log10
h
log 10
|Err
or|
α = 2.0
−2 −1 0−9
−8
−7
−6
−5
−4
−3
−2
log10
h
α = 2.1
−2 −1 0−10
−8
−6
−4
−2
log10
h
α = 2.9
28/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Heat Equation with u(0, t) = tα
−2 −1 0−8
−7
−6
−5
−4
−3
−2
−1lo
g 10|E
rror
|α = 1.1
−2 −1 0−9
−8
−7
−6
−5
−4
−3
−2Order Plot: α = 1.5
−2 −1 0−10
−8
−6
−4
−2α = 1.9
−2 −1 0−10
−8
−6
−4
−2
log10
h
log 10
|Err
or|
α = 2.0
−2 −1 0−12
−10
−8
−6
−4
−2
log10
h
α = 2.1
−2 −1 0−12
−10
−8
−6
−4
−2
log10
h
α = 2.9
Crank−NicolsonTDRK244sssGauss 2
28/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
TDRK methods are more efficient compared with somepopular RK methods for the stiff problems we tested.
The second derivative terms in TDRK give us more freedomand enable us to construct methods with higher stage order.
Although the cost of calculating the second derivatives maybe higher than the first derivatives, the advantage gainedmakes their use beneficial.
For ODE problems: Our study suggests it will be of interest toimplement a variable stepsize code for implicit TDRKmethods.
29/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
TDRK methods are more efficient compared with somepopular RK methods for the stiff problems we tested.
The second derivative terms in TDRK give us more freedomand enable us to construct methods with higher stage order.
Although the cost of calculating the second derivatives maybe higher than the first derivatives, the advantage gainedmakes their use beneficial.
For ODE problems: Our study suggests it will be of interest toimplement a variable stepsize code for implicit TDRKmethods.
29/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
TDRK methods are more efficient compared with somepopular RK methods for the stiff problems we tested.
The second derivative terms in TDRK give us more freedomand enable us to construct methods with higher stage order.
Although the cost of calculating the second derivatives maybe higher than the first derivatives, the advantage gainedmakes their use beneficial.
For ODE problems: Our study suggests it will be of interest toimplement a variable stepsize code for implicit TDRKmethods.
29/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
TDRK methods are more efficient compared with somepopular RK methods for the stiff problems we tested.
The second derivative terms in TDRK give us more freedomand enable us to construct methods with higher stage order.
Although the cost of calculating the second derivatives maybe higher than the first derivatives, the advantage gainedmakes their use beneficial.
For ODE problems: Our study suggests it will be of interest toimplement a variable stepsize code for implicit TDRKmethods.
29/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
We have developed a novel approach for the discretization ofPDEs.
This approach allows for more compact finite schemes for thehigher derivatives and will provide a systematic way to applyODE methods to PDEs.
Many classical PDE schemes can be interpreted in the sameway in terms of our new approach.
The order-4 TDRK244sss method only requires twice the costof the order-2 Crank-Nicolson method and is shown to bemore efficient.
We will further explore this type of numerical scheme forsolving diffusion and diffusion-advection equations of higherdimension in the future.
30/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
We have developed a novel approach for the discretization ofPDEs.
This approach allows for more compact finite schemes for thehigher derivatives and will provide a systematic way to applyODE methods to PDEs.
Many classical PDE schemes can be interpreted in the sameway in terms of our new approach.
The order-4 TDRK244sss method only requires twice the costof the order-2 Crank-Nicolson method and is shown to bemore efficient.
We will further explore this type of numerical scheme forsolving diffusion and diffusion-advection equations of higherdimension in the future.
30/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
We have developed a novel approach for the discretization ofPDEs.
This approach allows for more compact finite schemes for thehigher derivatives and will provide a systematic way to applyODE methods to PDEs.
Many classical PDE schemes can be interpreted in the sameway in terms of our new approach.
The order-4 TDRK244sss method only requires twice the costof the order-2 Crank-Nicolson method and is shown to bemore efficient.
We will further explore this type of numerical scheme forsolving diffusion and diffusion-advection equations of higherdimension in the future.
30/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
We have developed a novel approach for the discretization ofPDEs.
This approach allows for more compact finite schemes for thehigher derivatives and will provide a systematic way to applyODE methods to PDEs.
Many classical PDE schemes can be interpreted in the sameway in terms of our new approach.
The order-4 TDRK244sss method only requires twice the costof the order-2 Crank-Nicolson method and is shown to bemore efficient.
We will further explore this type of numerical scheme forsolving diffusion and diffusion-advection equations of higherdimension in the future.
30/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
Discussion/Conclusion
We have developed a novel approach for the discretization ofPDEs.
This approach allows for more compact finite schemes for thehigher derivatives and will provide a systematic way to applyODE methods to PDEs.
Many classical PDE schemes can be interpreted in the sameway in terms of our new approach.
The order-4 TDRK244sss method only requires twice the costof the order-2 Crank-Nicolson method and is shown to bemore efficient.
We will further explore this type of numerical scheme forsolving diffusion and diffusion-advection equations of higherdimension in the future.
30/31
TDRK MethodsTDRK Methods for ODEsTDRK Methods for PDEs
Discussion/Conclusion
31/31