MA557/MA578/CS557
Lecture 36
Spring 2003
Prof. Tim Warburton
timwar@math.unm.edu
1
Time Integration for PDEs with Terms of Mixed Stiffness
• Consider a generic PDE with the following form:
C
 f C   g C 
t
• Examples:
  2C  2C 
C
  C 1  C   D  2  2 
t
y 
 x
• Or:
• Or:
  2C  2C 
C
C
C
 x
y
 D 2  2 
t
x
y
y 
 x
C

t
 C 2 
x
  2C  2C 
 D 2  2 
y 
 x
2
Split Scheme
• If we split C into a stiff component Cs and a non-stiff
component Cns then we can solve the related system:
C NS
 f C 
t
CS
 g C 
t
C  CS  C NS
• We then use two compatible Runge-Kutta schemes to integrate
the equations in lock step.
• The conditions to ensure that the explicit and implicit schemes
advance at the same time in each substage have been
determined and satisfied for matched implicit and explicit (imex)
RK schemes which are both fourth order, six stage.
“Additive Runge–Kutta schemes for convection–diffusion–reaction
equations”, Christopher A. Kennedy, Mark H. Carpenter, Applied Numerical
Mathematics 44 (2003) 139–181.
3
Non-Stiff Component
• Note this is an explicit RK scheme:
f 1  f C n 
C1  C n
C1NS  C1NS
CNS
 f C 
t
for i=2:s
C
i
NS
C
n
NS
 j i 1
j 
 dt   aˆij f 
 j 1

 
f i : f C i
end
C
n 1
NS
C
n 1
NS
 j s ˆ j 
 dt   b j f 
 j 1

4
Stiff Component
• Note this is an implicit RK scheme:
C1  C n
CS1  CSn
for i=2:s
CS
 g C 
t
  
C  C  dt aii g C
i
S
n
S
i
 
 j i 1
j 
 dt   aij g 
 j 1

g i : g C i
end
C
n 1
S
 j s
j 
 C  dt   b j g 
 j 1

n
S
5
Adding The Schemes Together
f 1  f C n 
C1  C n
for i=2:s
  
C  C  dt aii g C
i
n
 
: f  C 
i
 j i 1
 j i 1
j 
j
 dt   aij g   dt   aˆij f 
 j 1

 j 1

g i : g C i
fi
i
end
j s
j s



n 1
n
j
j
ˆ
C  C  dt   b j g   dt   b j f 
 j 1

 j 1

6
Kennedy-Carpenter 6 Stage ARK4
f 1  f C n 
C1  C n

    dt  a g   dt  aˆ f 
 g C  , f  f C 
 C  dt  a g  C    dt  a g  a g   dt  aˆ
 g C  , f  f C 
C 2  C n  dt a22 g C 2
g2
C3
g3
2
1
21
2
n
1
33
...
31
3

21
2
3
3
1
2
32
1
2
ˆ
f

a
f

31
32
3
 
C 6  C n  dt a66 g C 6
 dt  a61 g 1  a62 g 2 ...  a65 g 5 
 dt  aˆ61 f 1  aˆ62 f 2 ...  aˆ65 f 5 
 
g6  g C6 ,
C
n 1
 
f 6  f C6
 j 6
 j 6 ˆ j 
j 
 C  dt   b j g   dt   b j f 
 j 1

 j 1

n
7
6 Stages
f 1  f C n 
C1  C n
    C  dt  a g   dt  aˆ f 
 dt  a g  C    C  dt  a g  a g   dt  aˆ
C 2  dt a22 g C 2
n
C3
n
...
3
33

   C
C
1
n
21
1
31
C 6  dt a66 g C 6
n 1
1
21
2
32
31
f 1  aˆ32 f 2 
 dt  a61 g 1  a62 g 2 ...  a65 g 5   dt  aˆ61 f 1  aˆ62 f 2 ...  aˆ65 f 5 
 j 6
 j 6 ˆ j 
j 
 C  dt   b j g   dt   b j f 
 j 1

 j 1

n
We now see that there is a sequence of solves. Each stage
2-6 requires the solution of a system (as before in the
ESDIRK4 scheme).
8
Linear g – discretized as g
f 1  f C n 
C1  C n
 I  dta22g  C 2  C n  dtg  a21C1   dt  aˆ21 f 1 
 I  dta33g  C 3  C n  dtg  a31C1  a32C 2   dt  aˆ31 f 1  aˆ32 f 2 
...
 I  dta66g  C 6  C n  dtg  a61C1  a62C 2 ...  a65C 5   dt  aˆ61 f 1  aˆ62 f 2 ...  aˆ65 f 5 
C
n 1
 j 6
 j 6 ˆ j 
j 
 C  dtg   b j C   dt   b j f 
 j 1

 j 1

n
We now see that there is a sequence of solves. Each stage
2-6 requires the solution of a system (as before in the
ESDIRK4 scheme).
9
Incorporating the Mass Matrix M for Symmetry
(specific to our DG formulations)
f 1  f C n 
C1  C n
 M  dta22g  C 2  MC n  dtg  a21C1   dt  aˆ21 f 1 
 M  dta33g  C 3  MC n  dtg  a31C1  a32C 2   dt  aˆ31 f 1  aˆ32 f 2 
.. .
 M  dta66g  C 6  MC n  dtg  a61C1  a62C 2 ...  a65C 5   dt  aˆ61 f 1  aˆ62 f 2 ...  aˆ65 f 5 
MC
n 1
 j 6
 j 6 ˆ j 
j 
 MC  dtg   b j C   dt   b j f 
 j 1

 j 1

n
Cn1  M 1  MC n1 
Where for example g is the diffusion operator
g  DM  D Dx D Nx  D Dy D Ny 
10
Kennedy-Carpenter Additive RK Schemes
f 1  f C n 
C1  C n
 M  dt g  C 2  MC n  dtg  a21C1   dt  aˆ21 f 1 
 M  dt g  C 3  MC n  dtg  a31C1  a32C 2   dt  aˆ31 f 1  aˆ32 f 2 
...
 M  dt g  C 6  MC n  dtg  a61C1  a62C 2 ...  a65C 5   dt  aˆ61 f 1  aˆ62 f 2 ...  aˆ65 f 5 
MC
n 1
 j 6
 j 6 ˆ j 
j 
 MC  dtg   b j C   dt   b j f 
 j 1

 j 1

n
Cn1  M 1  MC n1 
• K-C set up the imex schemes so that the Butcher block diagonal entries
• a22=a33=a44=a55=a66=gamma
• Thus the same matrix needs to be inverted at stages 2-5
11
Mixed Implicit Explicit Runge Kutta Scheme (IMEX RK)
• Many papers. Example:
• http://citeseer.nj.nec.com/cache/papers/cs/484/http:z
SzzSzwww.cs.ubc.cazSzspiderzSzascherzSzpapersz
Szars.pdf/ascher97implicitexplicit.pdf
-- or --
• “Additive Runge–Kutta schemes for convection–
diffusion–reaction equations”, Christopher A.
Kennedy, Mark H. Carpenter, Applied Numerical
Mathematics 44 (2003) 139–181.
12
Fourth Order, 6 Stage
Notes:
1) bhat = b
2) See p 176 of
Kennedy/Carpenter
13
K-C ARK4 IMEX Scheme
• 87-91 sum up stiff
terms
• 93-96 sum up non-stiff
terms
• 98-102 invert system
• 104 evaluate f(Ctilde_j)
• 107-119 piece together
C^{n+1}
• Dr Day will discuss the
use of umPERM stages.
14
Project Lab Time
• Start working on your projects.
15