High order explicit methods for parabolic equations
and stiff ODEs
(DUMKA3, DUMKA4, ROCK2, ROCK4)
Alexei Medovikov
Tulane University
A. Abdulle, A. Medovikov
Second order Chebyshev methods based on orthogonal polynomials. Numerische Mathematik
V90.1. pp.1-18
Medovikov A.A.
High order explicit methods for parabolic equations. BIT, V38,No2,pp.372-390
Lebedev V.I., Medovikov A.A.
Method of second order accuracy with variable time steps. Izv. Vyssh. Uchebn. Zaved. Mat.
no. 9, 52--60 (English translation).
WELCOME TO DUMKALAND
Explicit numerical methods for stiff differential equations
Dumka3 Examples
Download DUMKA3.cpp (C++)
Download DUMKA3.c (C)
Download DUMKA3.f (FORTRAN)
Download ROCK2/ROCK4 (rock.tar)(FORTRAN)
Applications (medicine, biology, apply math ...)
Motility of microorganisms
E-Mail:amedovik@math.tulane.edu
Phone:(504)862-8396
Address: Mathematics Department
Tulane University New Orleans, LA
Web: http://www.math.tulane.edu/~amedovik
DUMKA3 - integrates initial value problems for systems of first order ordinary
differential equations y'=f(y,t). It is based on a family of explicit Runge-KuttaChebyshev formulas of order three. It uses optimal third order accuracy stability
polynomials with the largest stability region along the negative real axis.
Examples of solution of stiff differential equations by explicit methods
Brusselator equation
  2u
u
 2u 
2
  f(x, y, t)
 1  u v  4.4u    2 
2 
t

x

y


  2v
v
 2v 
2

 3.4u  u v    2 
t
y 2 
 x
Nagumo nerve conduction equation
u
 2u

 f(u)  v
t
x 2
v
 n(u  bv)
t
 5 if (x  0.3)2  ( y  0.6)2  0.12 and t  1.1
f(u)  
else
0,
Burgers equation
duk
uk2  1  uk2 1
u
 2uk  uk 1
 
  k 1
dt
4x
x2
Summary
, due to conditional stability
1.
Stability: Explicit methods have small stepsize
2.
Variable steps can be used to maximize mean stepsize of a sequence of
n
explicit methods
    i / n  c pn2 / n  (n)
3.
Optimal sequence of explicit steps can be found in terms of roots of stability
polynomials, which approximate exponential function and possess
n
p
Chebyshev alternation
Rn(z)  1  z    z / p!  aizi
p 1
Asymptotic formulas and orthogonal polynomials can be used
to construct
such polynomials, even very high degree polynomials (n > 1000)
4.
5.
Accuracy: In order to construct high order explicit methods for non-linear
ODE, we start with stability polynomials and we use B-series in order to
satisfy order conditions, and build Runge-Kutta methods for non-liner ODEs
6.
Efficient stepsize control and step rejection procedure are achieved via
embedded methods
7.
For automatic computation of spectral radius we used non-linear power
method.
Stability analysis of explicit RK methods
dy
 f( y, t)
dt
ODEs:
Explicit Euler method:
Test equation:
Stability function:
Stability region:
Goal:
y 1  y0

y(t0 )  y0
 f( y0, t0 )
y'   λy
y1  y0  y0  (1  )y0
n
y n  (1  ) y0
yn  Rn(h)y0 where h is a total step
S  z  C : Rn(z)  1, z  h
Find stability polynomial which maximize average stepsize
{Sp( f / y)  S}
h / n , given
Stability analysis of explicit RK methods
Explicit Euler method:
y1  (1  )y0
Rn()  1    1
Stability condition of explicit Euler method:
h  
  2/ 
Linear stability analysis for non-linear ODEs:
~  y  err
y
0
0
0
~  ( y  f( y ))  ( y
~  f( y
~)
y1  y
1
0
0
0
0
~ )  J( y )(y  y
~)
 (y  y
0
0
0
0
0
err1  err0  J(y0)err0  (I  J(y0))err0
  2 / M where M  (f(y) / y)
{Sp( f / y)  stability region}
Linear stability RK methods vs. Stability RK methods?
R(z)  1  z, h  , z  
n
Rn(h)  (1   i)
i 1
h 
n
 i  
i 1
dRn( )
d  0
Can we solve stiff ordinary differential equations (ODE) by explicit
methods with stepsize larger than 2/M?
Example:
y'  500( y  cos(t ))
y(0)  0
  2 / 500
Consider two steps
1
and
 2 where 1  1 / 500
y1  (1  500 1 ) y0   1 cos(t0 )
y2  (1  500 2 ) y1   2 cos(t0   1 )
y2  (1  500 2 )(1  5001 ) y0  (1  500 2 )1 cos(t0 )   2 cos(t0  1 )
if 1   2    2 / 500 
instability
if  2  1  1 / 500  stability
(1  500 2)(1  5001)  0  1 for any  2
R1(h)  (1  ), h  
y'  500(y  cos(t))
R1(h)  (1  h1)(1  h2), h  (h1  h2)
n
R1(h)  (1  hi), h 
i 1
n
h
i
i 1
Original idea of Runge-Kutta-Chebyshev methods
Consider sequence of Euler steps and find an optimal polynomial
yn 
n
(1
  i )y0
h 
| Rn( ) ||
h 
n

(1
  i ) |   1
i 1
i
/ n  
i 1

i
i 1
i 1
n
n
dRn( )
d
 0
  Sp ( A)  [0, M ]  S

as large as possible
If we have found the optimal stability polynomial, the variable sequence of
steps can be found in terms of the roots of the stability polynomial
 i  1 / i
The solution for n-stage Runge-Kutta-Chebyshev method order p=1 is
given by Chebyshev stability polynomial.
1  x0
cos n arccosx 
Rn( ) 
, x  x0 
 , x0  1
cos n arccosx0 
M
0    M  1  x( )  x0, x(0)  x0
1
 1 
cos n arccosx0 
x 1
max Rn( )    1, 0
M    M
1  x0
sin(n arccos(x0 ))
1  x0
dRn( )
h(x0 )  
 n
d  0
1  x02 cos(n arccos(x0 )) M
 
2
h(1)  lim h(x0 )  n
x0 1
M
2
2 2
h  n
/n  n
 nheuler
M
M
2
Runge-Kutta methods
n
Yi  y0  h aijf(t0  cjh, Yj),
j 1
n
y1  y0  h bjf(t0  cjh, Yj).
j 1
c1
c2

cn  1
cn
a11
a21

an  11
an1
b1
a12
 a1n 1
a1n

a2n
  

an  12 
an 1n
an2  ann  1 ann
b2  bn  1 bn

cA

b
Stability function of explicit Runge-Kutta method
y'  y

Y  (Y1,  , Yn )t
n
Yi  y0  h aij( )Yj,
j 1
n

y1  y0  h bj( )Yj, y 0  ( y0,  , y0 )t
j 1
Y  y 0  zΑY
t
y 1  y0  z b Y
z  h


y1  (1  z b (I  z) 1)y0, 1  (1  1)t
t
1
if A n  0, aij  0, j  i,

t
t 
t
2
n
n 1
Rn(z)  1  z b 1  z b 1    z b  1
 1  z  z / 2!   z / p!
2
p
n
i
d
z
 i
i  p 1
Theorem (T. Riha): Among all polynomials of the order p the polynomial
which possess Chebyshev alternant, would maximize real stability
interval ln
Rn(z)  1  z  z / 2    z
2
p
/ p!
n

aizi
i  p 1
Rn(z) 1 for z  [ln,0]
or equivalently, the polynomial which possess Chebyshev alternant:
  
M
z
ln
2
ln
 ln
Rn( )  1 
  
  / 2! 
M
M


 l

  n  
M


p
/ p!
n

i  p 1
i
 l

di   n  
M


has maximal possible stepsize
dRn( )
ln
h  

d
M
, given stability
Rn()  1,   [0, M]
Theorem : Let w(x) be a positiveweight functionon the interval[1,1] which satisfies
0  c1  (x)  c2 and | (x   )  (x) || ln  |1    L(where c1,2, , L are fixed positive numbers).
Then the polynomialPn(x) of the least deviationfrom zero withthe
weight function (x)1 / 2 in the intrval[1,1] is expressedin asymptotical form as
(x)Pn(x) 
2

1
where ( ) 
2
cos(n  ( ))   n(x),   arccos(x)
ln (z)  ln (x) 1  x 2
dz

z  x
1  z2
1
1
Theorem : Let w(x) be a positiveweight function
on the interval[1,1] which satisfies0  c1  w(x)  c2 and
| w(x   )  w(x) || ln  |1    L
(where c1,2,
Then the orthogonal polynomials
w(x) / sqrt(1  x 2 ), satisfy
,
L
Pn(x) associated
uniformly
on
are fixed positive numbers).
with
[1,1] :
w(x)Pn(x)  cos(n  ( ))  (ln(n)  ),   arccos(x)
where
1
( ) 
2
ln w(z)  ln w(x) 1  x 2
dz
1
z  x
1  z2
1
the
weight
function
p
p
p
p




1
2
3
4
Rn(0)
Rn(0)
Rn(0)
Rn(0)




1
1
1
1
R'n(0)
R'n(0)
R'n(0)
R'n(0)




ln
ln R'n'(0)  ln2
ln R'n'(0)  ln2 R'n''(0)  ln3
ln R'n'(0)  ln2 R'n''(0)  ln3
(1)
R'n'''(0)  ln4
Two algorithms of computation of stability polynomials:
 p(x) 
(x
 xj)
DUMKA3,4
1. For given n calculate weight
p
j 1
and roots via asymptotic formula for polynomials Pn p (x)
of the least deviation from zero:
(n  p)k  ( k )  (k  0.5), k  1,  , n  p

j1
k

(k  0.5)
n  p

(kj)
, xk  cos(k )
n  p
2. For given n calculate weight
 p()Rn  p( ) satisfies (1),
w p(x) 
p
(x
 xj)
j 1
so that the polynomial R ( )  w ( )P ( )satisfies (1),
n
p
n p
where Pn p (x) is orthogonal polynomial with the weight
(w(x))2
1  x2
ROCK2,4, RKC
so that the polynomial Rn( ) 
w 2(x) 
2
(x
 xj)
j 1
Tk(x1) Tk(x2 ) Tk'(x1) Tk'(x2 ) Tk(x)
Tk  1(x1) Tk  1(x2 ) Tk'  1(x1) Tk'  1(x2 ) Tk  1(x)
Pk(x) 
C
'
'
T
(
x
)
T
(
x
)
T
(
x
)
T
k 2
1
k 2
2
k 2
1
k  2( x2 ) Tk  2( x)
2
w 2(x)
Tk  3(x1) Tk  3(x2 ) Tk'  3(x1) Tk'  3(x2 ) Tk  3(x)
Tk  4(x1) Tk  4(x2 ) Tk'  4(x1) Tk'  4(x2 ) Tk  4(x)
1
 P (x)P(x)
k
1
l
w(x)2
1 x
2
dx   kl
Rn(x)  w 2(x)Pn  2(x)  cos(n  ( ))
p  1 Rn(0)  1 R'n(0)  ln
p  2 Rn(0)  1 R'n(0)  ln R'n'(0)  ln2
Pk(z)  ( kz   k )Pk 1(z)   kPk  2(z), z  [ M,0]
Accuracy:
Order conditions of Runge-Kutta methods
Taylor expansions of the exact solution and numerical solution :
dy J
d 2 y J h2
d 3 y J h3
y (h)  y 
h 


2
3
dt
dt 2
dt 6
h2
h3
J
J
J I
J I K
J I K

 y0  f h 
f
f

f
f
f

f
 I
 IK
 I fK f   
2
6
J
J
0
h
h2
J I
bt31  fIKJ f If K  bt32  fIJ fKIf K   
 b( )f h  bt21  fI f 
2
6
~ (h)  y
y
J
J
0
J
I
where
b( ) 
J
f f
f J I
 2f J
f J f I K
J I K
I K
J I K

f , fIK f f 
f f , fI fK f 
f
y I
y I y K
y I y K
I
t
 b1
n
b
i
i 1
b(t21 )  2
n
ba
i ij
t 
 2b Α1
i,j  1
b(t31 )  3
n

biaijaik
b(t31 )  6
i,j,k  1
b(t41 )  4
n

biaijaikail
i,j,k,l  1

t
 24b Α 3 1
n
ba
a
i ij jk

t
 6 b Α 21
i,j,k  1
b(t42 )  8
n

biaijaikajl
i,j,k,l  1
b(t43 )  12
n

biaijajkajl
i,j,k,l  1
b(t44 )  24
n
ba
a a
i ij jk jl
i,j,k,l  1
Construction of pth order composition method
Let us consider two consecutive steps by Runge-Kutta methods A
and B. We call the method which is the result of one step of A
and one step of B as the composition method C=B(A)
Stability function of the composition method C is the product of
stability functions of the methods A and B
Theory of composition methods allows to calculate Taylor
expansion of composition methods:
c( )
c(t 21 )
c(t31 )
c(t32 )
c(t 41 )





c(t 42 )

c(t 43 )
c(t 44 )


b( )  a ( )
b(t 21 )  2b( )a ( )  a (t 21 )
b(t31 )  3b( )a 2 (t 21 )  3b(t 21 )a 2 ( )  a (t31 )
b(t32 )  3b( )a (t 21 )  3b(t 21 )a ( )  a (t32 )
b(t 41 )  4b( )a 3 ( )  6b(t 21 )a 2 ( )  4b(t31 )a ( )  a (t 41 )
b(t 42 )  4b( )a (t 21 )a ( )  6 1 / 3b(t 21 )a (t 21 )  2 / 3b(t 21 )a 2 ( )

 42 / 3b(t31 )a ( )  1 / 3b(t32 )a ( )   a (t 42 )
b(t 43 )  4b( )a (t31 )  6b(t 21 )a 2 ( )  4b(t31 )a ( )  a (t 43 )
b(t 44 )  4b( )a (t32 )  6b(t 21 )a (t 21 )  4b(t32 )a ( )  a (t 44 )

Given method A, define method B such that method C=B(A) will be
method of the order p and stability function of the method C will be
product of the stability functions of the methods A and B.
Coefficients of Taylor expansion of the method B can be
expressed in terms of coefficients of the methods C and B
p  1
p  2
p  3
c( )  1
c(t21 )  1
c(t31 )  1, c(t31 )  1
b( )
b(t21 )
b(t31 )
b(t32 )
b(t41 )
p  4
c(t41 )  1, c(t42 )  1,
c(t43 )  1, c(t44 )  1
1  a( )
1  (2b( )a( )  a(t21 ))
1  (3b( )a2(t21 )  3b(t21 )a2( )  a(t31 ))
1  (3b( )a(t21 )  3b(t21 )a( )  a(t32 ))
1  (4b( )a3( )  6 b(t21 )a2( )  4b(t31 )a( )  a(t41 ))
1  (4b( )a(t21 )a( )  6 1 / 3b(t21 )a(t21 )  2 / 3b(t21 )a2( )
b(t42 ) 
 42 / 3b(t31 )a( )  1 / 3b(t32 )a( )  a(t42 ))
b(t43 )  1  (4b( )a(t31 )  6 b(t21 )a2( )  4b(t31 )a( )  a(t43 ))
b(t44 )  1  (4b( )a(t32 )  6 b(t21 )a(t21 )  4b(t32 )a( )  a(t44 ))





Examples of the method A (RKC) and DUMKA:
Y0  y0
Y1  y0  h1f(Y0 )
Yj  hjf(Yj 1)   jYj 1   jYj  2, j  2,  , n  p
yi 1  yi  f(y0, t0), i  1,  , n  p
Equations for coefficients of the method B
p  2
b1  b 2  b3  b( )

b 2c2  b3 c3  b(t21 ) / 2
p  3
b1  b 2  b3  b( )

b 2c2  b3 c3  b(t21 ) / 2

2
2
b 2c2  b3 c3  b(t31 ) / 3

b 2 a32 c2  b(t32 ) / 6
p  4
b1  b 2  b3  b3  b( )

b 2c2  b3 c3  b4c4  b(t21 ) / 2

2
2
2
b 2c2  b3 c3  b4c4  b(t31 ) / 3

3
3
3
b 2c2  b3 c3  b4c4  b(t41 ) / 4

b3 a32 c2  b4(a42 c2  a43 c3 )  b(t32 ) / 6

b3 a32 c2c3  b4c4(a42 c2  a43 c3 )  b(t42 ) / 8

2
2
2
b3 a32 c2  b4(a42 c2  a43 c3 )  b(t43 ) / 6
b a a c  b(t ) / 24
44
 4 43 42 2
Embedded methods
hp d py
 y0   
 C0  (h p  1)
p
p! dh
y
p
1
y
p 1
1
h p 1
d p  1y
 y0   
 C1  ( h p  2 )
p 1
( p  1)! dh
h p 1
d p  1y
y(h)  y0   
 C 2  (h p  2 )
p 1
( p  1)! dh
err  y(h)  y
err  y
p 1
errnew  y(h)  y1p  C new  hnew
errnew  tol
C new  Cold
p 1
C new  err / hold
p 1
hold
err p  1  tol
hnew
1 / p 1
hnew
 tol 
 hold 

 err 
p 1
1
p
1
 y
p
1
1
d p  1y

 C0 h p  1  O(h p  2 )
p 1
( p  1)! dh
1
d p  1y

 C0 h p  1  O(h p  2 )
p 1
( p  1)! dh
Embedded methods
Yi

n
y0
 h aijf(t0
 cj  h, Yj),
j 1
y1

n
y0
 h bjf(t0
 cjh, Yj),
j 1
~
y
1

n
y0
~
 h bjf(t0
 cjh, Yj),
j 1
err

~
y1  y
1
 bi  1


and 
 bici  1 / 2

b(t )  1, b(t )  1
 31
32
~
bi  1
~
bci  1 / 2
Embedded composition method C’=B’(A)
0
0 0
c2 a21 0 0
c3 a31 a32 0
0
b1
~
b1
b1
~
b2
b1
~
b3
~
~
~
~
b
1  b2  b3  b4  b( )
~
~
~
b2c2  b3c2  b4  b( )  b(t21 ) / 2
~
b4
0 0
0
0
0
c2 a21
0
0
0
c3 a31
a32
0
0
c4 a41
a42
a43
0
b1 b2 b3 b4
~ ~ ~ ~ ~
b1 b2 b3 b4 b5
5 ~
 bi  b( )
i~1
~
~
~
b c  b
3c3  b4c4  b5  b( )  b(t21 ) / 2
 2 2
~ 2
~
~
~
b
c  b3c32  b4c42  b5  b 2( )  b(t31 ) / 3
 2 2
~
~
~
b
3 a32c2  b4( a42c2  a43c3 )  b5( b2c2  b3c3  b3c4 )  b(t32 ) / 6
Numerical results
duk
uk2  1  uk2 1
u
 2uk  uk 1
 
  k 1
dt
4x
x 2
u(0)  1.5x(1  x)2,   0.0005,
x  [0,1], t  [0,2.5]
| Rn ( ) | 1
Rn ( ) 
 [1,0] max | Rn (i ) | 
w p ( x) Pn  p ( x)
w p ( x0 ) Pn  p ( x0 )
,
dy
 f( y)
dt
x  x0  (1  x0 ) (2)