high order explicit methods for parabolic equations and stiff odes (dumka3, dumka4, rock2, rock4)...
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High order explicit methods for parabolic equationsHigh order explicit methods for parabolic equationsand stiff ODEsand stiff ODEs
(DUMKA3, DUMKA4, ROCK2, ROCK4)(DUMKA3, DUMKA4, ROCK2, ROCK4)
Alexei MedovikovTulane 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.372-390Lebedev 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 DUMKALANDExplicit numerical methods for stiff differential equations
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-Kutta-Chebyshev formulas of order three. It uses optimal third order accuracy stability polynomials with the largest stability region along the negative real axis.
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:[email protected]
Phone:(504)862-8396
Address: Mathematics Department Tulane University New Orleans, LA Web: http://www.math.tulane.edu/~amedovik
Examples of solution of stiff differential equations by explicit methods
Brusselator equation
Nagumo nerve conduction equation
Burgers equation
2
2
2
22
2
2
2
22
yv
xv
vuu4.3tv
)t,y,x(fyu
xu
u4.4vu1tu
else
1.1tand1.0)6.0y()3.0x(if,0
5)u(f
)bvu(ntv
v)u(fxu
tu
222
2
2
21kk1k
21k
21kk
xuu2u
x4uu
dtdu
SummarySummary
1. Stability: Explicit methods have small stepsize , due to conditional stability
2. Variable steps can be used to maximize mean stepsize of a sequence of explicit methods
3. Optimal sequence of explicit steps can be found in terms of roots of stability polynomials, which approximate exponential function and possess Chebyshev alternation
4. Asymptotic formulas and orthogonal polynomials can be used to construct such polynomials, even very high degree polynomials (n > 1000)
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.
)n(n/ncn/ 2p
n
i
n
1p
ii
pn za!p/zz1)z(R
00 y)t(y)t,y(fdtdy
λyy'
0001 y)1(yyy
0n
n y)1(y
)t,y(fyy
0001
Explicit Euler method:
Stability analysis of explicit RK methods
ODEs:
Test equation:
Stability function:
Stability region: hz,1)z(R:CzS n
Goal:
}S)y/f(Sp{ Find stability polynomial which maximize average stepsize , given
h0nn y)h(Ry is a total stepwhere
n/h
M/2
01 y)1(y
11)(Rn
000001 err))y(JI(err)y(Jerrerr )y~y)(y(J)y~y(
)y~(fy~())y(fy(y~y
00000
000011
Stability analysis of explicit RK methods
/2
Explicit Euler method:
Stability condition of explicit Euler method:
h
Linear stability analysis for non-linear ODEs:
000 erryy~
where )y/)y(f(M
Linear stability RK methods vs. Stability RK methods?
})y/f(Sp{ regionstability
z,h,z1)z(R
)1()h(R i
n
1in
0
nn
1ii d
)(dRh
Can we solve stiff ordinary differential equations (ODE) by explicit Can we solve stiff ordinary differential equations (ODE) by explicit methods with stepsize larger than 2/M?methods with stepsize larger than 2/M?
Example:Example:
500/2
0)0())cos((500'
ytyy
1 2Consider two steps and where 500/11
)cos()5001(
)cos()5001(
102122
01011
tyy
tyy
)cos()cos()5001()5001)(5001( 1020120122 ttyy
yinstabilitif 500/221 stabilityif 500/112
2anyfor 10)5001)(5001( 12
h),1()h(R1
)hh(h),h1)(h1()h(R 21211
n
1iii
n
1i1 hh),h1()h(R
))tcos(y(500'y
Original idea of Runge-Kutta-Chebyshev methodsOriginal idea of Runge-Kutta-Chebyshev methods
Consider sequence of Euler steps and find an optimal polynomial
n
1i 0
ni
n
1iin
n
1ii
n
1i0in
d)(dR
n/h
S]M,0[)A(Sp1|)1(||)(R|
hy)1(y
as large as possible
ii /1
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
The solution for n-stage Runge-Kutta-Chebyshev method order p=1 is given by Chebyshev stability polynomial.
1x,
M
x1xx,
xarccosncosxarccosncos
)(R 00
00
n
euler2
20
1x
0
02
0
0
0
n0
nhM2
nn/M2
nh
M2
n)x(hlim)1(h
Mx1
))xarccos(ncos(x1
))xarccos(nsin(n
d)(dR
)x(h
0
00 x)0(x,x)(x1M0
MMx1
1x,1)(Rmax
1xarccosncos
1
0
0n
0
Runge-Kutta methodsRunge-Kutta methods
n
1jjj0j01
n
1jjj0ij0i
).Y,hct(fbhyy
),Y,hct(fahyY
n1n21
nn1nn2n1n
n1n12n11n
n221
n11n11211
n
1n
2
1
bbbbaaaa
aaa
aa
aaaa
c
c
c
c
b
Ac
y'y
hz
zyy
z
)y,,y(,Y)(bhyy
)Y,,Y(,Y)(ahyY
t
01
t00
n
1jjj01
tn1
n
1jjij0i
Yb
YΑyY
y
Y
0
0
n
1pi
ii
p2
1ntnt2t
n
ijn
t0
1t
1
zd!p/z!2/zz1
zzz1)z(R
,ij,0a,
)11(,y))z(z1(y
1b1b1b
0A
11Ib
if
Stability function of explicit Runge-Kutta method
Theorem (T. Riha): Among all polynomials of the order p the polynomial which possess Chebyshev alternant, would maximize real stability interval
n
1pi
ii
p2n za!p/z2/zz1)z(R
]0,l[z1)z(R nn for
zlM
n
n
1pi
in
i
pn
2nn
n
Ml
d!p/Ml
!2/Ml
Ml
1)(R
or equivalently, the polynomial which possess Chebyshev alternant:
has maximal possible stepsize
M
l
d
)(dRh nn
, given stability ]M,0[,1)(Rn
nl
dzz1x1
xz)x(ln)z(ln
21
)(where
)xarccos(),x())(ncos(2
)x(P)x(
asformalasymptoticinexpressedis]1,1[intrvalthein)x(functionweight
thewithzerofromdeviationleasttheof)x(PpolynomialtheThen
).numberspositivefixedareL,,cwhere(L|ln||)x()x(|andc)x(c0
satisfieswhich]1,1[intervaltheonfunctionweightpositiveabe)x(wLet:Theorem
2
21
1
nn
2/1
n
2,11
21
dzz1x1
xz)x(wln)z(wln
21
)(where
)xarccos(),)n(ln())(ncos()x(P)x(w
:]1,1[onuniformlysatisfy),x1(sqrt/)x(w
functionweightthewithassociated)x(PspolynomialorthogonaltheThen
).numberspositivefixedareL,,cwhere(L|ln||)x(w)x(w|
andc)x(wc0satisfieswhich]1,1[intervaltheon
functionweightpositiveabe)x(wLet:Theorem
2
21
1
n
2
n
2,11
21
)1(
l)0(Rl)0(Rl)0(Rl)0(R1)0(R4p
l)0(Rl)0(Rl)0(R1)0(R3p
l)0(Rl)0(R1)0(R2p
l)0(R1)0(R1p
4n
''''n
3n
'''n
2n
''nn
'nn
3n
'''n
2n
''nn
'nn
2n
''nn
'nn
n'nn
Two algorithms of computation of stability polynomials:
1. For given n calculate weight
and roots via asymptotic formula for polynomials
of the least deviation from zero:
so that the polynomial satisfies (1),
2. For given n calculate weight
so that the polynomial satisfies (1),
where is orthogonal polynomial with the weight
)(P)(w)(R pnpn
p
1jjp )xx()x(w
)(xP pn 2
2
x1
))x(w(
)(R)()(R pnpn
)cos(x,pn)(
pn)5.0k(
pn,,1k),5.0k()()pn(
kk
jk1j
k
kk
p
1jjp )xx()x(
)(xP pn
DU
MK
A3,
4R
OC
K2,4,
RK
C
)x(T)x(T)x(T)x(T)x(T
)x(T)x(T)x(T)x(T)x(T
)x(T)x(T)x(T)x(T)x(T
)x(T)x(T)x(T)x(T)x(T
)x(T)x(T)x(T)x(T)x(T
)x(wC
)x(P
4k2'
4k1'
4k24k14k
3k2'
3k1'
3k23k13k
2k2'
2k1'
2k22k12k
1k2'
1k1'
1k21k11k
k2'k1
'k2k1k
22
k
kl2
21
1
lk dxx1
)x(w)x(P)x(P
]0,M[z),z(P)z(P)z()z(P 2kk1kkkk
2
1jj2 )xx()x(w
))(ncos()x(P)x(w)x(R 2n2n
2n
''nn
'nn
n'nn
l)0(Rl)0(R1)0(R2p
l)0(R1)0(R1p
Accuracy:Accuracy:Order conditions of Runge-Kutta methodsOrder conditions of Runge-Kutta methods
KIK
JI
KIJIK
3IJ
I
2JJ
0
3
3
J32
2
J2JJ0
J
ffffff6h
ff2h
hfy
6h
dtyd
2h
dtyd
hdtdy
y)h(y
KIK
JI32
KIJIK31
2IJ
I21JJ
0J ffftbffftb
6h
fftb2h
hf)(by)h(y~
Taylor expansions of the exact solution and numerical solution :
1Α
1Α
1Α
1
3t
n
1l,k,j,ijljkiji44
n
1l,k,j,ijljkiji43
n
1l,k,j,ijlikiji42
n
1l,k,j,iilikiji41
2tn
1k,j,ijkiji31
n
1k,j,iikiji31
t
ij
n
1j,ii21
tn
1ii
b24
aaab24)t(baaab12)t(baaab8)t(baaab4)t(b
b6aab6)t(baab3)t(b
b2ab2)t(b
bb)(b
KK
I
I
JKI
KJI
KIKI
J2KIJ
IKI
I
JIJ
I fyf
yf
fff,ffyyf
fff,fyf
ff
where
Construction of Construction of ppthth order composition method 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:
)()()(4)()(6)()(4)()(
)()()(4)()(6)()(4)()()()()(3/1)()(3/24
)()(3/2)()(3/16)()()(4)()(
)()()(4)()(6)()(4)()(
)()()(3)()(3)()(
)()()(3)()(3)()(
)()()(2)()(
)()()(
44322121324444
43312
21314343
423231
22121212142
42
41312
213
4141
3221213232
312
21212
3131
212121
taatbtatbtabtbtc
taatbatbtabtbtctaatbatb
atbtatbatabtbtc
taatbatbabtbtc
taatbtabtbtc
taatbtabtbtc
taabtbtc
abc
Given method A, define method B such that method C=B(A) will be 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 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.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
))t(a)(a)t(b4)t(a)t(b6)t(a)(b4(1)t(b
))t(a)(a)t(b4)(a)t(b6)t(a)(b4(1)t(b))t(a)(a)t(b3/1)(a)t(b3/24
)(a)t(b3/2)t(a)t(b3/16)(a)t(a)(b4(1)t(b
))t(a)(a)t(b4)(a)t(b6)(a)(b4(1)t(b
))t(a)(a)t(b3)t(a)(b3(1)t(b
))t(a)(a)t(b3)t(a)(b3(1)t(b
))t(a)(a)(b2(1)t(b
)(a1)(b
443221213244
43312
213143
423231
221212121
42
41312
213
41
32212132
312
21212
31
2121
1)t(c,1)t(c
,1)t(c,1)t(c1)t(c,1)t(c1)t(c1)(c
4p3p2p1p
4443
4241
313121
pn,,2j,YY)Y(fhY
)Y(fhyY
yY
2jj1jj1jjj
0101
00
pn,,1i),t,y(fyy 00i1i
Examples of the method A (RKC) and DUMKA:
Equations for coefficients of the method BEquations for coefficients of the method B
24/)t(bcaab
6/)t(b)caca(bcab
8/)t(b)caca(cbccab
6/)t(b)caca(bcab
4/)t(bcbcbcb
3/)t(bcbcbcb
2/)t(bcbcbcb
)(bbbbb
4p
6/)t(bcab
3/)t(bcbcb
2/)t(bcbcb
)(bbbb
3p
2/)t(bcbcb
)(bbbb
2p
44242434
43
2343
22424
22323
423432424432323
3234324242323
41
344
333
322
31
244
233
222
21443322
3321
322322
31
233
222
213322
321
213322
321
Embedded methodsEmbedded methods
)h(Cdh
yd)!1p(
hy)h(y
)h(Cdh
yd)!1p(
hyy
)h(Cdhyd
!ph
yy
2p21p
1p1p
0
2p11p
1p1p
01p
1
1p0p
pp
0p1
)h(OhCdh
yd)!1p(
1yyerr
)h(OhCdh
yd)!1p(
1y)h(yerr
2p1p01p
1pp1
1p1
2p1p01p
1pp1
1p/1
oldnew
1pnew
1pold
1poldnew
oldnew
new
1pnewnew
p1new
errtol
hh
tolhh
err
h/errC
CC
tolerr
hCy)h(yerr
Embedded methodsEmbedded methods
2/1cb~1b~
and
1)t(b,1)t(b
2/1cb
1b
i
i
3231
ii
i
11
n
1jjj0j01
n
1jjj0j01
n
1jjj0ij0i
y~yerr
),Y,hct(fb~hyy~
),Y,hct(fbhyy
),Y,hct(fahyY
Embedded composition method C’=B’(A)Embedded composition method C’=B’(A)
4321
111
3231
21
3
2
b~b~b~b~bbb
0aa
00a
000
c
c
0
2/)t(b)(bb~cb~cb~)(bb~b~b~b~
2142322
4321
6/)t(b)cbcbcb(b~)caca(b~cab~3/)t(b)(bb~cb~cb~cb~
2/)t(b)(bb~cb~cb~cb~
)(bb~
32433322534324242323
312
5244
233
222
215443322
5
1ii
54321
4321
434241
3231
21
4
3
2
b~b~b~b~b~bbbb
0aaa
00aa
000a
0000
c
c
c
0
Numerical resultsNumerical results
]5.2,0[t],1,0[x
,0005.0,)x1(x5.1)0(ux
uu2ux4uu
dtdu
2
21kk1k
21k
21kk
|)(|max]0,1[1|)(| inn RR
)2()1(,)()(
)()()( 00
00
xxxxPxw
xPxwR
pnp
pnpn
)y(fdtdy