time integration of differential...

PrefaceAnalytical framework

Exponential integratorsProjects

Time Integration of Differential Equations

Mechthild Thalhammer

Universität Innsbruck

June 12, 2006

Mechthild Thalhammer Time Integration of Differential Equations



General scopeResearch areaParabolic problemsDiscretisation

General scope





Mathematical models

Around 1960, things became completely different and everyone became aware thatthe world was full of stiff problems. (GERMUND DAHLQUIST, 1925-2005)

Mathematical models. Differential equations are used in thedescription of dynamical processes.

Natural sciencesPhysics, AstrophysicsBiologyChemistryMetereology

Engineering technologies

Medical sciences

Finance etc.





Mathematical models

Cardiovascular system Alinghi

A. Quarteroni (EPFL Lausanne, Politecnico di Milano)





Simulations

There are at least two ways to combat stiffness. One is to design a better computer,the other, to design a better algorithm. (HARVARD LOMAX, 1922-1999)

Mathematical models. Realistic models require computer-aidedsimulations.

Computer. Representation of numbers

π= 3.141592653589793238462643383279502884197169399. . .

Limited precision (efficiency, capacity).

Numerical mathematics. Construction and analysis of algorithmsfor numerical solution of mathematical problems.





Numerical mathematics

Mein Verzicht auf das Restglied war leichtsinnig. (W. ROMBERG, 1979)

Numerical mathematics. Construction and analysis of algorithms.

Precision (Convergence)

Influence of perturbations (Stability)

Computational effort (Efficiency)

Areas in numerical mathematics.

Linear equations

Nonlinear equations

Eigenvalue problems

Approximation, interpolation

Optimisation

Differential equations

Integral equations





Research area





Research area

Problem classes.

Ordinary differential equationsSingularly perturbed problems

Partial differential equationsParabolic problemsHyperbolic problems

Time integration.

Construction of favourable methodsError behaviourQualitative behaviour





Parabolic problems





Parabolic initial-boundary value problems

Allen-Cahn equation

∂tU = 1100 ∂x

2U +U(1−U 2

) Kuramoto-Sivashinsky equation

∂tU =− ∂x4U − ∂x

2U −U ∂xU





Chaotic behaviour





Spatial and time discretisationof parabolic problems





Numerical solution of partial differential equations

Continued study of special problems is still a commendable way towards greaterinsight. (EBERHARD HOPF, 1902-1983)

Parabolic initial boundary-value problem. Model problem

∂tU (x, t ) = ∂x2U (x, t ), 0 < x < 1, 0 < t ≤ T,

U (0, t ) = 0 =U (1, t ), 0 ≤ t ≤ T, U (x,0) =U0(x), 0 ≤ x ≤ 1.

Discretisation in space and time.

Spatial discretisation yields(high dimensional) system ofordinary differential equations.

Apply numerical method forthe time integration and studyits quantitative and qualitativebehaviour.





Spatial discretisation


∂tU (x, t ) = ∂x2U (x, t ), 0 < x < 1, 0 < t ≤ T,

U (0, t ) = 0 =U (1, t ), 0 ≤ t ≤ T, U (x,0) =U0(x), 0 ≤ x ≤ 1.

Spatial discretisation. Use approximation by finite differences

∂x2U (x j , ·) = U (x j +∆x, ·)−2U (x j , ·)+U (x j −∆x, ·)

(∆x)2 +O((∆x)2)

at grid points x j = j∆x = j (M +1)−1 for 1 ≤ j ≤ M .





Spatial discretisation


∂tU (x, t ) = ∂x2U (x, t ), 0 < x < 1, 0 < t ≤ T,

U (0, t ) = 0 =U (1, t ), 0 ≤ t ≤ T, U (x,0) =U0(x), 0 ≤ x ≤ 1.

Spatial discretisation. Use approximation by finite differences

∂x2U (x j , ·) = U (x j +∆x, ·)−2U (x j , ·)+U (x j −∆x, ·)

(∆x)2 +O((∆x)2)

at grid points x j = j∆x = j (M +1)−1 for 1 ≤ j ≤ M . Obtain (large)system of ordinary differential equations for U j ≈U (x j , ·)

U ′1(t )...

U ′M (t )

= 1

(∆x)2

−2 11 −2 1

. . .

1 −2 1

U1(t )

...UM (t )

.





Time discretisation

Situation. Spatial discretisation of initial-boundary value problemyields initial value problem of large dimension

u′(t ) = A u(t ), 0 < t ≤ T, u(0) given.

Standard numerical method classes.

Runge-Kutta methodsRunge-Kutta methods prove popular at IMA conference on numerical ODEs.

(SIAM NEWS, March 1990)

Linear multistep methods

Explicit Euler method. Approximation un ≈ u(nh) for h > 0

un+1 −un

h= A un , n ≥ 0, u0 given.





Numerical example

Integration without preparation is frustration. (REVEREND LEON SULLIVAN)

Time integration of spatially discretised test equation (M = 100) byexplicit Euler method for time-steps h = 1

10 , h = 120300 , h = 1

20400 .

Explicit Euler solution

0.2 0.4 0.6 0.8−2

0

2x 10

30

0.2 0.4 0.6 0.8−2

0

2x 10

67

0.2 0.4 0.6 0.80

0.5

1

1.5x 10

−5

Exact solution

0.2 0.4 0.6 0.80

0.5

1

1.5x 10

−5

0.2 0.4 0.6 0.80

0.5

1

1.5x 10

−5

0.2 0.4 0.6 0.80

0.5

1

1.5x 10

−5





Stability and convergence

Test equation (DAHLQUIST). Scalar differential equation

u′(t ) =λu(t ), 0 < t ≤ T, u(0) = 1, Reλ≤ 0,

with exact solution and explicit Euler solution

u(nh) = enhλ, un = (1+hλ

)n , n ≥ 0.

Local error. Difference between numerical and exact solution

u1 −u(h) = 1+hλ−ehλ =O(h2) .





Stability and convergence

Test equation (DAHLQUIST). Scalar differential equation

u′(t ) =λu(t ), 0 < t ≤ T, u(0) = 1, Reλ≤ 0,

with exact solution and explicit Euler solution

u(nh) = enhλ, un = (1+hλ

)n , n ≥ 0.

Local error. Difference between numerical and exact solution

u1 −u(h) = 1+hλ−ehλ =O(h2) .

Stability. Explicit Euler solution remains bounded for n ≥ 0 if

|1+hλ| ≤ 1. −1−1−1−1−1−1





Time integration of test equation

Semidiscretised model problem. Numerical solution of

u′(t ) = A u(t ), 0 < t ≤ T, u(0) given.

Spectral properties. Eigenvalues of A ∈RM×M satisfy

−4(M +1)2 <λ< 0.

Stability. Explicit Euler solution remains bounded if

−2 < hλ< 0.−1−1−1−1−1−1





Time integration of test equation

Semidiscretised model problem. Numerical solution of

u′(t ) = A u(t ), 0 < t ≤ T, u(0) given.

Spectral properties. Eigenvalues of A ∈RM×M satisfy

−4(M +1)2 <λ< 0.

Stability. Explicit Euler solution remains bounded if

−2 < hλ< 0.−1−1−1−1−1−1

Consequence. Severe stepsize restriction for explicit methods!





Implicit time integration schemes

Explicit Euler method

un+1 −un

h= A un

hopt ≈ 5 ·10−5, h ≈ hopt +10−8

Implicit Euler method

un+1 −un

h= A un+1

h = 10−1





Summary

Situation. Spatial discretisation of partial differential equationleads to stiff ordinary differential equation

u′(t ) = F(t ,u(t )

), 0 < t ≤ T.

Severe stepsize restriction for explicit methods!

Need to construct efficient numerical methods with favourablestability and convergence properties.





Summary

Situation. Spatial discretisation of partial differential equationleads to stiff ordinary differential equation

u′(t ) = F(t ,u(t )

), 0 < t ≤ T.

Severe stepsize restriction for explicit methods!

Need to construct efficient numerical methods with favourablestability and convergence properties.

Established schemes.

Implicit Runge-Kutta methods

Implicit linear multistep methods

Recent research activities.

Explicit exponential integration methods




General scopeAbstract setting

Analytical framework





Numerical methods for differential equations

Theory without practice cannot survive and dies as quickly as it lives.(LEONARDO DA VINCI, 1452-1519)

Numerical methods for differential equations. Interplay between

Theory Applicationsand

(Functional analysis) (Physics)

He who loves practice without theory is like the sailor who boards ship without arudder and compass and never knows where he may cast.

(LEONARDO DA VINCI, 1452-1519)





Semilinear parabolic initial-boundary value problems

In 1971, I read the beautiful paper of Kato and Fujita on the Navier-Stokes equationsand was delighted to find that, properly viewed, it looked like an ordinary differentialequation, and the analysis proceeded in ways familiar for ODEs. (DAN HENRY, 1981)

General form. Semilinear parabolic partial differential equation

∂tU (x, t ) =A (x) U (x, t )+F(t , x,U (x, t ),∇U (x, t )

)involving strongly elliptic differential operator A with smoothspace-dependent coefficients and nonlinear part F of lower order.Additional boundary and initial condition.





Semilinear parabolic initial-boundary value problemsIn 1971, I read the beautiful paper of Kato and Fujita on the Navier-Stokes equationsand was delighted to find that, properly viewed, it looked like an ordinary differentialequation, and the analysis proceeded in ways familiar for ODEs. (DAN HENRY, 1981)

General form. Semilinear parabolic partial differential equation

∂tU (x, t ) =A (x) U (x, t )+F(t , x,U (x, t ),∇U (x, t )

)involving strongly elliptic differential operator A with smoothspace-dependent coefficients and nonlinear part F of lower order.Additional boundary and initial condition.

Abstract formulation. Interprete partial differential equation asabstract differential equation for u(t ) =U (·, t )

u′(t ) = A u(t )+ f(t ,u(t )

).

Employ framework of sectorial operators and analytic semigroups.Mechthild Thalhammer Time Integration of Differential Equations




Sectorial operators

Abstract initial value problem on Banach space(X ,‖·‖X

)u′(t ) = A u(t )+ f

(t ,u(t )

), 0 < t ≤ T, u(0) given.

Hypothesis

Linear operator A : D ⊂ X → X is sectorial. It holds∥∥∥(λI − A

)−1∥∥∥

X←X≤ M

|λ−a| , λ ∈C\ Sφ(a),

where Sφ(a) = {λ ∈C : |arg(a −λ)| ≤φ}∪ {a} with φ ∈ (0,π/2).

Define linear operator et A : X → X by integral formula of Cauchy

et A = 1

2πi

∫Γ

eλ(λI − t A

)−1 dλ, t > 0, et A = I , t = 0.





Analytic semigroups & Variation-of-constants formula

Fundamental relations for analytic semigroup(et A

)t≥0 generated

by sectorial operator A : D → X

es A et A = e(s+t )A ,d

dtet A = A et A .

Homogeneous equation. Exact solution given through semigroup

u′(t ) = A u(t ), 0 < t ≤ T, u(t ) = et A u(0).





Analytic semigroups & Variation-of-constants formula

Fundamental relations for analytic semigroup(et A

)t≥0 generated

by sectorial operator A : D → X

es A et A = e(s+t )A ,d

dtet A = A et A .

Homogeneous equation. Exact solution given through semigroup

u′(t ) = A u(t ), 0 < t ≤ T, u(t ) = et A u(0).

Semilinear equation. Justify variation-of-constants formula onintermediate space D ⊂ Xα ⊂ X

u′(t ) = A u(t )+ f(t ,u(t )

), 0 < t ≤ T,

u(t ) = et A u(0)+∫ t

0e(t−τ)A f

(τ,u(τ)

)dτ.




Order conditionsConvergence result

Exponential integrators





Exponential integration methods

Although most of these methods appear at the moment to be largely of theoreticalinterest ... (BYRON EHLE, 1968)

Homogeneous problem. Exact solution of initial value problem

u′(t ) = A u(t ), 0 < t ≤ T, u(0) given,

is given by analytic semigroup u(t ) = et A u(0).

Presumption. Efficient computation of et A feasible.

MOLER & VAN LOAN, Nineteen dubious ways to compute the exponential ofa matrix, twenty-five years later (2003).





Exponential integration methods

Presumption. Efficient computation of et A v feasible.

Realisation.

Spectral techniques

Krylov subspace methodsHOCHBRUCK & HOCHSTENBACH, Subspace extraction for matrixfunctions (2006).

Interpolation methodsBERGAMASCHI ET AL., A parallel exponential integrator for large-scalediscretizations of advection-diffusion models (2005).





Exponential integrators for semilinear problems

Semilinear problem. Exact solution of initial value problem

u′(t ) = A u(t )+ f(t ,u(t )

), 0 < t ≤ T, u(0) given,

is represented by variation-of-constants formula

u(tn+1) = eh A u(tn)+∫ h

0e(h−τ)A f

(τ,u(tn +τ)

)dτ.





Exponential integrators for semilinear problems

Semilinear problem. Exact solution of initial value problem

u′(t ) = A u(t )+ f(t ,u(t )

), 0 < t ≤ T, u(0) given,

is represented by variation-of-constants formula

u(tn+1) = eh A u(tn)+∫ h

0e(h−τ)A f

(τ,u(tn +τ)

)dτ.

Explicit exponential Euler method. Approximation un ≈ u(tn) by

un+1 = eh A un +h ϕ1(h A) f (tn ,un), ϕ1(h A) = 1

h

∫ h

0e(h−τ)A dτ.

Higher-order schemes. Determine un+1 by means of previousnumerical solution values un−i for 0 ≤ i ≤ q −1 and auxiliaryapproximations Uni ≈ u(tn + ci h) where 0 ≤ ci ≤ 1 for 1 ≤ i ≤ s.





Example method

Man hüte sich, auf Grund einzelner Beispiele allgemeine Schlüsse über den Wertoder Unwert einer Methode zu ziehen. Dazu gehört sehr viel Erfahrung.

(LOTHAR COLLATZ, 1910-1990)

Two-step two-stage scheme. Determine approximation through

Un = eh A un +h a(h A) f (un)+h v(h A) f (un−1),

un+1 = eh A un +h b1(h A) f (un)+h b2(h A) f (Un)+h w(h A) f (un−1).

0

1 ϕ1 +ϕ2 −ϕ2

ϕ1 −2ϕ312 ϕ2 +ϕ3 − 1

2 ϕ2 +ϕ3

Nice features.

Nonlinear part is integrated explicitly.

Good stability and convergence behaviour in numericalexamples.





Exponential general linear methods

General linear methods were originally introduced as a means of unifying andgeneralizing existing theories for traditional methods. (JOHN BUTCHER, 1987)

Semilinear problem. Initial value problem

u′(t ) = A u(t )+ f(t ,u(t )

), 0 < t ≤ T, u(0) given.

Numerical scheme. Combine one-step and multistep method

Uni = eci h A un +hi−1∑j=1

ai j (h A) f (tn j ,Un j )+hq−1∑k=1

vi k (h A) f (tn−k ,un−k ),

un+1 = eh A un +hs∑

i=1bi (h A) f (tni ,Uni )+h

q−1∑k=1

wk (h A) f (tn−k ,un−k ).

Coefficient functions are combinations of exponential functions.

Aim. Stability and convergence analysis for parabolic problems.





Derivation of order conditions

Order and stage order. Numerical scheme for u′ = A u + f (t ,u)has quadrature order P and stage order Q if defects of numericalsolution dn+1 and defects of internal stages Dni satisfy

dn+1 =O(hP+1), Dni =O

(hQ+1).

Strategy. Employ reasonable assumptions that exact solution uand nonlinear part f are sufficiently regular.

Substituting solution into numerical scheme defines defects.

Represent solution values by variation-of-constants formula.

Employ Taylor series expansions of solution and nonlinearity.





Order conditions

Stage order conditions. Scheme has stage order Q if for 1 ≤ `≤Q

c`i ϕ`(ci h A) =i−1∑j=1

c`−1j

(`−1)!ai j (h A)+

q−1∑k=1

(−k)`−1

(`−1)!vi k (h A).

Order conditions. Scheme has quadrature order P if for 1 ≤ `≤ P

ϕ`(h A) =s∑

i=1

c`−1i

(`−1)!bi (h A)+

q−1∑k=1

(−k)`−1

(`−1)!wk (h A).

Questions.

Connection with convergence order for semilinear problems?

Solvability of stage order and quadrature order conditions?





Convergence result

Das besondere Schmerzenskind sind die Fehlerabschätzungen.(LOTHAR COLLATZ, 1910-1990)

Theorem (Ostermann, Th. & Wright, 2006)

Assume that the explicit exponential general linear method hasorder P and stage order Q. Then, the convergence order forsemilinear parabolic problems u′(t ) = A u(t )+ f

(t ,u(t )

)is

p = min{P,Q +1}, provided that g (t ) = f(t ,u(t )

)is sufficiently

regular. That is, the estimate

‖un −u(tn)‖Xα≤C

q−1∑`=0

‖u`−u(t`)‖Xα+C hQ+1 sup

0≤t≤tn

‖g (Q)(t )‖Xα

+C hP sup0≤t≤tn

‖g (P )(t )‖X , tq ≤ tn ≤ T,

holds with some constant C > 0 independent of n and h.




Former projectsFuture prospectsFuture projectsConclusions

Projects





Former projects.

Established methods

Exponential methods





Established methods





Long-term behaviour

Semilinear parabolic problems.

u′(t ) = A u(t )+ f(t ,u(t )

)

OSTERMANN & TH., Non-smooth data error estimates for linearlyimplicit Runge-Kutta methods (2000).





Fully nonlinear equations

Nonlinear parabolic problems.

u′(t ) = F(t ,u(t )

)Nonlinear diffusion processes

∂tU = ∂x(α(U ,∂xU )∂xU

)+ f(U ,∂xU

)Displacement of a shock

∂tU = lnexp

(U ∂x

2 U)−1

∂x2 U

− 12

(∂xU

)2

GONZÁLEZ, OSTERMANN, PALENCIA & TH., Backward Eulerdiscretization of fully nonlinear parabolic problems (2001).

OSTERMANN & TH., Convergence of Runge-Kutta methods fornonlinear parabolic equations (2002).

OSTERMANN, TH. & KIRLINGER, Stability of linear multistepmethods and applications to nonlinear parabolic problems (2004).





Singularly perturbed problems

I have a theory that whenever you want to get in trouble with a method, look for theVan der Pol equation. (PEDRO ZADUNAISKY, 1982)

Singularly perturbed problems. ε<< 1

y ′(t ) = f(y(t ), z(t )

)εz ′(t ) = g

(y(t ), z(t )

)0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−3

−2

−1

0

1

2

3

Van der Pol equation

z ′′+µ(z2−1

)z ′+z = 0

TH., On the convergence behaviour of variable stepsize multistepmethods for singularly perturbed problems (2004).





Exponential methods





Quasilinear problems

Quasilinear parabolic problems.

u′(t ) = A(u(t )

)u(t )

Un = eh/2 A(un ) un , un+1 = eh A(Un ) un

Fluid through

porous medium

GONZÁLEZ, OSTERMANN & TH., A second-order Magnus integratorfor nonautonomous parabolic problems (2006).

GONZÁLEZ & TH., A second-order Magnus type integrator forquasilinear parabolic problems (2006).





Magnus type methods

Non-autonomous parabolic problems.

u′(t ) = A(t )u(t )

un+1 = eh(α2 A1+α1 A2) eh(α1 A1+α2 A2) un , Ai = A(ci h)

TH., A fourth-order commutator-free exponential integrator fornon-autonomous differential equations (2006).





High-order schemes

Semilinear parabolic problems.

u′(t ) = A u(t )+ f(t ,u(t )

)Allen-Cahn equation

∂tU = 1100 ∂x

2U +U(1−U 2

)OSTERMANN, TH. & WRIGHT, A class of explicit exponential generallinear methods (2006).





Positivity

Linear parabolic problems.

u′(t ) = A u(t )+ f (t )

OSTERMANN & TH., Positivity of exponential multistepmethods (2006).





Future prospects?





Issues

There are, however, still many hard problems that can not be adequately solved.(BJÖRN ENGQUIST AND GENE GOLUB, Mathematics Unlimited – 2001 and Beyond)

Complex mathematical models arise in applications.

Life Sciences





Issues

There are, however, still many hard problems that can not be adequately solved.(BJÖRN ENGQUIST AND GENE GOLUB, Mathematics Unlimited – 2001 and Beyond)

Complex mathematical models arise in applications.

Life Sciences

Weather forecast

Mr. Dahlquist, when is the spring coming? – Tomorrow, at two o’clock.(WHEATHER FORECAST, STOCKHOLM, 1955)

Endlich viel Sonne! (RADIO TIROL WETTER, 8. Juni 2006)





Numerical analysis

Es fehlt indessen noch der Beweis dass diese Näherungs-Verfahren convergent sindoder, was practisch wichtiger ist, es fehlt ein Kriterium, um zu ermitteln, wie kleindie Schritte gemacht werden müssen, um eine vorgeschriebene Genauigkeit zuerreichen. (CARL RUNGE, 1856-1927)

The benefit of the numerical analysis is then to understand the inherent potentialand limitation of the method and its generalization in order to increase its range ofapplicability. (BJÖRN ENGQUIST AND GENE GOLUB, 2001)

Development and analysis of advanced numerical methods.

Numerical simulation of complex models.

Permanent improvement of basic algorithms.

Open question. Geometric time integration methods for partialdifferential equations.

LUBICH, HAIRER & WANNER, Geometric Numerical Integration. StructurePreserving Algorithms for Ordinary Differential Equations (2002).





Future projects





Splitting methods

Splitting methods.

Widely used methods for partial differential equations.

Error analysis available for special cases only.





Splitting methods

Splitting methods.Widely used methods for partial differential equations.Error analysis available for special cases only.

Approach. For differential equation of form

u′(t ) = F(u(t )

)= F1(u(t )

)+F2(u(t )

)consider associated equations

v ′(t ) = F1(v(t )

), v(t ) =Φ1(t )

(v(0)

),

w ′(t ) = F2(w(t )

), w(t ) =Φ2(t )

(w(0)

).

Determine approximation to u by means ofΦ1 andΦ2.

Example. Numerical solution given by

un+1 =Φ1(h/2

)(Φ2(h)

(Φ1

(h/2

)(un)

)).





Splitting methods

Objectives. Error analysis for different problem classes.

Nonlinear reaction-diffusion equations

∂tU (x, t ) =A (x) U (x, t )+F(t , x,U (x, t )

)Schrödinger type equations

i∂tU (x, t ) =−∆U (x, t )+V (x) U (x, t )

Justify practical use of methods proposed in literature.BLANES ET AL., Symplectic splitting operator methods for thetime-dependent Schrödinger equation (2006).

Employ analytic framework of C0-semigroups.Derive stiff order conditions and high-order error bounds.





Splitting methods9

TABLE III: Coefficients for the symmetric kernel (18) with m = 19, corresponding to the processed method P382.

a1 = 0.0215672851797585075705350295278 b1 = 0.0431461454881086359990876258277a2 = 0.0431726343853101639735369714998 b2 = 0.0431853234593364152087490292063a3 = 0.0431324297795690599949127838602 b3 = 0.0429704744650982147539363885468a4 = 0.0427852961505675320118200419401 b4 = 0.0430364300871454499243887883740a5 = 0.0449747930772476869948630891275 b5 = 0.0532805678508921227350798781968a6 = 0.521477840977180737598212898081 b6 = −0.0000741632590652008982349604299511a7 = −0.460297865581209561666776462059 b7 = 0.0549252685049280768846009673282a8 = 0.0476657723717784446737564703982 b8 = 0.0572922318289063436814214008313a9 = −0.299809415632442402707251772031 b9 = −0.000216083699929765754852184048464a10 = 0.360890555491738732398154005651 b10 = 0.0429262827299850710231689679598a11 = 0.0355310860247975525993505717327 b11 = 0.0509590583382259625517957082533a12 = 0.0451459109591929143698396854787 b12 = 0.0125876466303119396367352929903a13 = 0.151663982419594313475358779605 b13 = −0.00110143601875055751217588524309a14 = −0.122723981192628473398202625228 b14 = 0.0589864485893508739845735668507a15 = −0.0342003644722802255132523920962 b15 = −0.00393919091210338198661577774009a16 = 0.0514702802470565594888643277103 b16 = 0.0909189791588641823686791563103a17 = −0.00346916149683374374401491713903 b17 = −0.107654717879545729464023522278a18 = 0.0201046430669616823814202845610 b18 = 0.0254278113893309936197644680648a19 = −0.0245251277750599926319683675996 b19 = 1

2− (b1 + · · · + b18)

a20 = 1 − 2(a1 + · · · + a19)

TABLE IV: Coefficients for the polynomials P1(x) and P2(x) in (20) with s = 21 for P382 (c0 = d0 = 1).

c1 = 0.0001162512086847406211140814 c2 = 3.376774894743804480444394 · 10−8c3 = 1.176364067599484205038903 · 10−11

c4 = 4.437111761894176717316941 · 10−15c5 = 1.749973819201524252032138 · 10−18

c6 = 7.101748878564126570715907 · 10−22

c7 = 2.939931769324440416879823 · 10−25c8 = 1.235098758247133102034345 · 10−28

c9 = 5.248386453665149303792009 · 10−32

c10 = 2.250866251009862206361312 · 10−35c11 = 9.727578606034733795739798 · 10−39

c12 = 4.231641947350449068306722 · 10−42

c13 = 1.851409459980067426102173 · 10−45c14 = 8.141553608452406208018081 · 10−49

c15 = 3.596667466064486029961227 · 10−52

c16 = 1.595498786085559337026367 · 10−55c17 = 7.104576813414967870669619 · 10−59

c18 = 3.174598116648571190359996 · 10−62

c19 = 1.423077177952293495040530 · 10−65c20 = 6.398117951527209690698617 · 10−69

c21 = 2.884478510968248948572185 · 10−72

d1 = −0.0001162512086847406211140814 d2 = −2.025340542677493159320967 · 10−8d3 = −5.483616185447620695388045 · 10−12

d4 = −1.748185395473289243875044 · 10−15d5 = −6.075023900031386380514259 · 10−19

d6 = −2.227092296947007254380344 · 10−22

d7 = −8.469091056567204221082539 · 10−26d8 = −3.308402509398670050765033 · 10−29

d9 = −1.319641733480979355653975 · 10−32

d10 = −5.353346141747406366467657 · 10−36d11 = −2.202620915392627214792992 · 10−39

d12 = −9.173684223172953098611281 · 10−43

d13 = −3.861783526343716602117122 · 10−46d14 = −1.641163468907425875108297 · 10−49

d15 = −7.033925071359782763595843 · 10−53

d16 = −3.037693851132668729625454 · 10−56d17 = −1.320846410906512328044568 · 10−59

d18 = −5.778602796374270082897366 · 10−63

d19 = −2.542100400250845548947583 · 10−66d22 = −1.123916118043500908715140 · 10−69

d21 = −4.991692562368483793888509 · 10−73

comparison with other symplectic integrators publishedin the literature is manifest through numerical experi-ments. It is especially remarkable the high performanceshown by the second order processed methods.

It is important to mention that for non-processedschemes the coefficients ai, bi have to solve a relativelylarge system of non-linear equations, and then their nu-merical solution is not so straightforward. However, itmight be the case that most techniques used to obtainefficient kernels could also be used to find highly efficientand stable non-processed methods, being this an inter-esting problem to be analyzed.

In conclusion, we claim that the processing techniqueleads to extraordinarily efficient symplectic split operatormethods por the Schrodinger equation, and thus they de-

serve further analysis and study. In particular, it wouldbe very interesting to analyze under which conditionsthis technique is superior to other schemes used in prac-tice, such as the Chebyshev scheme or methods based onKrylov subspace techniques [3].

Acknowledgments

This work has been partially supported by Ministeriode Educacion y Ciencia (Spain) under project MTM2004-00535 (co-financed by the ERDF of the European Union).The work of SB has also been supported by a contract inthe Pogramme Ramon y Cajal 2001.

[1] M.D. Feit, J.A. Fleck, Jr, and A. Steiger, J. Comput.Phys. 47, 412 (1982).

[2] R. Kosloff, J. Phys. Chem. 92, 2087 (1988).





Time integrators for partial differential equations

... methods for stiff problems, we are just beginning to explore them ...(LAWRENCE SHAMPINE, 1985)

A good numerical method is . . .

. . . reliable in demanding moments.


time integration of differential...

Documents