exponential peer methods - tu dresden€¦ · martin-luther-university halle-wittenberg april 30,...

Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary

Exponential Peer Methods

Tamer El-Azab & Rüdiger Weiner

Institute of MathematicsMartin-Luther-University Halle-Wittenberg

April 30, 2010

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods


1 IntroductionWhat are exponential integrators?φ-functions

Computing the φ-functions

2 Short historical overview

3 Expint Matlab package

4 Exponential Peer Methods (EPM)ConsistencyConvergenceChoosing α values

5 Numerical Tests

6 Summary



What are exponential integrators?

What are exponential integrators?

Exponential integrators are those integrators which use the exponentialfunction (and related functions) of the Jacobian or an approximation to it,inside the numerical method.

An alternative to implicit methods for the numerical solution of stiff orhighly oscillatory differential equations.

Many exponential integrators are designed for solving differentialequations of the form

y ′ (t) = f (t ,y (t)) = Ty (t)+g (t ,y (t)) (1)

Exponential integrators have two main features:

1 If T = 0, then the scheme reduces to a standard scheme.2 If g(t ,y ) = 0 for all y and t , then the scheme reproduces the

exact solution of (1).



φ-functions

φ-functions

The most common related functions used in exponential integrators are theso called φ-functions, which are defined as

φl (z) = ∫ 1

0e(1−θ)z θ l−1(l−1)!dθ, l ≥ 1, φ0 (z) = ez .

The φ-functions are related by the recurrence relation

φl+1 (z) = φl (z)−φl (0)z

, φl (0) = 1l!



φ-functions


The hard part of implementing exponential integrators is the evaluation of(linear combinations of) φ-functions.Some methods for evaluating the φ-function :

Krylov-subspace methods. (Friesner, Tuckerman & Dornblaser 1989,Hochbruck & Lubich 1995)

Leja point interpolation (Caliari & Ostermann).

Using contour integrals (Schmelzer & Trefethen).

RD-rational approximations (Moret & Novati 2004).

Rational Krylov (Grimm & Hochbruck).

Using Padè approximation combined with scaling-and-squaring.(Higham 2005)



Outline






5 Numerical Tests

6 Summary



Short historical overview

1 In 1960 Certaine,Adams Moulton methods of order 2 and 3.

2 In 1967 Lawson,Generalized RK Processes (IF methods).

3 In 1978 Friedli,ETD based on explicit RK Methods of order 5.

4 In 1998 Hochbruck and Lubich,Exponential Integrators (EXP4) with inexact Jacobian.

5 In 2003 Hochbruck and Ostermann,Exponential collocation methods, convergence analysis.

6 In 2006 Ostermann and Wright,A Class of Explicit Exponential General Linear Methods.

7 In 2009 Hochbruck, Ostermann, and Schweitzer,Exponential Rosenbrock-type methods.



Outline






5 Numerical Tests

6 Summary



Expint Matlab package

Expint is a Matlab package designed as a tool for the numerical testingof various exponential integrators.

Runge-Kutta type.Multistep type and.General Linear type.

Designed by Berland, H., Skaflestad, B., and Wright, W.M. 2005.

Aims of the Expint package.Create a uniform environment which enables the comparison ofvarious integrators.Provide tools for easy visualizing of numerical behavior.Users can include problems and integrators of their own.

Expint includes test problems and time stepping methods with constantstep size.

Computing φ-functions by using Padè approximation combined withscaling-and-squaring.

We will use this package for the test and comparison of Exponential Peermethods.



Outline






5 Numerical Tests

6 Summary



Exponential Peer Methods (EPM)

We consider y ′ = f (t ,y (t)) and Ymi ≈ y (tm +cih) i = 1, ...,s1 s-stage Peer methods.

All stages have the same properties.Explicit and implicit Peer methods (Podhaisky, Schmitt & Weiner2004 – 2009).No order reduction for stiff systems observed for implicit Peermethods.

2 Here Exponential Peer Methods.



Exponential Peer Methods (EPM) Con.

Ymi = φ0(αihT ) s∑j=1

bijYm−1,j +hs∑

j=1

Aij (αihT )[fm−1,j −TYm−1,j ] (2)

+hi−1∑j=1

Rij (αihT )[fm,j −TYm,j ], i = 1, ...,s.

fm−1,j = f (tm−1 +cjh,Ym−1,j ).T ≈ fy for stability reasons & if T = 0 we get explicit Peer Methods.

The coefficients Aij (αihT ) and Rij (αihT ) are linear combinations ofthe φ-functions and B = (bij )si ,j=1 ∈ Rs×s.

c = (ci )si=1 ∈ Rs and vector α= (αi )si=1 ∈ Rs is chosen to have asmall number of different arguments.



Consistency

Consistency

Definition 1The exponential peer method (3) is consistent of nonstiff order p if there areconstants h0,C > 0 such that

‖∆m,i‖ ≤ Chp+1 for all h ≤ h0, and for all 1≤ i ≤ s.

The method is consistent of stiff order p, if C and h0 may depend on ω, Lgand bounds for derivatives of the exact solution, but are independent of ‖T‖.

where

The nonlinear part satisfies a global Lipschitz condition

‖g(t ,u)−g(t ,v )‖ ≤ Lg‖u−v‖

T has a bounded logarithmic norm

µ(T )≤ ω..



Consistency

Consistency Con.

Theorem 1

Consider y ′ = Ty . If the exponential Peer method satisfies the conditions

s∑j=1

bij (cj −1)l = (ci −αi )l , l = 0,1, ...,q. (3)

then it is of stiff order of consistency p = q for the linear equation y ′ = Ty .

Theorem 2

Let the condition (3) be satisfied for l = 0, ...,q. Let further

s∑j=1

Aij(cj −1

)r + i−1∑j=1

Rijcrj = r∑

j=0

αj+1i

(rj

) (ci −αi )r−j j!φj+1. (4)

for r = 0, ...,q.Then the EPM is at least of stiff order of consistency p = qfor (1).



Consistency

Consistency Con.

Theorem 3

Let the solution y (t) be (q +2)-times continuously differentiable. Let theconditions (3) be satisfied for l = 0, ...,q +1, and (4) for l = 0, ...,q. Thenthe method is of nonstiff order p = q +1.

Definition 2The exponential peer method (2) is zero-stable if the spectral radius of thestability matrix at z = 0 is one ( i.e. ρ(M (0)) = 1) and all eigenvalues on theunit circle are simple.

whereM(z) = Φ(B⊗ I); z = hT ;



Convergence

Convergence

Theorem 4Let the exponential peer method be consistent of nonstiff order p and

zero-stable. Let for the starting values hold Y0i −y (t0 +cih) = O(hp). Thenthe method is convergent of nonstiff order p.

Theorem 5Let the exponential peer method be consistent of stiff order p and

zero-stable. Let for the starting values hold Y0i −y (t0 +cih) = O(hp). Letbij ≥ 0 for all 1≤ i , j ≤ s. Then the method is convergent of stiff order p.



Choosing α values

Choosing α values

We chose two different possibilities for the values of α.

1 Natural Choice is α= c ÍÑ s different values are required.

2 α= (α∗, ...,α∗,1)T ÍÑ only 2 different arguments

α= (α∗, ...,α∗,1)T , ci = (s− i)(αi −1)+1, i = 1, ...,s. (5)

This gives by (3)

B =

0 1 0 . . . 0 00 0 1 . . . 0 0...

... . . . . . ....

...

0 0 0 . . . . . . 00 0 0 . . . 0 10 0 0 . . . 0 1

(6)

Ñ The methods are optimally zero-stable.Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods


Choosing α values

A special case con.

For the choice (5):

Theorem 6For

s−2s−1

≤ α∗ < 1

the nodes ci are distinct and satisfy 0≤ ci ≤ 1 with cs = 1. Themethod is of order p ≥ s−1 for y ′ = Ty .

Theorem 7

Let the starting values Y0i be exact. Then Y1i = e(1+ci )hT y (t0), i.e.the exact solution of y ′ = Ty .



Outline






5 Numerical Tests

6 Summary



Numerical Tests

Expint package is usedφ-functions.Test examples.Methods for comparison.

Constructed Methodsepm3, epm4, epm5, epm6, epm7 with 3, 4, 5, 6, 7 –stages.

We modified some Expint files and added our own code.

Starting values are computed by ode15s.

Figures



Numerical Tests Con.

Example

Method epm4 with 4 stages of stiff order p ≥ 3:

α= [ 34, 3

4, 3

4,1]T

, C = [ 14, 1

2, 3

4,1]T

,

B =

0 1 0 00 0 1 00 0 0 10 0 0 1

, A =

A11 A12 A13 A140 A11 A12 A130 0 A11 A120 0 0 A44

, R =

0 0 0 0A14 0 0 0A13 A14 0 0R41 R42 R43 0

where

A11 =− 34φ2 + 27

4φ3−

814φ4 , A12 = 3

4φ1−

98φ2−

272φ3 + 243

4φ4,

A13 = 94φ2 + 27

4φ3−

2434φ4 , A14 =− 3

8φ2 + 81

4φ4

A44 = φ1−223φ2 +32φ3−64φ4 , R41 = 12φ2−80φ3 +192φ4

R42 =−6φ2 +64φ3−192φ4, R43 = 43φ2−16φ3 +64φ4 .

Here, in Aij , Rij the argument of the φ-functions is αi hT .



10−3

10−2

10−1

10−15

10−10

10−5

Timestep h

Err

or Gray−Scott, ND=128, IC: Smooth, α=0.035, β=0.065

3

4

5

67

epm3epm4epm5epm6epm7



10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

100

Timestep h

Err

or Allen−Cahn, ND=64, IC: 0.53x+0.47sin(−1.5*pi*x), λ=0.001

4

5

epm4epm5ablawson4lawson4etd4rkstrehmelweinerhochost4rkmk4tetd5rkf



10−3

10−2

10−1

100

10−15

10−10

10−5

100

Timestep h

Err

or Gray−Scott, ND=128, IC: Smooth, α=0.035, β=0.065

4

5




10−3

10−2

10−1

100

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Timestep h

Err

or Hochbruck−Ostermann, ND=200, IC: x(1−x)

4

5




10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

100

102

Timestep h

Err

or Hyperbolic test, ND=200, IC: x(1−x)

4

5




10−3

10−2

10−1

100

10−15

10−10

10−5

Timestep h

Err

or Kuramoto−Sivashinsky, ND=128, IC: cos(x/16)(1+sin(x/16))

4

5




10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

100

Timestep h

Err

or Nonlinear Schrödinger, ND=128, IC: exp(sin(2x)), Pot: 1overSinSqr, λ=1

4

5




10−3

10−2

10−1

100

10−12

10−10

10−8

10−6

10−4

10−2

100

Timestep h

Err

or Parabolic Test, ND=200, IC: x(1−x)

4

5




10−3

10−2

10−1

10−12

10−10

10−8

10−6

10−4

10−2

Timestep h

Err

or Parabolic Test, ND=200, IC: x(1−x)

3

4

5

67

epm3epm4epm5epm6epm7



Outline






5 Numerical Tests

6 Summary



Summary

Exponential Peer methods show good accuracy.

No order reduction observed.

It is easy to obtain methods with 8, 9, ...– stages.Current work.

Variable step sizes.Order conditions for variable step sizes.Implementation (Step size control).Krylov techniques for large dimensions.


exponential peer methods - tu dresden€¦ · martin-luther-university halle-wittenberg april 30,...

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