Exponential Peer Methods Tamer El-Azab & Rdiger Weiner - - PowerPoint PPT Presentation

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

Exponential Peer Methods

Tamer El-Azab & Rüdiger Weiner

Institute of Mathematics Martin-Luther-University Halle-Wittenberg

April 30, 2010

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary What are exponential integrators?

What are exponential integrators?

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

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

Introduction Short historical overview Expint Matlab package Exponential Peer Methods (EPM) Numerical Tests Summary What are exponential integrators?

What are exponential integrators?

Exponential integrators are those integrators which use the exponential function (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 or highly oscillatory differential equations. Many exponential integrators are designed for solving differential equations 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).

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

φ-functions

The most common related functions used in exponential integrators are the so called φ-functions, which are defined as φl (z) = 1 e(1−θ)z θl−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) = 1 l!

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Computing the 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)

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Outline

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Outline

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Expint Matlab package

Expint is a Matlab package designed as a tool for the numerical testing

• f 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 of various 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 constant step size. Computing φ-functions by using Padè approximation combined with scaling-and-squaring. We will use this package for the test and comparison of Exponential Peer

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Outline

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Exponential Peer Methods (EPM)

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

1

s-stage Peer methods. All stages have the same properties. Explicit and implicit Peer methods (Podhaisky, Schmitt & Weiner 2004 – 2009). No order reduction for stiff systems observed for implicit Peer methods.

2

Here Exponential Peer Methods.

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Exponential Peer Methods (EPM) Con.

Ymi = φ0(αihT)

s

• j=1

bijYm−1,j +h

s

• j=1

Aij(αihT)[fm−1,j −TYm−1,j] (2) +h

i−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 of the φ-functions and B = (bij)s

i,j=1 ∈ Rs×s.

c = (ci)s

i=1 ∈ Rs

and vector α = (αi)s

i=1 ∈ Rs is chosen to have a

small number of different arguments.

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Consistency

Definition 1 The exponential peer method (3) is consistent of nonstiff order p if there are constants 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, Lg and 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) ≤ Lgu −v T has a bounded logarithmic norm µ(T) ≤ ω. .

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Rijcr

j = r

• j=0

αj+1

i

r j

• (ci −αi)r−j j!φj+1.

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

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Consistency Con.

Theorem 3 Let the solution y(t) be (q +2)-times continuously differentiable. Let the conditions (3) be satisfied for l = 0,...,q +1, and (4) for l = 0,...,q. Then the method is of nonstiff order p = q +1. Definition 2 The exponential peer method (2) is zero-stable if the spectral radius of the stability matrix at z = 0 is one ( i.e. ρ(M (0)) = 1) and all eigenvalues on the unit circle are simple. where M(z) = Φ(B ⊗I); z = hT ;

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Convergence

Theorem 4 Let 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). Then the method is convergent of nonstiff order p. Theorem 5 Let 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). Let bij ≥ 0 for all 1 ≤ i,j ≤ s. Then the method is convergent of stiff order p.

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Choosing α values

We chose two different possibilities for the values of α.

1

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

2

α = (α∗,...,α∗,1)T only 2 different arguments α = (α∗,...,α∗,1)T , ci = (s −i)(αi −1)+1, i = 1,...,s. (5) This gives by (3) B =           1 ... 1 ... . . . . . . ... ... . . . . . . ... ... ... 1 ... 1           (6) The methods are optimally zero-stable.

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

A special case con.

For the choice (5): Theorem 6 For s −2 s −1 ≤ α∗ < 1 the nodes ci are distinct and satisfy 0 ≤ ci ≤ 1 with cs = 1. The method is of order p ≥ s −1 for y′ = Ty. Theorem 7 Let the starting values Y0i be exact. Then Y1i = e(1+ci)hTy(t0), i.e. the exact solution of y′ = Ty.

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Outline

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Numerical Tests

Expint package is used

φ-functions. Test examples. Methods for comparison.

Constructed Methods epm3, 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

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Numerical Tests Con.

Example Method epm4 with 4 stages of stiff order p ≥ 3: α = 3 4 , 3 4 , 3 4 ,1 T , C = 1 4 , 1 2 , 3 4 ,1 T , B =     1 1 1 1    , A =     A11 A12 A13 A14 A11 A12 A13 A11 A12 A44    , R =     A14 A13 A14 R41 R42 R43     where A11 = − 3 4 φ2 + 27 4 φ3 − 81 4 φ4, A12 = 3 4 φ1 − 9 8 φ2 − 27 2 φ3 + 243 4 φ4, A13 = 9 4 φ2 + 27 4 φ3 − 243 4 φ4, A14 = − 3 8 φ2 + 81 4 φ4 A44 = φ1 − 22 3 φ2 +32φ3 −64φ4, R41 = 12φ2 −80φ3 +192φ4 R42 = −6φ2 +64φ3 −192φ4, R43 = 4 3 φ2 −16φ3 +64φ4. Here, in Aij , Rij the argument of the φ-functions is αi hT. Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

10

−3

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Timestep h Error Gray−Scott, ND=128, IC: Smooth, α=0.035, β=0.065

3 4 5 6 7

epm3 epm4 epm5 epm6 epm7

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

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

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

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Timestep h Error Gray−Scott, ND=128, IC: Smooth, α=0.035, β=0.065

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

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Timestep h Error Hochbruck−Ostermann, ND=200, IC: x(1−x)

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epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

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Timestep h Error Hyperbolic test, ND=200, IC: x(1−x)

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

10

−3

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−2

10

−1

10 10

−15

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

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

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

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

10

−3

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−2

10

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10 10

−12

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Timestep h Error Parabolic Test, ND=200, IC: x(1−x)

4 5

epm4 epm5 ablawson4 lawson4 etd4rk strehmelweiner hochost4 rkmk4t etd5rkf

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

10

−3

10

−2

10

−1

10

−12

10

−10

10

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10

−4

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−2

Timestep h Error Parabolic Test, ND=200, IC: x(1−x)

3 4 5 6 7

epm3 epm4 epm5 epm6 epm7

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Outline

1

Introduction What are exponential integrators? φ-functions

Computing the φ-functions

2

Short historical overview

3

Expint Matlab package

4

Exponential Peer Methods (EPM) Consistency Convergence Choosing α values

5

Numerical Tests

6

Summary

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods

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

Tamer El-Azab & Rüdiger Weiner Exponential Peer Methods