Parallel Numerical Methods for Ordinary DifferentialEquations

S. I. Solodushkin1,2, I. F. Yumanova1

1 Ural Federal University2 Institute of Mathematics and Mechanics

Ural HPC 2016, Ekaterinburg, Russia

2016, October 06.

The problem

y ′(x) = f (x , y(x)),y(x0) = y0,

(1)

where x ∈ [x0;X ], y ∈ Rm.

Gear classification:1 parallelism across the system or, that is the same, parallelism across

the space;2 parallelism across the method or, that is the same, parallelism across

the time.

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Predictor-corrector type methods

W. L. MIRANKER AND W. LINIGER, Parallel methods for the numericalintegration of ordinary differential equations, Math. Comp., 91 (1967)

ypn+1 = y c

n +h

2

(3f (xn, y

cn )− f (xn−1, y

cn−1)

),

y cn+1 = y c

n +h

2

(f (xn, y

cn ) + f (xn+1, y

pn+1)

),

(2)

ypn+1 = y c

n−1 + 2hf (xn, ypn ),

y cn = y c

n−1 +h

2

(f (xn−1, y

cn−1) + f (xn, y

pn ))),

(3)

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Runge–Kutta methods

Consider an explicit s-stage Runge–Kutta method

k1 = f (xn, yn),

k2 = f (xn + c2h, yn + ha21k1),

k3 = f (xn + c3h, yn + h(a31k1 + a32k2)),

...

ks = f (xn + csh, yn + h(as1k1 + ...+ as,s−1ks−1)),

yn+1 = yn + h(b1k1 + ...+ bsks)

(4)

0c2 a21c3 a31 a32...cs as1 as2 ... as,s−1

b1 b2 ... bs−1 bs

k1 = f (xn, yn)

k2 = f (xn + h2 , yn + h

2k1)

k3 = f (xn + h2 , yn + h

2k2)

k4 = f (xn + h, yn + hk3)

yn+1 = yn + h6 (k1 + 2k2 + 2k3 + k4)

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Internal parallelism of explicit Runge–Kutta methods

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Internal parallelism of explicit Runge–Kutta methods

Let us consider RK matrix A which could be partitioned (possibly after apermutation of the stages) as

01

A21 02

A31 A32 03...

.... . .

Aσ1 Aσ2 . . . Aσ2 0σ

TheoremFor an explicit Runge–Kutta method with σ sequential stages1) the order p 6 σ for any number of available processors,2) if p = σ, the stability region is

{z : |

∑pk=1

zk

k! | 6 1}.

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Internal parallelism of implicit Runge–Kutta methods

Consider an implicit s−stage Runge–Kutta method

ki = f

xn + cih, yn + hs∑

j=1

aijkj

, i = 1, ..., s

yn+1 = yn + hs∑

i=1

biki ,

(5)

D1

A21 D2

A31 A32 D3...

.... . .

Aσ1 Aσ2 . . . Aσ2 Dσ

1/2 1/2 0 0 02/3 0 2/3 0 01/2 -5/2 5/2 1/2 01/3 -5/3 4/3 0 2/3

-1 3/2 -1 3/2

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Optimal with respect to the order methods

Whether it is always possible to find ERK method of order p using notmore than p effective stages, assuming that suffucient number ofprocessors are available?

These methods are called P-optimal.

6 4 5 6 7 8 9 10Sequential ERKsmin p 6 7 9 11 > 12 > 13S p 5 6 7 8 9 10Optimal RKSeff p 6 7 9 11 > 12 > 13Num. of proc - 3 3 4 4 5 5

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Optimal with respect to the order methods

Method (5) can be interpreted as an ERK method with scheme

0 0c A 0c 0 A 0...

.... . .

c 0 0 0 A 00 . . . 0 0 bT

(6)

We assume that σ sequential stages are performed and s processors areavailable.

TheoremThe parallel iterated Runge–Kutta method (5) in form (6) is of orderp = min(p0, σ), where p0 denotes the order of the basic method.

The choice σ = p0 yields P-optimal ERK methods.

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How many processors do we need?

What is the least number of processors needed to implement an optimalERK method.

The basic method is the s-stage Gaussian–Legendre type RK method,which has the smallest number of stages with respect to their order.

This allowed to construct the method of order p = 2s which is P-optimalon s processors.

Van Der Houwen P. J., Sommeijer B. P. Parallel iteration of high-order Runge-Kuttamethods with stepsize control // Journal of Computational and Applied Mathematics,1990. Volume 29. Issue 1. pp. 111–127.

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Block methods

[yn+1

yn+2

]= h

[2/3 −1/124/3 1/3

] [fn+1

fn+2

]+

[yn

yn

]+ h

[5/121/3

] [fnfn

].

εn =

(h4

24u(4)(ζn+1), −

h5

90u(5)(ζn+2)

)

Shampine L. F. and Watts H. A. Block Implicit One-Step Methods // Math. Comp.,1969. V. 23. pp. 731–740.

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Extrapolation methods

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Multiple shooting algorithms

y ′0(x) = f (x , y0(x)), y ′1(x) = f (x , y1(x)), . . . y ′n−1(x) = f (x , yn−1(x)),y0(0) = y0 y1(x1) = U1 yn−1(xn−1) = Un−1

matching conditions U1 = y0(x1), . . . ,Un−1 = yn−2(xn−1),U1 − y0(x1) = 0U2 − y1(x2) = 0. . .Un−1 − yn−1(xn−1) = 0

Uk+10 = y0

Uk+1i+1 = yi (xi+1,U

ki ) +

∂yi (xi+1,Uki )

∂Ui(Uk+1

i − Uki ), i = 0, 1, 2, . . . , n − 2

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Thank you for attention

Parallel Numerical Methods for Ordinary Differential Equations:a Survey

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