Design of High-Order Multirate General Additive Runge Kutta Schemes - - PowerPoint PPT Presentation

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Design of High-Order Multirate General Additive Runge Kutta Schemes Arash Sarshar, Steven Roberts, and Adrian Sandu Computational Science Laboratory Compute the Future!, Department of Computer Science, Virginia Polytechnic Institute and

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Design of High-Order Multirate General Additive Runge Kutta Schemes

Arash Sarshar, Steven Roberts, and Adrian Sandu

Computational Science Laboratory “Compute the Future!”, Department of Computer Science, Virginia Polytechnic Institute and State University Blacksburg, VA 24060

Feb 26, 2019

Design of High-Order MRGARK Schemes. . [1/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Outline

Multirate GARK methods Accuracy and order conditions Coupled and decoupled MR GARK methods Adaptivity and error control Numerical experiments Conclusions Bibliography

Design of High-Order MRGARK Schemes. . [2/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Multirate systems involve mixed dynamics.

y′ =

N

  • σ=1

f {σ}(y), y(t0) = y0, y(t) ∈ Rd. y′ = f (y) = f {s}(y) + f {f}(y), y(t0) = y0, y(t) ∈ Rd, M = H/h. ◮ Systems driven by hybrid dynamics that incur different time-scales. ◮ Fast dynamics (shock waves, diffusion, electro/magneto waves) interact with slow ones (long range transport, reaction processes, nuclear decay). ◮ Multiple discretization lead to varying stiffness and evaluation costs of the right hand side partitions.

Design of High-Order MRGARK Schemes. Multirate GARK methods. [3/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Desired features of a multirate time-stepping method

◮ Flexible methods that work at different rates (M dependent coefficients). ◮ Treat different partitions of the system according to their stiffness (couple implicit and explicit methods). ◮ Avoid unnecessary computation cost (decouple stage computations across different partitions). ◮ Control both the error and the rates of integration in different partitions (H − M adaptivity).

Design of High-Order MRGARK Schemes. Multirate GARK methods. [4/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 5

We design of multirate methods using the GARK framework.

◮ Multirating body of work is rich: Rice1, Andrus2, Gear and Wells, Kværnø and Rentrop3 Bartel4, . . . ◮ GARK framework developed by Sandu and G¨ unther5 describes a general methodology and order condition theory for partitioned Runge-Kutta schemes ◮ Multirate GARK framework6 lays out the order condition theory for MR GARK methods. ◮ We will consider methods discrete in all partitions

1Rice, “Split Runge-Kutta methods for simultaneous equations”. 2Andrus, “Numerical Solution for Ordinary Differential Equations Separated into Subsystems”; Andrus, “Stability of a

multirate method for numerical integration of ODEs”.

3Kværnø, “Stability of multirate Runge-Kutta schemes”; Kværnø and Rentrop, Low order multirate Runge-Kutta methods in

electric circuit simulation.

4Bartel and G¨

unther, “A multirate W-method for electrical networks in state-space formulation”.

5Sandu and G¨

unther, “A generalized-structure approach to additive Runge-Kutta methods”.

6G¨

unther and Sandu, “Multirate generalized additive Runge-Kutta methods”. Design of High-Order MRGARK Schemes. Multirate GARK methods. [5/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 6

One step of a MR GARK scheme:

Y {s}

i

= yn + H

s{s}

  • j=1

a{s,s}

i,j

f {s} Y {s}

j

  • + h

M

  • λ=1

s{f}

  • j=1

a{s,f,λ}

i,j

f {f} Y {f,λ}

j

  • ,

i = 1, . . . , s{s}, Y {f,λ}

i

= yn+(λ−1)/M + H

s{s}

  • j=1

a{f,s,λ}

i,j

f {s} Y {s}

j

  • + h

s{f}

  • j=1

a{f,f}

i,j

f {f} Y {f,λ}

j

  • ,

i = 1, . . . , s{f},

  • yn+λ/M =

yn+(λ−1)/M + h

s{f}

  • i=1

b{f}

i

f {f} Y {f,λ}

i

  • ,

λ = 1, . . . , M, yn+1 = yn+M/M + H

s{s}

  • i=1

b{s}

i

f {s} Y {s}

i

  • .

tn λ = 1 λ = 2 λ = 3

c1 c2 c3

{s} {s} {s}

tn+1 Slow stage Fast stage

M = 3

Design of High-Order MRGARK Schemes. Multirate GARK methods. [6/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Butcher tableau for a MR GARK method:

A{f,f} A{f,s} A{s,f} A{s,s} b{f} T b{s} T :=

1 M A{f,f}

· · · A{f,s,1} . . . ... . . . . . .

1 M ✶ b{f} T

· · ·

1 M A{f,f}

A{f,s,M}

1 M A{s,f,1}

· · ·

1 M A{s,f,M}

A{s,s}

1 M b{f} T

· · ·

1 M b{f} T

b{s} T .

Design of High-Order MRGARK Schemes. Multirate GARK methods. [7/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Order 3 coupling conditions for Internally consistent MR GARK

M 6 =

M

  • λ=1

b{f} T A{f,s,λ} c{s}, (order 3) M2 6 =

M

  • λ=1

b{s} TA{s,f,λ} (λ − 1)✶ + c{f} , (order 3)

Design of High-Order MRGARK Schemes. Accuracy and order conditions. [8/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Order 4 coupling conditions for Internally consistent MR GARK

M2 8 =

M

  • λ=1

(λ − 1) b{f} T A{f,s,λ} c{s} +

M

  • λ=1

b{f} T c{f} × A{f,s,λ} c{s} , (order 4) M2 8 = b{s} T

M

  • λ=1
  • c{s} ×
  • A{s,f λ}

(λ − 1) ✶ + c{f} , (order 4) M 12 =

M

  • λ=1

b{f} T A{f,s,λ}c{s}×2, (order 4) M3 12 =

M

  • λ=1

b{s} T A{s,f,λ} c{f}×2 +

M

  • λ=1

(λ − 1)2b{s} T A{s,f,λ} ✶ + 2

M

  • λ=1

(λ − 1)b{s} T A{s,f,λ} c{f}, (order 4) M2 24 =

M

  • λ=1

b{s} T A{s,s} A{s,f,λ} (λ − 1)✶ + c{f} , (order 4) Design of High-Order MRGARK Schemes. Accuracy and order conditions. [9/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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More order 4 coupling conditions for Internally consistent MR GARK

M 24 =

M

  • λ=1

b{s} T A{s,f,λ} A{f,s,λ} c{s}, (order 4) M3 24 =

M

  • λ=1

(λ − 1)2 2 b{s} T A{s,f,λ} ✶ +

M

  • λ=1

(λ − 1)b{s} T A{s,f,λ} c{f} +

M

  • λ=1

b{s} T A{s,f,λ} A{f,f} c{f}, (order 4) M2 24 =

M

  • λ=1

λ−1

  • k=1

b{f} T A{f,s,k} c{s} +

M

  • λ=1

b{f} T A{f,f} A{f,s,λ} c{s}, (order 4) M 24 =

M

  • λ=1

b{f} T A{f,s,λ} A{s,s} c{s}, (order 4) M3 24 =

M

  • λ=1

M

  • k=1

b{f} T A{f,s,λ} A{s,f,k} (k − 1)✶ + c{f} . (order 4) Design of High-Order MRGARK Schemes. Accuracy and order conditions. [10/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 11

Coupling order conditions in 2-color tree representation.

fast slow j k l j k l j k l m j k l m j k l m j k l m j k l m j k l m j k l m j k l m j k l m j k l m

Design of High-Order MRGARK Schemes. Accuracy and order conditions. [11/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 12

Examining the coupling structure reveals a pattern.

A{s,f} A{f,f} A{s,s} A{f,s}

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

(a) decoupled A

7 1 2 8 3 4 9 5 6 7 1 2 8 3 4 9 5 6

(b) permuted decoupled A{s,f} A{f,f} A{s,s} A{f,s}

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

(c) coupled MrGARK A

7 1 2 8 3 4 9 5 6 7 1 2 8 3 4 9 5 6

(d) permuted coupled

Design of High-Order MRGARK Schemes. Coupled and decoupled MR GARK methods. [12/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Decoupled methods are computationally more efficient.

◮ Opting in for better computational efficiency at the cost of losing some stability. ◮ Decoupled methods have complementary structure in the slow-to-fast and fast-to-slow coupling: A{s,f} × A{f,s} T = 0s{s}×Ms{f}.

Design of High-Order MRGARK Schemes. Coupled and decoupled MR GARK methods. [13/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Isolating the slow, fast and coupling error is a challenging task.

◮ 4 order 4 slow trees: , , , ◮ 4 order 4 fast trees: , , , ◮ 10 order 4 coupling trees: , , , , , , , , ,

Design of High-Order MRGARK Schemes. Adaptivity and error control. [14/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Controlling the step sizes by balancing the projected errors:

◮ Choose macro step size according to traditional error control theory. ◮ Choose step size ratio M to balance the projected slow and fast errors. Hnew = fac · H · ( εn+1)− 1

p ,

  • ε{s}

n+2 =

ε{f}

n+2,

Mnew ≈ M ·

  • ε{f}

n+1

  • ε{s}

n+1

1

q

.

Design of High-Order MRGARK Schemes. Adaptivity and error control. [15/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Controlling the step sizes by balancing the projected errors vs computational cost:

◮ Monitor the computational cost of evaluating right-hand-side pieces ◮ Solve an online univariate optimization to minimize function evaluation cost along with error min

Hnew,Mnew

t{s} + Mnew t{f} Hnew subject to εn+2 = 1, min

Mnew

t{s} + Mnew t{f} H

  • ε{s}

n+1 +

ε{f}

n+1 · Mq

Mq

new

  • 1

q+1

.

Design of High-Order MRGARK Schemes. Adaptivity and error control. [16/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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Overview of type-A Multirate GARK methods developed:

Order Fast Slow Stiff FSAL H − M High-Order Method Method Accuracy FSAL Adaptive Coupling 2 Ralston’s ERK2(1)2 {Ralston, 1962} Ralston’s ERK2(1)2 {Ralston, 1962}

  • 2

SDIRK2(1)2 {Alexander, 1977} Ralston’s ERK2(1)2 {Ralston, 1962}

  • 2

Ralston’s ERK2(1)2 {Ralston, 1962} SDIRK2(1)2 {Alexander, 1977}

  • 3

Ralston’s ERK3(2)3 {Ralston, 1962} Ralston’s ERK3(2)3 {Ralston, 1962}

  • 3

SDIRK3(2)3 {Alexander, 1977} Ralston’s ERK3(2)3 {Ralston, 1962}

  • 3

Ralston’s ERK3(2)3 {Ralston, 1962} SDIRK3(2)3 {Alexander, 1977}

  • 3

Custom ERK3(2)3 Custom ERK3(2)3

  • 4

ERK4(3)5 {Sofroniou and Spaletta, 2004} ERK4(3)5 {Sofroniou and Spaletta, 2004}

  • 4

Fehlberg’s ERK4(5)6 {Fehlberg, 1969} SDIRK4(3)5 {Kennedy and Carpenter, 2016}

  • 4

ERK4(3)4 {Sofroniou and Spaletta, 2004} Custom SDIRK4(3)6

  • Design of High-Order MRGARK Schemes. Numerical experiments.

[17/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 18

Overview of type-S Multirate GARK methods developed:

Order Fast Slow H − M Method Method Adaptivity 2 Ralston’s ERK2(1)2 {Ralston, 1962} Ralston’s ERK2(1)2 {Ralston, 1962}

  • 3

Ralston’s ERK3(2)3 Ralston’s ERK3(2)3

  • 3

SDIRK3(2)3 Ralston’s ERK3(2)3 {Ralston, 1962}

  • 3

Ralston’s ERK3(2)3 {Ralston, 1962} SDIRK3(2)3

  • Design of High-Order MRGARK Schemes. Numerical experiments.

[18/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 19

Component partitioning test

We use the reaction-diffusion equation: ut = ∇ · (D(x, y)∇u) + 10

  • 1 − u2

(u + 0.6), x, y ∈ Ω, u(x, y, 0) = 2 exp

  • −10(x − 0.5)2 − 10(y + 0.1)2

, D(x, y) = 0.1

3

  • i=1

e−100(x−0.5)2+(y−yi)2, D(x, y)∇u · n = 0, x, y ∈ ∂Ω, t ∈ [0, tF].

Design of High-Order MRGARK Schemes. Numerical experiments. [19/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 20

Slow and fast partitions are defined on mesh points.

Design of High-Order MRGARK Schemes. Numerical experiments. [20/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 21

Convergence diagram for component partitioning test

500 1000 2000 4000

Macro Steps

10-13 10-12 10-11 10-10 10-9 10-8 10-7

Error

M = 2

, = 3 , = 4 , = 2

EX2-EX2 2(1) EX3-EX3 3(2) EX5-EX5 4(3) 500 1000 2000 4000

Macro Steps

10-12 10-11 10-10 10-9 10-8 10-7

Error

M = 4

, = 3 , = 4 , = 2

EX2-EX2 2(1) EX3-EX3 3(2) EX5-EX5 4(3) Design of High-Order MRGARK Schemes. Numerical experiments. [21/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 22

Additive partitioning test

We use the advection-diffusion equation: ut − ε∇2u + w · ∇u = 0 in Ω, u = 0 on ∂Ω, w = 2y(1 − x2) −2x(1 − y2)

  • .

A Streamline Upwind Petrov-Galerkin (SUPG) spatial discretization is used, which leads to a semi-discrete system of linear ODEs: Mhuh

t = A uh + (

n + nstab) uh,

Design of High-Order MRGARK Schemes. Numerical experiments. [22/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 23

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 0.00 [s]

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 1.00 [s]

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 2.00 [s]

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 3.00 [s]

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 4.00 [s]

0.25 0.5 0.75 1

  • 0.04

1.05

u Time: 5.00 [s]

Design of High-Order MRGARK Schemes. Numerical experiments. [23/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 24

Convergence diagram for additive partitioning test

10 1 10 2 10 3

Macro Steps

10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2

Error

M = 2

, = 4 , = 3 , = 2

EX3-EX3 3(2) IM3-EX3 3(2) IM6-EX5 4(2) EX3-IM3 3(2) EX6-IM5 4(3) EX5-EX5 4(3) EX2-EX2 2(1) EX2-IM2 2(1)

10 1 10 2 10 3

Macro Steps

10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2

Error

M = 4

, = 4 , = 3 , = 2

EX3-EX3 3(2) IM3-EX3 3(2) IM6-EX5 4(2) EX3-IM3 3(2) EX6-IM5 4(3) EX5-EX5 4(3) EX2-EX2 2(1) EX2-IM2 2(1) Design of High-Order MRGARK Schemes. Numerical experiments. [24/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 25

Conclusions

◮ MR GARK EX-EX, EX-IM and IM-EX methods of up to order 4 ◮ Applied to component and operator splitting systems ◮ Developed adaptivity strategies ◮ What about stability considerations and Implicit-Implicit methods ?

◮ Steven Robert’s talk on Friday (MS390)

◮ What if you need full control over the fast system ?

◮ Adrian Sandu’s talk on Friday (MS358) ◮ arxiv.org/abs/1802.07188 ◮ arxiv.org/abs/1812.00808

◮ Where can I find the coefficients ?

◮ arxiv.org/abs/1804.07716 ◮ wolfr.am/BsWkAHiM ◮ MatlODE package

Design of High-Order MRGARK Schemes. Conclusions. [25/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 26

Bibliography I

J.F. Andrus. “Numerical Solution for Ordinary Differential Equations Separated into Subsystems”. In: SIAM Journal on Numerical Analysis 16.4 (1979), pp. 605–611. J.F. Andrus. “Stability of a multirate method for numerical integration of ODEs”. In: Computers Math. Applic 25 (1993),

  • pp. 3–14.
  • A. Bartel and M. G¨
  • unther. “A multirate W-method for electrical

networks in state-space formulation”. In: Journal of Computational Applied Mathematics 147.2 (2002), 411–425. issn: 0377-0427. doi: 10.1016/S0377-0427(02)00476-4.

  • M. G¨

unther and A. Sandu. “Multirate generalized additive Runge-Kutta methods”. In: Numerische Mathematik 133.3 (2016),

  • pp. 497–524. doi: 10.1007/s00211-015-0756-z.

Design of High-Order MRGARK Schemes. Bibliography. [26/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)

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SLIDE 27

Bibliography II

  • A. Kværnø. “Stability of multirate Runge-Kutta schemes”. In:

International Journal of Differential Equations and Applications 1.1 (2000), 97–105.

  • A. Kværnø and P. Rentrop. Low order multirate Runge-Kutta

methods in electric circuit simulation. 1999. url: citeseer.ist.psu.edu/629589.html. J.R. Rice. “Split Runge-Kutta methods for simultaneous equations”. In: Journal of Research of the National Institute of Standards and Technology 60.B (1960).

  • A. Sandu and M. G¨
  • unther. “A generalized-structure approach to

additive Runge-Kutta methods”. In: SIAM Journal on Numerical Analysis 53.1 (2015), pp. 17–42. doi: 10.1137/130943224.

Design of High-Order MRGARK Schemes. Bibliography. [27/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)