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)
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)
Multirate systems involve mixed dynamics. � N y ′ = f { σ } ( y ) , y ( t ) ∈ R d . y ( t 0 ) = y 0 , σ =1 y ′ = f ( y ) = f { s } ( y ) + f { f } ( y ) , y ( t ) ∈ R d , y ( t 0 ) = y 0 , 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)
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)
We design of multirate methods using the GARK framework. ◮ Multirating body of work is rich: Rice 1 , Andrus 2 , Gear and Wells, Kværnø and Rentrop 3 Bartel 4 , . . . unther 5 describes a ◮ GARK framework developed by Sandu and G¨ general methodology and order condition theory for partitioned Runge-Kutta schemes ◮ Multirate GARK framework 6 lays out the order condition theory for MR GARK methods. ◮ We will consider methods discrete in all partitions 1 Rice, “Split Runge-Kutta methods for simultaneous equations”. 2 Andrus, “Numerical Solution for Ordinary Differential Equations Separated into Subsystems”; Andrus, “Stability of a multirate method for numerical integration of ODEs”. 3 Kværnø, “Stability of multirate Runge-Kutta schemes”; Kværnø and Rentrop, Low order multirate Runge-Kutta methods in electric circuit simulation . 4 Bartel and G¨ unther, “A multirate W-method for electrical networks in state-space formulation”. 5 Sandu and G¨ unther, “A generalized-structure approach to additive Runge-Kutta methods”. 6 G¨ 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)
One step of a MR GARK scheme: s { s } s { f } f { s } � � M f { f } � � � � � Y { s } a { s , s } Y { s } a { s , f ,λ } Y { f ,λ } i = 1 , . . . , s { s } , = y n + H + h , i i , j j i , j j j =1 λ =1 j =1 s { s } s { f } f { s } � � f { f } � � � � Y { f ,λ } a { f , s ,λ } Y { s } a { f , f } Y { f ,λ } i = 1 , . . . , s { f } , = � y n +( λ − 1) / M + H + h , i i , j j i , j j j =1 j =1 s { f } � f { f } � � b { f } Y { f ,λ } y n + λ/ M = � � y n +( λ − 1) / M + h , λ = 1 , . . . , M , i i i =1 s { s } f { s } � � � b { s } Y { s } y n +1 = � y n + M / M + H . i i i =1 M = 3 Slow stage λ = 1 λ = 2 λ = 3 Fast stage {s} {s} {s} c 1 c 2 c 3 tn tn+1 Design of High-Order MRGARK Schemes. Multirate GARK methods. [6/27] SIAM CSE19. Feb 26, 2018. Computational Science Lab (http://csl.cs.vt.edu)
Butcher tableau for a MR GARK method: M A { f , f } 1 A { f , s , 1 } 0 · · · . . . ... . . . A { f , f } A { f , s } . . . A { s , f } A { s , s } := M ✶ b { f } T 1 M A { f , f } 1 A { f , s , M } . · · · b { f } T b { s } T M A { s , f , 1 } 1 M A { s , f , M } 1 A { s , s } · · · M b { f } T 1 M b { f } T 1 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)
Order 3 coupling conditions for Internally consistent MR GARK M � M b { f } T A { f , s ,λ } c { s } , 6 = (order 3) λ =1 b { s } T A { s , f ,λ } � ( λ − 1) ✶ + c { f } � M � M 2 = , (order 3) 6 λ =1 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)
Order 4 coupling conditions for Internally consistent MR GARK M 2 M � ( λ − 1) b { f } T A { f , s ,λ } c { s } = 8 λ =1 (order 4) M b { f } T � c { f } × A { f , s ,λ } c { s } � � + , λ =1 M 2 M � � A { s , f λ } � ( λ − 1) ✶ + c { f } � �� � c { s } × = b { s } T (order 4) , 8 λ =1 M � M b { f } T A { f , s ,λ } c { s }× 2 , = (order 4) 12 λ =1 M 3 M M � � b { s } T A { s , f ,λ } c { f }× 2 + ( λ − 1) 2 b { s } T A { s , f ,λ } ✶ = 12 λ =1 λ =1 (order 4) M � ( λ − 1) b { s } T A { s , f ,λ } c { f } , + 2 λ =1 M 2 M b { s } T A { s , s } A { s , f ,λ } � ( λ − 1) ✶ + c { f } � � = , (order 4) 24 λ =1 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)
More order 4 coupling conditions for Internally consistent MR GARK M � M b { s } T A { s , f ,λ } A { f , s ,λ } c { s } , = (order 4) 24 λ =1 M 3 M ( λ − 1) 2 � b { s } T A { s , f ,λ } ✶ = 24 2 λ =1 (order 4) M M � � ( λ − 1) b { s } T A { s , f ,λ } c { f } + b { s } T A { s , f ,λ } A { f , f } c { f } , + λ =1 λ =1 M 2 M λ − 1 M � � � b { f } T A { f , s , k } c { s } + b { f } T A { f , f } A { f , s ,λ } c { s } , = (order 4) 24 λ =1 k =1 λ =1 M � M b { f } T A { f , s ,λ } A { s , s } c { s } , = (order 4) 24 λ =1 M 3 M M b { f } T A { f , s ,λ } A { s , f , k } � ( k − 1) ✶ + c { f } � � � = . (order 4) 24 λ =1 k =1 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)
Coupling order conditions in 2-color tree representation. fast slow l l l l m m k k k k j j j j m m m m l l l l k k k k j j j j m m m m l l l l k k k k j j j j 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)
Examining the coupling structure reveals a pattern. 1 7 2 1 A {f,f} 3 A {f,s} 2 4 8 A 5 3 6 4 7 9 A {s,s} A {s,f} 8 5 9 6 1 2 3 4 5 6 7 8 9 7 1 2 8 3 4 9 5 6 (a) decoupled (b) permuted decoupled 1 7 2 1 A {f,s} A {f,f} 3 2 4 8 A 5 3 6 4 7 9 A {s,f} A {s,s} 8 5 9 6 1 2 3 4 5 6 7 8 9 7 1 2 8 3 4 9 5 6 (c) coupled MrGARK (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)
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 = 0 s { 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)
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)
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. ε n +1 ) − 1 p , H new = fac · H · ( � ε { s } ε { f } � n +2 = � n +2 , � � 1 ε { f } q � n +1 . M new ≈ M · ε { s } � n +1 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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