Design of High-Order Decoupled Multirate GARK Schemes

Sarshar, Arash; Roberts, Steven; Sandu, Adrian

doi:10.1137/18m1182875

Cited by 18 publications

(14 citation statements)

References 29 publications

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“…Examples of partitioned methods developed in the GARK framework include [15,27,38]. MR-GARK and MRI-GARK frameworks [15,16,25,26,29,34] define general classes of multirate Runge-Kutta methods based on the GARK formalism.…”

Section: Introductionmentioning

confidence: 99%

Multirate linearly-implicit GARK schemes

Günther

Sandu

2021

Bit Numer Math

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Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

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Section: Introductionmentioning

confidence: 99%

Multirate linearly-implicit GARK schemes

Günther

Sandu

2021

Bit Numer Math

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“…IMEX methods [3, 4, 6-9, 12, 15, 24, 28, 56-59] treat a split IVP with a pair of coupled methods, one implicit, the other explicit. Multirate methods [1,2,11,13,14,16,20,21,34,36,40,42,44] use different timestep sizes to solve different components of the problem. W-methods (first introduced in the context of Rosenbrock schemes [32,33,45,46,53]) use the problem Jacobian, f y (t, y) = J, explicitly in the computational process (in either a linearly-implicit or matrix exponential formula) and use additional order conditions to eliminate the errors associated with using an approximate Jacobian.…”

Section: Introductionmentioning

confidence: 99%

Biorthogonal Rosenbrock-Krylov time discretization methods

Glandon

Sandu

2020

Applied Numerical Mathematics

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Many scientific applications require the solution of large initial-value problems, such as those produced by the method of lines after semi-discretization in space of partial differential equations. The computational cost of implicit time discretizations is dominated by the solution of nonlinear systems of equations at each time step. In order to decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time integration methods extend the classical linearly-implicit Rosenbrock(-W) methods, and make use of a Krylov subspace approximation to the Jacobian computed via an Arnoldi process. Since the ROK order conditions rely on the construction of a single Krylov space, no restarting of the Arnoldi process is allowed, and the iterations quickly become expensive with increasing subspace dimensions. This work extends the ROK framework to make use of the Lanczos biorthogonalization procedure for constructing Jacobian approximations. The resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos procedure's short two-term recurrence allows BOROK methods to utilize larger subspaces for the Jacobian approximation, resulting in increased numerical stability of the time integration at a reduced computational cost. Adaptive subspace size selection and basis extension procedures are also developed for the new schemes. Numerical experiments show that for stiff problems, where a large subspace used to approximate the Jacobian is required for stability, the BOROK methods outperform the original ROK methods.

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“…Günther and Sandu continue in [9] where many variants of multirate Runge-Kutta methods are cast as GARK methods. Multirate GARK (MrGARK) methods up to order four are derived in [23]. These include methods that are explicit in both partitions and methods that combine explicit and implicit methods.…”

Section: Introductionmentioning

confidence: 99%

Implicit multirate GARK methods

Roberts,

Loffeld,

Sarshar

et al. 2019

Preprint

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This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

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Design of High-Order Decoupled Multirate GARK Schemes

Cited by 18 publications

References 29 publications

Multirate linearly-implicit GARK schemes

Multirate linearly-implicit GARK schemes

Biorthogonal Rosenbrock-Krylov time discretization methods

Implicit multirate GARK methods

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