Linearly implicit GARK schemes

Sandu, Adrian; Günther, Michael; Roberts, Steven

doi:10.1016/j.apnum.2020.11.014

Cited by 2 publications

(3 citation statements)

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“…In the previous section we have seen that MR-GARK-ROS/ROW schemes can be interpreted as partitioned GARK-ROS/ROW schemes. Thus the order conditions of these multirate schemes can be easily derived form the underlying GARK-ROS/ROW schemes derived in [32]. In the following we focus on pure multirate methods (3.5).…”

Section: Order Conditions For Mr-gark-ros/row Schemesmentioning

confidence: 99%

“…Rosenbrock-Wanner (ROS) methods [17,28] are linearly implicit Runge-Kutta schemes that use the exact Jacobian of the right hand side function in the computational process; Rosenbrock-W methods [36] allow arbitrary approximations of the Jacobian. In a recent paper [32] the authors have generalized the GARK approach to partitioned linearly-implicit schemes based on using exact (GARK-ROS) and inexact (GARK-ROW) Jacobian information.…”

Section: Introductionmentioning

confidence: 99%

“…(2.5d) Theorem 2.2 (GARK-ROW order conditions [32]) The GARK-ROW order conditions Eq. (2.1) are the same as the Rosenbrock-W order conditions [17], except that the method coefficients are also labelled according to partition indices.…”

Section: Introductionmentioning

confidence: 99%

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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: Order Conditions For Mr-gark-ros/row Schemesmentioning

confidence: 99%