Multirate linearly-implicit GARK schemes. (English) Zbl 1507.65122
The efficient solution of initial value problems relies on the use of variable stepsize methods, which adapt to the dynamics of the problem. Multirate schemes exploit this different dynamical behavior by applying small step sizes to the fast part and large step sizes to the slow part.
On the other hand, linearly implicit methods enjoy the same stability properties as the implicit schemes, but solve only linear systems of equations at each step. This results in an improvement in efficiency.
The paper presents a new MR-GARK ROS/ROW formalism for these kind of methods (GARK-ROS schemes use the exact Jacobian information while GARK-ROW schemes allow any approximation of the Jacobian). Using the partitioned GARK ROS/ROW framework, general order conditions are derived. A linear stability analysis is also performed to validate the applicability of the proposed schemes. Various slow-fast coupling strategies for an efficient computation are presented. Some multirate infinitesimal step (MRI-GARK ROS/ROW) schemes are developed and multirate infinitesimal step SPC methods.
On the other hand, linearly implicit methods enjoy the same stability properties as the implicit schemes, but solve only linear systems of equations at each step. This results in an improvement in efficiency.
The paper presents a new MR-GARK ROS/ROW formalism for these kind of methods (GARK-ROS schemes use the exact Jacobian information while GARK-ROW schemes allow any approximation of the Jacobian). Using the partitioned GARK ROS/ROW framework, general order conditions are derived. A linear stability analysis is also performed to validate the applicability of the proposed schemes. Various slow-fast coupling strategies for an efficient computation are presented. Some multirate infinitesimal step (MRI-GARK ROS/ROW) schemes are developed and multirate infinitesimal step SPC methods.
Reviewer: Higinio Ramos (Zamora)
MSC:
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
multirate integration; generalized additive Runge-Kutta schemes; linear implicitness; GARK ROS/ROW methods; stability analysisReferences:
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