VTechWorks staff will be away for the Thanksgiving holiday beginning at noon on Wednesday, November 27, through Friday, November 29. We will resume normal operations on Monday, December 2. Thank you for your patience.

Browsing by Author "Guenther, Michael"

Now showing 1 - 6 of 6
  • Thumbnail Image
    A class of generalized additive Runge-Kutta methods
    Sandu, Adrian; Guenther, Michael (2013-10-22)
    This work generalizes the additively partitioned Runge-Kutta methods by allowing for different stage values as arguments of different components of the right hand side. An order conditions theory is developed for the new family of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit and implicit-implicit,methods in the new framework. The new family, named GARK, introduces additional flexibility when compared to traditional partitioned Runge-Kutta methods, and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.
  • Thumbnail Image
    Linearly implicit GARK schemes
    Sandu, Adrian; Guenther, Michael; Roberts, Steven (Elsevier, 2021-03-01)
    Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge–Kutta (GARK) framework, which constructs multimethods based on Runge–Kutta schemes. This paper constructs the new GARK-ROS/GARK-ROW families of multimethods based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary differential equation models, we develop a general order condition theory for linearly implicit methods with any number of partitions, using exact or approximate Jacobians. We generalize the order condition theory to two-way partitioned index-1 differential-algebraic equations. Applications of the framework include decoupled linearly implicit, linearly implicit/explicit, and linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of order up to four are constructed.
  • Thumbnail Image
    Multirate generalized additive Runge Kutta methods
    Guenther, Michael; Sandu, Adrian (Springer, 2016-07-01)
  • Thumbnail Image
    Multirate implicit Euler schemes for a class of differential-algebraic equations of index-1
    Hachtel, Christoph; Bartel, Andreas; Guenther, Michael; Sandu, Adrian (Elsevier, 2021-05-15)
    Systems of differential equations which consist of subsystems with widely different dynamical behaviour can be integrated by multirate time integration schemes to increase the efficiency. These schemes allow the usage of inherent step sizes according to the dynamical properties of the subsystem. In this paper, we extend the multirate implicit Euler method to semi-explicit differential–algebraic equations of index-1 where the algebraic constraints only occur in the slow changing subsystem. We discuss different coupling approaches and show that consistency and convergence order 1 can be reached. Numerical experiments validate the analytical results.
  • Thumbnail Image
    Multirate linearly-implicit GARK schemes
    Guenther, Michael; Sandu, Adrian (Springer, 2021-12-28)
    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.
  • Thumbnail Image
    Symplectic GARK methods for Hamiltonian systems
    Guenther, Michael; Sandu, Adrian; Zanna, Antonella (2021-03-06)
    Generalized Additive Runge-Kutta schemes have shown to be a suitable tool for solving ordinary differential equations with additively partitioned right-hand sides. This work generalizes these GARK schemes to symplectic GARK schemes for additively partitioned Hamiltonian systems. In a general setting, we derive conditions for symplecticeness, as well as symmetry and time-reversibility. We show how symplectic and symmetric schemes can be constructed based on schemes which are only symplectic. Special attention is given to the special case of partitioned schemes for Hamiltonians split into multiple potential and kinetic energies. Finally we show how symplectic GARK schemes can use efficiently different time scales and evaluation costs for different potentials by using different order for these parts.