1,249 research outputs found
Error Analysis of Explicit Partitioned Runge-Kutta Schemes for Conservation Laws
An error analysis is presented for explicit partitioned Runge-Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation
In this paper we consider the development of Implicit-Explicit (IMEX)
Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such
systems the scaling depends on an additional parameter which modifies the
nature of the asymptotic behavior which can be either hyperbolic or parabolic.
Because of the multiple scalings, standard IMEX Runge-Kutta methods for
hyperbolic systems with relaxation loose their efficiency and a different
approach should be adopted to guarantee asymptotic preservation in stiff
regimes. We show that the proposed approach is capable to capture the correct
asymptotic limit of the system independently of the scaling used. Several
numerical examples confirm our theoretical analysis
A class of implicit-explicit two-step Runge-Kutta methods
This work develops implicit-explicit time integrators based on two-step Runge-Kutta methods.
The class of schemes of interest is characterized by linear invariant
preservation and high stage orders. Theoretical consistency and stability analyses are performed to reveal the properties of these methods. The new framework offers extreme flexibility
in the construction of partitioned integrators, since no coupling conditions are necessary.
Moreover, the methods are not plagued by severe order reduction, due to their high stage orders.
Two practical schemes of orders four and six are constructed, and are used to solve several test problems.
Numerical results confirm the theoretical findings
Asymptotic Preserving time-discretization of optimal control problems for the Goldstein-Taylor model
We consider the development of implicit-explicit time integration schemes for
optimal control problems governed by the Goldstein-Taylor model. In the
diffusive scaling this model is a hyperbolic approximation to the heat
equation. We investigate the relation of time integration schemes and the
formal Chapman-Enskog type limiting procedure. For the class of stiffly
accurate implicit-explicit Runge-Kutta methods (IMEX) the discrete optimality
system also provides a stable numerical method for optimal control problems
governed by the heat equation. Numerical examples illustrate the expected
behavior
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