72 research outputs found
Multirate timestepping methods for hyperbolic conservation laws
This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different time-steps to be used in different parts of the spatial domain. The discretization is second order accurate in time and preserves the conservation and stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global time-steps (restricted by the largest value of the Courant number on the grid) and therefore results in more efficient algorithms
Update on Multirate Timestepping Methods for Hyperbolic Conservation Laws
This paper constructs multirate time discretizations for hyperbolic conservation laws that allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps (restricted by the largest value of the Courant number on the grid) and therefore results in more efficient algorithms. Numerical results obtained for the advection and Burgers equations confirm the theoretical findings
Multirate explicit Adams methods for time integration of conservation laws
This paper constructs multirate linear multistep time discretizations based on Adams-Bashforth methods. These methods are aimed at solving conservation laws and allow different timesteps to be used in different parts of the spatial domain. The proposed family of discretizations is second order accurate in time and has conservation and linear and nonlinear stability properties under local CFL conditions. Multirate timestepping avoids the necessity to take small global timesteps - restricted by the largest value of the Courant number on the grid - and therefore results in more efficient computations. Numerical results obtained for the advection and Burgers' equations confirm the theoretical findings
On Extrapolated Multirate Methods
In this manuscript we construct extrapolated multirate discretization methods that allow to efficiently solve problems that have components with different dynamics. This approach is suited for the time integration of multiscale ordinary and partial differential equations and provides highly accurate discretizations. We analyze the linear stability properties of the multirate explicit and linearly implicit extrapolated methods. Numerical results with multiscale ODEs illustrate the theoretical findings
A conservative implicit multirate method for hyperbolic problems
This work focuses on the development of a self adjusting multirate strategy
based on an implicit time discretization for the numerical solution of
hyperbolic equations, that could benefit from different time steps in different
areas of the spatial domain. We propose a novel mass conservative multirate
approach, that can be generalized to various implicit time discretization
methods. It is based on flux partitioning, so that flux exchanges between a
cell and its neighbors are balanced. A number of numerical experiments on both
non-linear scalar problems and systems of hyperbolic equations have been
carried out to test the efficiency and accuracy of the proposed approach
Multirate Timestepping for the Incompressible Navier-Stokes Equations in Overlapping Grids
We develop a multirate timestepper for semi-implicit solutions of the
unsteady incompressible Navier-Stokes equations (INSE) based on a
recently-developed multidomain spectral element method (SEM). For {\em
incompressible} flows, multirate timestepping (MTS) is particularly challenging
because of the tight coupling implied by the incompressibility constraint,
which manifests as an elliptic subproblem for the pressure at each timestep.
The novelty of our approach stems from the development of a stable overlapping
Schwarz method applied directly to the Navier-Stokes equations, rather than to
the convective, viscous, and pressure substeps that are at the heart of most
INSE solvers. Our MTS approach is based on a predictor-corrector (PC) strategy
that preserves the temporal convergence of the underlying semi-implicit
timestepper. We present numerical results demonstrating that this approach
scales to an arbitrary number of overlapping grids, accurately models complex
turbulent flow phenomenon, and improves computational efficiency in comparison
to singlerate timestepping-based calculations.Comment: 40 pages, 13 figure
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