Implicit-Explicit Methods for Time-Dependent Partial Differential Equations

Ascher, Uri M.; Ruuth, Steven J.; Wetton, Brian

doi:10.1137/0732037

Cited by 883 publications

(977 citation statements)

References 19 publications

(43 reference statements)

Supporting

Mentioning

942

Contrasting

Unclassified

Order By: Relevance

“…The pressure is approximated by polynomials of two units lower order than for the velocity, in order to compute a pressure unpolluted by spurious modes with only one collocation grid (Botella 1997). The time marching combines a fourth-order Adams-Bashforth scheme and a fourth-order backward differentiation scheme, with the viscous term treated implicitly (Ascher, Ruuth & Wetton 1995). The numerical parameters are y ∞ = 10, N y = 100 and t = 10 −5 .…”

Section: Methodsmentioning

confidence: 99%

Transient growth of perturbations in Stokes oscillatory flows

Biau

2016

J. Fluid Mech.

View full text Add to dashboard Cite

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Oscillatory Stokes flows, with zero mean, are subjected to subcritical transition to turbulence. The maximal energy growth of perturbations is computed in the subcritical regime through an optimisation method. The results show strong amplifications during half a period. The energy transfer from the base flow involves an Orr mechanism with two-dimensional vorticity waves, and the maximum energy scales exponentially with the Reynolds number. Nonlinear simulations show that low-energy perturbations are sufficient to trigger turbulent flow.

show abstract

Section: Methodsmentioning

confidence: 99%

Transient growth of perturbations in Stokes oscillatory flows

Biau

2016

J. Fluid Mech.

View full text Add to dashboard Cite

show abstract

“…As a spatial discretization we use a Fourier collocation method with 128 points [14] and an IMEX scheme [1] as a time integrator, treating the linear terms implicitly and the nonlinear term explicitly. The parameters in the parareal method are taken as N c = 100, N it = 5, ∆t = 10 −2 , δt = 10 −4 .…”

Section: The Original Parareal Methodsmentioning

confidence: 99%

“…We use a first order local discontinuous Galerkin method (LDG) with 100 elements in space [26,15] and an IMEX scheme in time [1], with the linear terms treated implicitly and the nonlinear term explicitly. We set the tolerance for POD and EIM to be 10 −13 and 10 −8 respectively.…”

Section: Allan-cahn Equation: Nonlinear Sourcementioning

confidence: 99%

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

Chen

Hesthaven

Zhu

2014

Reduced Order Methods for Modeling and Computational Reduction

View full text Add to dashboard Cite

We propose a modified parallel-in-time -parareal -multi-level time integration method which, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built from the information obtained from the fine solver at each iteration. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method for similar problems and the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation methods (EIM), we develop this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate the advantages through algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.

show abstract

“…Both subcategories, multistage Runge-Kutta and linear multistep methods, can be fullyimplicit (the current time-level is obtained by solving a nonlinear problem that uses information from the current time-step) and fully-explicit (the current time-level is calculated using information coming from the previous time-steps only). Representative classes of time-integration schemes embedded in the GL method consist of implicit multistep methods such as Adams-Moulton (AM) [22] and backward differentiation (BDF) methods [13,20,21], implicit multistage Runge-Kutta schemes such as diagonally (DIRK) and singly-diagonally (SDIRK) implicit Runge-Kutta schemes [3,19,59], explicit multistep methods, such as leapfrog and Adams-Bashforth methods [28,43], explicit Runge-Kutta schemes, such as the fourthorder Runge-Kutta scheme [55] and partitioned methods, such as Implicit-Explicit (IMEX) schemes, whereby the operators are linearized in some fashion with-e.g., two Butcher tableaux, one explicit and one implicit [5,40,106]. While EBTI schemes are widely used in computational fluid dynamics, especially in the engineering sector [18,52], their adoption in the weather and climate communities has been less widespread, with SE schemes [54,88,107] and horizontally-explicit vertically-implicit schemes [8,40,63]-i.e., schemes where the horizontal direction is treated explicitly and the vertical is treated implicitly-becoming more prominent but still confined mainly to research and limited-area models (with very few exceptions-see Table 1).…”

Section: Eulerian-based Time-integration (Ebti)mentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

View full text Add to dashboard Cite

The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.

show abstract

Implicit-Explicit Methods for Time-Dependent Partial Differential Equations

Cited by 883 publications

References 19 publications

Transient growth of perturbations in Stokes oscillatory flows

Transient growth of perturbations in Stokes oscillatory flows

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Contact Info

Product

Resources

About