A unified computational framework for developing general multi-time-step (MTS) coupling schemes for time-dependent problems

An explicit multi-time step integration method for second order equations

Simulating the dynamics of structural systems containing both stiff and flexible parts with a time integration scheme that uses a uniform time-step for the entire system is challenging because of the presence of multiple spatial and temporal scales in the response. We present, for the first time, a multi-time-step (MTS) coupling method for composite time integration schemes that is well suited for such stiff-flexible systems. Using this method, the problem domain is divided into smaller subdomains that are integrated using different time-step sizes and/or different composite time integration schemes to achieve high accuracy at a low computational cost. In contrast to conventional MTS methods for single-step (SS) schemes, a key challenge with coupling composite schemes is that multiple constraint conditions are needed to enforce continuity of the solution across subdomains. We develop the constraints necessary for achieving unconditionally stable coupling of the composite ρ<sub>∞</sub>-Bathe schemes and prove this property analytically. Further, we conduct a local truncation error (LTE) analysis and study the period elongation and amplitude decay characteristics of the proposed method. Lastly, we demonstrate the performance of the method for linear and nonlinear stiff-flexible systems to show that the proposed MTS method can achieve higher accuracy than existing methods for time integration, for the same computational cost.

Stability of multitime step integration methods for finite-element computations in structural dynamics is analyzed. Multitime step procedures based on the Newmark family of methods are described. The basic idea of multitime step methods is to utilize various time step sizes in different domains of an element mesh. Interpolated nodal values from the large time step domain are used in the computation of displacements, velocities, and accelerations in the small time step domain. An analytical study of the errors introduced by the interpolation is given. The analysis shows instability because of the resonance phenomena and because of the propagation of spurious high frequencies caused by unbalanced forces originating from interpolation errors. Numerical results exhibiting instability problems are presented. To complete the study, examples of solutions stabilized with numerical damping of high frequencies are included in the presentation. The conclusion from the present study is that multitime step methods are unstable in their nature. Numerical damping can stabilize the computations but alters the response of free vibrating undamped systems.

Two FETI-based heterogeneous time step coupling methods for Newmark and [formula omitted]-schemes derived from the energy method

Hybrid Asynchronous Perfectly Matched Layer for seismic wave propagation in unbounded domains

Computing a flow system a number of times with different samples of flow parameters is a common practice in many uncertainty quantification applications, which can be prohibitively expensive for complex nonlinear flow problems. This report presents two second order, stabilized, scalar auxiliary variable (SAV) ensemble algorithms for fast computation of the Navier--Stokes flow ensembles: Stab-SAV-CN and Stab-SAV-BDF2. The proposed ensemble algorithms are based on the ensemble timestepping idea which makes use of a quantity called the ensemble mean to construct a common coefficient matrix for all realizations at the same time step after spatial discretization, in which case efficient block solvers, e.g., block GMRES, can be used to significantly reduce both storage and computational time. The adoption of a recently developed SAV approach that treats the nonlinear term explicitly results in a constant shared coefficient matrix among all realizations at different time steps, which further cuts down the computational cost, yielding an extremely efficient ensemble algorithm for simulating nonlinear flow ensembles with provable long time stability without any time step conditions. The SAV approach for the Navier--Stokes equations for a single realization was proved to be unconditionally stable in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491]. However we found the SAV approach has very low accuracy that compromises its stability in our initial numerical investigations for several commonly tested benchmark flow problems. In this report, we propose to use the stabilization $-\alpha h \Delta (u^{n+1}-u^n)$ in Stab-SAV-CN and $-\alpha h \Delta (3u^{n+1}-4 u^{n}+u^{n-1})$ in Stab-SAV-BDF2 to address this issue. We prove that both of our ensemble algorithms are long time stable under one parameter fluctuation condition, without any time step constraints. For a single realization, both algorithms are unconditionally stable and have better accuracy than the SAV methods studied in [L. Lin, Z. Yang, and S. Dong, J. Comput. Phys., 388 (2019), pp 1--22; X. Li and J. Shen, SIAM J. Numer. Anal., 58 (2020), pp. 2465--2491] for our test problems. Extensive numerical experiments are performed to show the efficiency of the proposed ensemble algorithms and the effectiveness of the stabilization for increasing accuracy and stability.

ABSTRACTImplicit‐explicit (IMEX) time integration schemes are well suited for non‐linear structural dynamics because of their low computational cost and high accuracy. However, the stability of IMEX schemes cannot be guaranteed for general non‐linear problems. In this article, we present a scalar auxiliary variable (SAV) stabilization of high‐order IMEX time integration schemes that leads to unconditional stability. The proposed IMEX‐BDFk‐SAV schemes treat linear terms implicitly using kth‐order backward difference formulas (BDFk) and non‐linear terms explicitly. This eliminates the need for iterations in non‐linear problems and leads to low computational costs. Truncation error analysis of the proposed IMEX‐BDFk‐SAV schemes confirms that up to kth‐order accuracy can be achieved and this is verified through a series of convergence tests. Unlike existing SAV schemes for first‐order ordinary differential equations (ODEs), we introduce a novel SAV for the proposed schemes that allows direct solution of the second‐order ODEs without transforming them into a system of first‐order ODEs. Finally, we demonstrate the performance of the proposed schemes by solving several non‐linear problems in structural dynamics and show that the proposed schemes can achieve high accuracy at a low computational cost while maintaining unconditional stability.

Implicit/explicit multi-time step co-computations for predicting reinforced concrete structure response under earthquake loading

Time adaptive conservative finite volume method

1L1530 高速多重極子展開法(FMM)とマルチ時間ステップ法(MTS)を用いた損傷DNAの分子動力学

Unsteady flow computations typically rely on time integration schemes which employ the same step size everywhere. However, in cases with strong variations of wave speed and/or spatial resolution, local stability criteria and truncation errors would allow for much larger time steps in parts of the domain. To exploit this potential of improving the computational efficiency, adaptive time-stepping methods have been developed. Recently, an adaptive conservative time integration (ACTI) scheme for explicit finite volume methods was devised. Opposed to previous methods, ACTI guarantees exact conservation and periodic synchronization. Both are achieved by splitting a global time step into 2L (L≥0 is a cell dependent level) sub-steps, whereas the order of cell updates is critical. It has been demonstrated for flows in fractured porous media and for the Euler equations that ACTI can reduce the computational cost dramatically while preserving second order accuracy in space and time. In this paper, the ACTI scheme is extended to the compressible Navier-Stokes-Fourier system, and special attention is required for the spatio-temporal discretization of the convective and viscous fluxes. Numerical studies of unsteady flows involving shock boundary layer interaction demonstrate that the same accuracy is achieved with the new compressible ACTI flow solver as with classical time integration, but at much lower cost. Moreover, since the method is explicit, it has the potential for efficient computations on multiple CPUs and GPUs.

Scalar Auxiliary Variable/Lagrange multiplier based pseudospectral schemes for the dynamics of nonlinear Schrödinger/Gross-Pitaevskii equations

Adaptive conservative time integration for transport in fractured porous media

Consistency-enhanced SAV BDF2 time-marching method with relaxation for the incompressible Cahn–Hilliard–Navier–Stokes binary fluid model

We propose an implementation of a general purpose multi-spatial method, multi-time scheme subdomain Differential Algebraic Equations (DAE) framework allowing a mix of different space discretization methods while interfacing altogether different time integration algorithms on a single body analysis for the linear first order transient systems. With the concept of subdomains and DAE framework which allows constraint in time as well as space, we can divide a body into multiple subdomains and implement different spatial methods (finite elements, particles, etc.) in different regions of the body including altogether different time integrators and enable targeting an area with a specific method which can fully utilize its best features. The robust Generalized Single Step Single Solve (GS4) family of Linear Multistep (LMS) framework with second order time accuracy encompasses most of the developments over the last 50 years or so including new and optimal designs, and provides a wide variety of choices available to the analyst in a single analysis setting. We also advance, analyze and present the GS4 parameter studies regarding parameter selection. The GS4 readily allows flexibility in matching different time integration schemes, both existing traditional time schemes and other new and optimal competitive designs while maintaining second order time accuracy. Such implementation of coupling a wide variety of different spatial and time integration algorithms is not possible (especially implicit-implicit algorithm couplings) in the current state of the technology. Existing traditional practices of coupling multiple spatial and time algorithms show limitations in reduced order of accuracy, and consistency and the like in various field variables typical of, and including, temperature, temperature rate, and Lagrange multipliers. Various combinations of spatial methods and time algorithms between subdomains are tested with simple linear first order system problems for readers to readily mimic the results and demonstrate the overall robustness and efficacy for numerical computations.