Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “multirate integration”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

A framework for multirate time integration of interface-coupled problems

The research described here was performed as part of the DOE SciDAC project Coupling Approaches for Next Generation Architectures (CANGA). A framework was developed for the derivation of novel algorithms for the multirate time integration of two-component systems coupled across an interface between spatial domains. The multirate aspect means that different time steps are allowed by each component integrator. The framework provides a way to construct multirate integrators with desirable properties related to stability, accuracy and preservation of system invariants. This report describes the framework and summarizes the major results, examples and research products.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Further, multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

42 ENGINEERING↗
DOE OSTI2022

Solving Coupled Surface and Subsurface Flow with Multirate Time Integration [Slides]

This report details an end of summer internship. The report lists the objective, "Add more multirate time integration methods to Amanzi" and concludes that, "Multirate methods can be used to speed up simulations and to get higher orders of accuracy", and "Coupled surface and subsurface simulations could benefit from using multirate schemes".

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Advancing simulations of coupled electron and phonon nonequilibrium dynamics using adaptive and multirate time integration

Electronic structure calculations in the time domain provide a deeper understanding of nonequilibrium dynamics in materials. The real-time Boltzmann equation (rt-BTE), used in conjunction with accurate interactions computed from first principles, has enabled reliable predictions of coupled electron and lattice dynamics. However, the timescales and system sizes accessible with this approach are still limited, with two main challenges being the different timescales of electron and phonon interactions and the cost of computing collision integrals. As a result, only a few examples of these calculations exist, mainly for two-dimensional (2D) materials. Here we leverage adaptive and multirate time integration methods to achieve a major step forward in solving the coupled rt-BTEs for electrons and phonons. Relative to conventional (non-adaptive) time-stepping, our approach achieves a 10x speedup for a target accuracy, or greater accuracy by 3–6 orders of magnitude for the same computational cost, enabling efficient calculations in both 2D and bulk materials. This efficiency is showcased by computing the coupled electron and lattice dynamics in graphene up to ~100 ps, as well as modeling ultrafast lattice dynamics and thermal diffuse scattering maps in bulk materials (silicon and gallium arsenide). In addition to improved efficiency, our adaptive method can resolve the characteristic rates of different physical processes, thus naturally bridging different timescales. This enables simulations of longer timescales and provides a framework for modeling multiscale dynamics of coupled degrees of freedom in matter. Our work opens new opportunities for quantitative studies of nonequilibrium physics in materials, including driven lattice dynamics with phonons coupled to electrons, spin, and other degrees of freedom.

Yao, Jia [California Institute of Technology (CalT↗
DOE OSTI2023

An assessment of the utility of multirate time integration for the modeling of laser powder bed fusion

Finite element simulation of the laser powder bed fusion process is made challenging by the inherently multiscale nature of the process. When using the typical global time stepping techniques, slowly-evolving regions of the domain receive the same numerical treatment as the regions with the highest temperature rates. The current work details the implementation and evaluation of an implicit multirate method which is able to advance different regions of the domain with distinct time steps, depending on their current solution rate. Previous work indicates that this representation of the temporal scales of the problem can lead to significant reductions in wall clock run time, and it is shown herein that multirate time integration, when used with a uniform process-scale mesh, can result in speedups of approximately 19, 42, and 87 times, for domains with edge lengths of approximately 1 mm, 2 mm, and 5 mm, respectively. When used in conjunction with $\ h$-refinement (limited to two levels of refinement), the resulting speedups (taken relative to the uniform mesh) are around 28, 85, and 665 times. The method is demonstrated to converge as indicated by the literature, and its use with an AM-Bench domain is demonstrated.

42 ENGINEERING↗
DOE OSTI2024

Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks

Large-scale multiphysics simulations are computationally challenging due to the coupling of multiple processes with widely disparate time scales. The advent of exascale computing systems exacerbates these challenges since these systems enable ever-increasing size and complexity. In recent years, there has been renewed interest in developing multirate methods as a means to handle the large range of time scales, as these methods may afford greater accuracy and efficiency than more traditional approaches of using implicit-explicit (IMEX) and low-order operator splitting schemes. However, to date there have been few performance studies that compare different classes of multirate integrators on complex application problems. In this work, we study the performance of several newly developed multirate infinitesimal (MRI) methods, implemented in the SUNDIALS solver package, on two reacting flow model problems built on structured mesh frameworks. The first model revisits prior work on a compressible reacting flow problem with complex chemistry that is implemented using BoxLib but where we now include comparisons between a new explicit MRI scheme with the multirate spectral deferred correction (SDC) methods in the original paper. The second problem uses the same complex chemistry as the first problem, combined with a simplified flow model, but runs at a large spatial scale where explicit methods become infeasible due to stability constraints. Two recently developed IMEX MRI multirate methods are tested. These methods rely on advanced features of the AMReX framework on which the model is built, such as multilevel grids and multilevel preconditioners. The results from these two problems show that MRI multirate methods can offer significant performance benefits on complex multiphysics application problems and that these methods may be combined with advanced spatial discretization to compound the advantages of both.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multirate linearly-implicit GARK schemes

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.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Implicit-Explicit Multirate Infinitesimal GARK Methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called infinitesimal formulation for the fast time scale through definition of a sequence of modified “fast" initial-value problems that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high-order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix φ functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called multirate ExpRB (MERB) methods. They consist of the solution to a sequence of modified “fast” initial-value problems, which may themselves be approximated through subcycling any desired initial-value problem solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders 2 through 6 (the highest-order infinitesimal multirate methods to date). Lastly, we then present numerical simulations to confirm these theoretical convergence rates and to compare the efficiency of MERB methods against other recently introduced high-order multirate methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Efficient and flexible multirate temporal adaptivity

In this work we present two new families of multirate time step adaptivity controllers, that are designed to work with embedded multirate infinitesimal (MRI) time integration methods for adapting time steps when solving problems with multiple time scales. We compare these controllers against competing approaches on two benchmark problems, showing that the proposed methods offer dramatically improved performance and flexibility. The combination of embedded MRI methods and the proposed controllers enable adaptive simulations of problems with a potentially arbitrary number of time scales, achieving high accuracy while maintaining low computational cost. Additionally, we introduce a new set of embeddings for the family of explicit multirate exponential Runge–Kutta (MERK) methods of orders 2 through 5, resulting in the first-ever fifth-order embedded MRI method. Finally, we compare the performance of a wide range of embedded MRI methods on our benchmark problems to provide guidance on how to select an appropriate MRI method and multirate controller.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Multirate partitioned Runge–Kutta methods for coupled Navier–Stokes equations

Earth system models are complex integrated models of atmosphere, ocean, sea ice, and land surface. Coupling the components can be a significant challenge due to the difference in physics, temporal, and spatial scales. Further, this study explores multirate partitioned Runge-Kutta methods for the fluid-fluid interaction problem and demonstrates its parallel performance by using the PETSc library. We consider compressible Navier-Stokes equations with gravity coupled through a rigid-lid interface. Our large-scale numerical experiments reveal that multirate partitioned Runge-Kutta coupling schemes (1) can conserve total mass; (2) have second-order accuracy in time; and (3) provide favorable strong- and weak-scaling performance on modern computing architectures. We also show that the speedup factors of multirate partitioned Runge-Kutta methods match theoretical expectations over their base (single-rate) method.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Adaptive Time Step Control for Multirate Infinitesimal Methods

Multirate methods have been used for decades to temporally evolve initial-value problems in which different components evolve on distinct time scales, and thus use of different step sizes for these components can result in increased computational efficiency. Generally, such methods select these different step sizes based on experimentation or stability considerations. For problems that evolve on a single time scale, adaptivity approaches that strive to control local temporal error are widely used to achieve numerical results of a desired accuracy with minimal computational effort, while alleviating the need for manual experimentation with different time step sizes. However, there is a notable gap in the publication record on the development of adaptive time step controllers for multirate methods. In this paper, we extend the single-rate controller work of Gustafsson [ACM Trans. Math. Software, 20 (1994), pp. 496-517] to the multirate method setting. Specifically, we develop controllers based on polynomial approximations to the principal error functions for both the "fast" and "slow" time scales within multirate infinitesimal (MRI) methods. We additionally investigate a variety of approaches for estimating the errors arising from each time scale within MRI methods. We then numerically evaluate the proposed multirate controllers and error estimation strategies on a range of multirate test problems, comparing their performance against an estimated optimal performance. Through this work, we combine the most performant of these approaches to arrive at a set of multirate adaptive time step controllers that robustly achieve desired solution accuracy with minimal computational effort.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
DOE OSTI2022

A Fast Time-Stepping Strategy for Dynamical Systems Equipped with a Surrogate Model

Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge--Kutta framework. The inexpensive surrogate model is integrated with a small time step to guide the solution trajectory, and the full model is treated with a large time step to occasionally correct for the surrogate model error and ensure convergence. Here, we provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.

Surrogate models↗
DOE OSTI2026

“Frameworks, Algorithms and Scalable Technologies for Mathematics (FASTMath) SciDAC Institute” (Final Technical Report)

SMU personnel formed a portion of the overall “Time Integration” team within the FASTMath SciDAC-5 Institute, and we interacted very closely with team members from collaborating institutions. The major goals of our team within the FASTMath institute may be categorized into two groups: the development of advanced and application-aware time integration methods and software, and close interactions with DOE application scientists to facilitate their use of these new methods and software. The two goals are intimately linked, since our research and development of novel tools is informed by the needs of our application partners, who in turn benefit from subsequent mathematical and software advances. While the SMU personnel collaborated with the rest of the Time Integration FASTMath team on most of our shared deliverables within the larger FASTMath institute, SMU personnel primarily contributed to the following subset of those goals: ● expanding capabilities for higher-order and solve-decoupled multirate methods, ● enhancing support for temporal adaptivity within multirate methods, and ● adding structure-aware time integration methods. Each of these goals focused both on intellectual contributions through journal articles or research presentations, and on enhancements to the open-source SUNDIALS library of time integrators and nonlinear solvers, of which the SMU PI Reynolds is a core developer.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

ARKODE: A Flexible IVP Solver Infrastructure for One-step Methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge–Kutta methods, ARKODE supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge–Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

New Time Integrators and Capabilities in SUNDIALS Versions 6.2.0-7.4.0

SUNDIALS is a well-established numerical library that provides robust and efficient time integrators and nonlinear solvers. This article overviews several significant improvements and new features added over the last 3 years to support scientific simulations run on high-performance computing systems. Notably, three new classes of one-step methods have been implemented: low storage Runge–Kutta, symplectic partitioned Runge–Kutta, and operator splitting. In addition, we describe new timestep adaptivity support for multirate methods, adjoint sensitivity analysis capabilities for explicit Runge–Kutta methods, additional options for Anderson acceleration in nonlinear solvers, and improved error handling and logging.

Computer science↗