Search NASA⌕ Search

SEARCH · Search NASA

Results for “Time 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 OSTI2022

Comparison of exponential integrators and traditional time integration schemes for the shallow water equations

We report the time integration scheme is probably one of the most fundamental choices in the development of an ocean model. In this paper, we investigate several time integration schemes when applied to the shallow water equations. This set of equations is accurate enough for the modeling of a shallow ocean and is also relevant to study as it is the one solved for the barotropic (i.e. vertically averaged) component of a three dimensional ocean model. We analyze different time stepping algorithms for the linearized shallow water equations. High order explicit schemes are accurate but the time step is constrained by the Courant-Friedrichs-Lewy stability condition. Implicit schemes can be unconditionally stable but, in practice lack accuracy when used with large time steps. In this paper we propose a detailed comparison of such classical schemes with exponential integrators. The accuracy and the computational costs are analyzed in different configurations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Optimizing time integration for accurate recovery of shockwave interface location in radiography

We present simulations and experiments of time integrated radiographic imaging of a moving 1D shock wave front and a quantitative method for determining the statistical error in locating the shock front as a function of integration time and noise in the radiograph. We discuss the trade-off between increasing motion blur, which leads to decreased shock front location certainty, and increasing signal-to-noise, which leads to improved image quality with increasing integration time. We find an optimum integration time between a short integration time, where noise limits the error, and a long integration time, where motion blurring limits the error. This methodology can be used to tune experimental configurations to obtain the highest quality radiograph for a given experimental configuration.

Bremsstrahlung↗
DOE OSTI2021

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional step, alternating direction implicit, operator splitting, and locally one dimensional methods partition the system of interest into components, and solve individual components implicitly in a cost-effective way. Here this work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge–Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions for this class, and explain their application to partitioned systems of ordinary differential equations. We show that classical splitting methods belong to the IMIM GARK family, and therefore can be studied in this unified framework. New IMIM-GARK splitting methods are developed and tested using parabolic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Interface flux recovery framework for constructing partitioned heterogeneous time-integration methods

A common approach for the development of partitioned schemes employing different time integrators on different subdomains is to lag the coupling terms in time. This can lead to accuracy issues, especially in multistage methods. Here, in this article, we present a novel framework for partitioned heterogeneous time-integration methods, which allows the coupling of arbitrary multistage and multistep methods without reducing their order of accuracy. At the core of our approach are accurate estimates of the interface flux obtained from the Schur complement of an auxiliary monolithic system. We use these estimates to construct a polynomial-in-time approximation of the interface flux over the current time coupling window. This approximation provides the interface boundary conditions necessary to decouple the subdomain problems at any point within the coupling window. In so doing our framework enables a flexible choice of time-integrators for the individual subproblems without compromising the time-accuracy at the coupled problem level. This feature is the main distinction between our framework and other approaches. To demonstrate the framework, we construct a family of partitioned heterogeneous time-integration methods, combining multistage and multistep methods, for a simplified tracer transport component of the coupled air-sea system in Earth system models. We report numerical tests evaluating accuracy and flux conservation for different pairs of time-integrators from the explicit Runge-Kutta and Adams-Moulton families.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Conservative High-Order Time Integration for Lagrangian Hydrodynamics

In tis study, we develop novel time integration methods for the compressible Euler equations in the Lagrangian frame that are of arbitrary high order and exactly preserve the mass, momentum, and total energy of the system. The equations are considered in nonconservative form, that is, common for staggered grid hydrodynamics (SGH) methods; namely, the evolved quantities are mass, momentum, and internal energy. A general family of time integration schemes is formulated, and practical pairs for orders three and four are derived. Numerical results on standard hydrodynamics benchmarks confirm the high-order convergence on smooth problems and the exact numerical preservation of all physically conserved quantities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

An Adaptive Timestepping Algorithm for Particle Time Integration in Coupled CFD-DEM Simulations

Coupled Computational Fluid Dynamics(CFD) - Discrete Element Method (DEM) models provide an accurate description of multiphase physical systems where a granular phase exists in an underlying continuous medium. The time integration of the granular phase in these simulations are typically handled using an explicit time integration scheme with a global time step that is much smaller than fluid time scales, in order to resolve inter-particle collisions. An adapative time integration technique that avoids this global constraint for the granular phase is presented in this work. The algorithm involves the use of a local time stepping approach to resolve collisional time scales for only a subset of particles that are in close proximity to potential collision partners, thereby resulting in substantial reduction of computational cost. This approach is observed to be 2-3X faster than traditional explicit methods for problems that involve both dense and dilute regions, while maintaining the same level of accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Jacobian-free Newton–Krylov method for the simulation of non-thermal plasma discharges with high-order time integration and physics-based preconditioning

A preconditioning framework for the numerical simulation of non-thermal streamer discharges is developed using the Jacobian-free Newton-Krylov (JFNK) method. A reduced plasma fluid model is considered, consisting of electrons, one positive ion, one negative ion, and the electrostatic potential. Here, the plasma kinetics model includes ionization, electron-ion recombination, electron attachment, electron detachment, and ion-ion recombination. The governing equations are made dimensionless, discretized in space with finite differences, and integrated in time with a fully implicit method based on high-order backward differentiation formulas. The preconditioning framework is based on a linearized form of the governing equations and physics-based operator splitting. The efficiency of the preconditioning strategy is assessed through two test cases: streamer propagation between parallel plates and an axisymmetric pin-to-pin discharge. The fully implicit approach overcomes traditional restrictions in the time step size due to processes such as electron drift, electron diffusion, and dielectric relaxation. Excellent performance is observed through relevant statistics of the JFNK solver, although the number of linear iterations increases for the pin-to-pin discharge when nonlinear numerical boundary conditions are imposed at the electrodes. Performance studies show scalability with O(100-1000) processors for O(10M) unknowns with ample room for optimization.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2019

Multi-rate time integration on overset meshes

Overset meshes are an effective tool for the computational fluid dynamic simulation of problems with complex geometries or multiscale spatio-temporal features. When the maximum allowable timestep on one or more meshes is significantly smaller than on the remaining meshes, standard explicit time integrators impose inefficiencies for time-accurate calculations by requiring that all meshes advance with the smallest timestep. With the targeted use of multi-rate time integrators, separate meshes can be time-marched at independent rates to avoid wasteful computation while maintaining accuracy and stability. This work applies time-explicit multi-rate integrators to the simulation of the compressible Navier-Stokes equations discretized on overset meshes using summation-by-parts (SBP) operators and simultaneous approximation term (SAT) boundary conditions. We introduce a class of multi-rate Adams-Bashforth (MRAB) schemes that offer significant stability improvements and computational efficiencies for SBP-SAT methods. We present numerical results that confirm the efficacy of MRAB integrators, outline a number of implementation challenges, and demonstrate a reduction in computational cost enabled by MRAB. We also look at the use of our method in the setting of a large-scale distributed-memory parallel implementation where we discuss concerns involving load balancing and communication efficiency.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

The Crank Nicolson Time Integrator for EMPHASIS

We investigate the use of implicit time integrators for finite element time domain approximations of Maxwell's equations in vacuum. We discretize Maxwell's equations in time using Crank-Nicolson and in 3D space using compatible finite elements. We solve the system by taking a single step of Newton's method and inverting the Eddy-Current Schur complement allowing for the use of standard preconditioning techniques. This approach also generalizes to more complex material models that can include the Unsplit PML. We present verification results and demonstrate performance at CFL numbers up to 1000.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Parallel-in-Time Integration for Nonlinear Hyperbolic Problems (Final Report)

The work for the subcontract was situated in the area of parallel-in-time integration for hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to diffusion-dominated PDEs, with some recent success in scalar linear hyperbolic PDEs. Given the ubiquity of numerical simulations of hyperbolic PDEs throughout the sciences, in particular, nonlinear hyperbolic systems, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems beyond simple scalar and linear cases, which is the main topic of this subcontract. The main focus of the work was to further develop and perfect coarse-grid operators for the Multigrid Reduction-inTime (MGRIT) method applied to hyperbolic PDEs that were recently proposed in PhD thesis, based on a modified semi-Lagrangian approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Parallel-In-Time Integration for Hyperbolic Problems (Final Report)

The work for the subcontract was situated in the area of parallel-in-time integration for advection-dominated and hyperbolic partial differential equations (PDEs). Parallel-in-time integration is an active area of research due to its ability to enable faster numerical simulations for applications throughout many areas of science. Over the past two decades, much progress has been made in this area; however, this progress has largely been limited to non-hyperbolic, diffusion-dominated PDEs. Given the ubiquity of numerical simulations of advection-dominated hyperbolic PDEs throughout the sciences, there is a strong need to develop efficient parallel-in-time techniques for hyperbolic problems, which was the main topic of this subcontract.

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 OSTI2018

An exponential time-integrator scheme for steady and unsteady inviscid flows

In this report an exponential time-integrator scheme of second-order accuracy based on the predictor–corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge–Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A family of second-order fully explicit time integration schemes

A new family of fully explicit time integration methods is proposed, which has second-order accuracy for structures with and without damping. Using a diagonal mass matrix, the suggested scheme remains fully explicit not only for structures with a non-diagonal damping matrix, but also when the internal force vector is a nonlinear function of velocity. The present algorithm has an acceptable domain of stability, and it is self-starting. This technique introduces effectively numerical dissipation to suppress the high-frequency spurious modes, while at the same time the lower modes are not affected too much. In addition, numerical dispersion error of the scheme is considerably smaller than that of the central difference method. Solving several linear and nonlinear problems highlights the superior performance of the authors’ approach. Findings demonstrate that solution time for the suggested scheme is much less than that of the central difference technique. The related algorithm can be easily implemented into programs, which already contain the central difference method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Efficient Integration of Algebraic Constraint for Exponential Time Integration

This work is a continuation of a previous project where the high efficiency of exponential integration methods for DRMHD systems was demonstrated. Often algebraic constraints must also be enforced for DRMHD systems of interest. Straightforward application of exponential methods to DRMHD equations with constraints leads to prohibitively computationally expensive methods. In this work, we propose new exponential schemes that allow the constraints to be removed from evaluation of exponential matrix functions which drastically reduces computational complexity by eliminating the need to perform computations enforcing constraints in exponential calculations while still preserving a high order of accuracy and allowing for a large time step even when the problem is stiff. This idea is similar to the W-methods and is achieved by carefully designing a method with the desired order of accuracy, even with an incomplete Jacobian matrix used as an argument of exponential-like functions. The constraints are accounted for by including them in the evaluation of the right-hand-side forcing function of the spatially discretized system. We study performance of the new methods on test problems and outline future research directions that this work opens.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
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 OSTI2019

Multirate Time Integration for ODEs and DAEs (Final Report)

This project investigated multirate time integration methods. These methods evolve a differential equation system in time by advancing parts of the problem with fast time scales at small time steps and parts of the problem with slow time scales with larger time steps. The project developed new methods and theory as well as a set of prototype software implementations. Multirate methods were tested on combustion and climate applications. A series of five presentations were given on project results and two papers are planned.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields

Here we present a new implicit asymptotic preserving time integration scheme for charged-particle orbit computation in arbitrary electromagnetic fields. The scheme is built on the Crank-Nicolson integrator and continues to recover full-orbit motion in the small time-step limit, but also recovers all the first-order guiding center drifts as well as the correct gyroradius when stepping over the gyration time-scale. In contrast to previous efforts in this direction, the new scheme also features exact energy conservation. In the derivation of the scheme, we find that a new numerical time-scale is introduced. This scale is analyzed and the resulting restrictions on time-step are derived. Based on this analysis, we develop an adaptive time-stepping strategy the respects these constraints while stepping over the gyration scale when physically justified. It is shown through numerical tests on single-particle motion that the scheme's energy conservation property results in tremendous improvements in accuracy, and that the scheme is able to transition smoothly between magnetized and unmagnetized regimes as a result of the adaptive time-stepping.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗