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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

High order explicit local time stepping methods for hyperbolic conservation laws

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RK-DG) methods and design LTS algorithms based on the strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integration in different regions of the computational domain. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of the interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms.

58 GEOSCIENCES↗
DOE OSTI2021

Local time stepping for the shallow water equations in MPAS

In this work we assess the performance of a set of local time-stepping (LTS) schemes for the shallow water equations implemented in the Model for Prediction Across Scales (MPAS). The goal of LTS is to speed up the simulation by allowing different time-steps on different regions of the computational grid. The LTS schemes considered here were originally introduced by Hoang et al. (2019) [26], who laid out the mathematical foundation of the methods. Here, the authors take on the task of presenting a fast, efficient and scalable parallel implementation of these LTS methods on high performance computing machines, with the aim to provide a recipe for other climate modeling groups that may be interested in employing LTS algorithms in their codes. As a matter of fact, even if MPAS is our framework of choice, our approach is general enough and could be of interest to other groups beyond the MPAS community. Due to their nature, LTS methods possess an inherent load imbalance that needs to be carefully addressed in order to obtain efficient scalability. Even more important is the far from trivial task of computing the right-hand side terms only on specific LTS regions during the time-stepping procedure. An inefficient handling of this task causes a drastic decay of the CPU time performance, making the LTS algorithms practically of no use. The emphasis of the present work is therefore on the computational and parallel aspects of the LTS methods, whose proper treatment is crucial to make the methods run faster against existing strategies, such as for instance high-order explicit global time-stepping schemes. This is in fact the ultimate goal of using an LTS procedure and it is the one to which we direct all our optimization efforts.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2013

Time-Accurate Local Time Stepping and High-Order Time CESE Methods for Multi-Dimensional Flows Using Unstructured Meshes

With the wide availability of affordable multiple-core parallel supercomputers, next generation numerical simulations of flow physics are being focused on unsteady computations for problems involving multiple time scales and multiple physics. These simulations require higher solution accuracy than most algorithms and computational fluid dynamics codes currently available. This paper focuses on the developmental effort for high-fidelity multi-dimensional, unstructured-mesh flow solvers using the space-time conservation element, solution element (CESE) framework. Two approaches have been investigated in this research in order to provide high-accuracy, cross-cutting numerical simulations for a variety of flow regimes: 1) time-accurate local time stepping and 2) highorder CESE method. The first approach utilizes consistent numerical formulations in the space-time flux integration to preserve temporal conservation across the cells with different marching time steps. Such approach relieves the stringent time step constraint associated with the smallest time step in the computational domain while preserving temporal accuracy for all the cells. For flows involving multiple scales, both numerical accuracy and efficiency can be significantly enhanced. The second approach extends the current CESE solver to higher-order accuracy. Unlike other existing explicit high-order methods for unstructured meshes, the CESE framework maintains a CFL condition of one for arbitrarily high-order formulations while retaining the same compact stencil as its second-order counterpart. For large-scale unsteady computations, this feature substantially enhances numerical efficiency. Numerical formulations and validations using benchmark problems are discussed in this paper along with realistic examples.

Chang, Chau-Lyan↗
NASA NTRS1995

A far-field non-reflecting boundary condition for two-dimensional wake flows

Far-field boundary conditions for external flow problems have been developed based upon long-wave perturbations of linearized flow equations about a steady state far field solution. The boundary improves convergence to steady state in single-grid temporal integration schemes using both regular-time-stepping and local-time-stepping. The far-field boundary may be near the trailing edge of the body which significantly reduces the number of grid points, and therefore the computational time, in the numerical calculation. In addition the solution produced is smoother in the far-field than when using extrapolation conditions. The boundary condition maintains the convergence rate to steady state in schemes utilizing multigrid acceleration.

Danowitz, Jeffrey S.↗
DOE OSTI2023

Storm Surge Modeling as an Application of Local Time-Stepping in MPAS-Ocean

This paper presents the first practical application of local time-stepping (LTS) schemes in the Model for Prediction Across Scales-Ocean (MPAS-O). We use LTS schemes in a single-layer, global ocean model that predicts the storm surge around the eastern coast of the United States during Hurricane Sandy. The variable-resolution meshes used are of unprecedentedly high resolution in MPAS-O, containing cells as small as 125 m wide in Delaware Bay. It is shown that a particular, third-order LTS scheme (LTS3) produces sea-surface height solutions that are of comparable quality to solutions produced by the classical four-stage, fourth-order Runge-Kutta method (RK4) with a uniform time step on the same meshes. Furthermore, LTS3 is up to 35% faster in the best cases considered, where the number of cells using the coarse time-step relative to those using the fine time-step is as low as 1:1. This shows that LTS schemes are viable for use in MPAS-O with the added benefit of substantially less computational cost. The results of these performance experiments inform us of the requirements for efficient mesh design and configuration of LTS regions for LTS schemes. In particular, we see that for LTS to be efficient on a given mesh, it is important to have enough cells using the coarse time-step relative to those using the fine time-step, typically at least 1:5 to see an increase in performance.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

LANL Institutional Computing Highlight: Local Time-Stepping in Global to Regional Ocean Modeling (Annual Report) [Slides]

Multi-resolution global ocean simulations are possible with the Model for Prediction Across Scales-Ocean (MPAS-Ocean), although the model is currently constrained to a uniform time step. The time step is determined by the size of the smallest cell in the grid, according to the CFL condition. To overcome this issue, a local time stepping (LTS) scheme has been designed for MPAS-Ocean. We have made progress on implementing the LTS scheme, as well as quantitative results and performance tests.

58 GEOSCIENCES↗
NASA NTRS1989

An efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

Radespiel, R.↗
NASA NTRS1989

A cell-vertex multigrid method for the Navier-Stokes equations

A cell-vertex scheme for the Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing, a multigrid method, and carefully controlled artificial dissipative terms, very good convergence rates are obtained for a wide range of two- and three-dimensional flows over airfoils and wings. The accuracy of the code is examined by grid refinement studies and comparison with experimental data. For an accurate prediction of turbulent flows with strong separations, a modified version of the nonequilibrium turbulence model of Johnson and King is introduced, which is well suited for an implementation into three-dimensional Navier-Stokes codes. It is shown that the solutions for three-dimensional flows with strong separations can be dramatically improved, when a nonequilibrium model of turbulence is used.

Radespiel, R.↗
NASA NTRS1991

A multiple-block multigrid method for the solution of the three-dimensional Euler and Navier-Stokes equations

A multiple block multigrid method for the solution of the three dimensional Euler and Navier-Stokes equations is presented. The basic flow solver is a cell vertex method which employs central difference spatial approximations and Runge-Kutta time stepping. The use of local time stepping, implicit residual smoothing, multigrid techniques and variable coefficient numerical dissipation results in an efficient and robust scheme is discussed. The multiblock strategy places the block loop within the Runge-Kutta Loop such that accuracy and convergence are not affected by block boundaries. This has been verified by comparing the results of one and two block calculations in which the two block grid is generated by splitting the one block grid. Results are presented for both Euler and Navier-Stokes computations of wing/fuselage combinations.

Atkins, Harold↗
NASA NTRS1991

A multi-block multigrid method for the solution of the Euler and Navier-Stokes equations for three-dimensional flows

A multi-block multigrid method for the solution of the three-dimensional Euler and Navier-Stokes equations is presented. The basic flow solver is a cell-vertex method which employs central-difference spatial approximations and Runge-Kutta time stepping. The use of local time stepping, implicit residual smoothing, multigrid techniques, and variable-coefficient numerical smoothing results in an efficient and robust scheme. The multi-block strategy places the block loop within the Runge-Kutta loop such that accuracy and convergence are not affected by block boundaries. This has been verified by comparing the results of one- and two-block calculations in which the two-block grid is generated by splitting the one-block grid. Results are presented for both Euler and Navier-Stokes computations of wings and wing-fuselage combinations.

Atkins, H. L.↗
NASA NTRS2010

Adaptive Numerical Algorithms in Space Weather Modeling

Space weather describes the various processes in the Sun-Earth system that present danger to human health and technology. The goal of space weather forecasting is to provide an opportunity to mitigate these negative effects. Physics-based space weather modeling is characterized by disparate temporal and spatial scales as well as by different physics in different domains. A multi-physics system can be modeled by a software framework comprising of several components. Each component corresponds to a physics domain, and each component is represented by one or more numerical models. The publicly available Space Weather Modeling Framework (SWMF) can execute and couple together several components distributed over a parallel machine in a flexible and efficient manner. The framework also allows resolving disparate spatial and temporal scales with independent spatial and temporal discretizations in the various models. Several of the computationally most expensive domains of the framework are modeled by the Block-Adaptive Tree Solar wind Roe Upwind Scheme (BATS-R-US) code that can solve various forms of the magnetohydrodynamics (MHD) equations, including Hall, semi-relativistic, multi-species and multi-fluid MHD, anisotropic pressure, radiative transport and heat conduction. Modeling disparate scales within BATS-R-US is achieved by a block-adaptive mesh both in Cartesian and generalized coordinates. Most recently we have created a new core for BATS-R-US: the Block-Adaptive Tree Library (BATL) that provides a general toolkit for creating, load balancing and message passing in a 1, 2 or 3 dimensional block-adaptive grid. We describe the algorithms of BATL and demonstrate its efficiency and scaling properties for various problems. BATS-R-US uses several time-integration schemes to address multiple time-scales: explicit time stepping with fixed or local time steps, partially steady-state evolution, point-implicit, semi-implicit, explicit/implicit, and fully implicit numerical schemes. Depending on the application, we find that different time stepping methods are optimal. Several of the time integration schemes exploit the block-based granularity of the grid structure. The framework and the adaptive algorithms enable physics based space weather modeling and even forecasting.

Toth, Gabor↗
DOE OSTI2022

H-AMR: A New GPU-accelerated GRMHD Code for Exascale Computing with 3D Adaptive Mesh Refinement and Local Adaptive Time Stepping

General relativistic magnetohydrodynamic (GRMHD) simulations have revolutionized our understanding of black hole accretion. Here, we present a GPU-accelerated GRMHD code H-AMR with multifaceted optimizations that, collectively, accelerate computation by 2–5 orders of magnitude for a wide range of applications. First, it introduces a spherical grid with 3D adaptive mesh refinement that operates in each of the three dimensions independently. This allows us to circumvent the Courant condition near the polar singularity, which otherwise cripples high-resolution computational performance. Second, we demonstrate that local adaptive time stepping on a logarithmic spherical-polar grid accelerates computation by a factor of ≲10 compared to traditional hierarchical time-stepping approaches. Jointly, these unique features lead to an effective speed of ~10 9 zone cycles per second per node on 5400 NVIDIA V100 GPUs (i.e., 900 nodes of the OLCF Summit supercomputer). We illustrate H-AMR's computational performance by presenting the first GRMHD simulation of a tilted thin accretion disk threaded by a toroidal magnetic field around a rapidly spinning black hole. With an effective resolution of 13,440 × 4608 × 8092 cells and a total of ≲22 billion cells and ~0.65 × 10 8 time steps, it is among the largest astrophysical simulations ever performed. We find that frame dragging by the black hole tears up the disk into two independently precessing subdisks. The innermost subdisk rotation axis intermittently aligns with the black hole spin, demonstrating for the first time that such long-sought alignment is possible in the absence of large-scale poloidal magnetic fields.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1987

Computation of steady and unsteady vortex dominated flows

The unsteady Euler equations have been written in the conservation form for the relative motion in a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-stage Runge-Kutta time stepping. For steady flow problems, local time stepping is used and for unsteady flow problems, the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wing. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Kandil, Osama A.↗
NASA NTRS1988

Computation of steady and unsteady vortex-dominated flows with shock waves

The unsteady Euler equations have been derived in the conservation form for the flow relative motion with respect to a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-state Runge-Kutta time stepping. For steady flow applications local time stepping is used, and for unsteady applications the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wings. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Kandil, Osama A.↗
NASA NTRS1993

The effect of grid aspect ratio on convergence

Vector stability theory and computational experiments are used to examine the effects of high aspect ratio grids on algorithm convergence. Emphasis is placed on identifying the degree of convergence deterioration that ADI schemes experience as grid aspect ratio increases and on deriving methods for circumventing this problem. Approximately factored implicit schemes are found to experience convergence deterioration because of nonoptimum local time stepping procedures and the increased need for viscous preconditioning. An enhanced algorithm is devised on the basis of improved selection of the local time step, appropriate definition of viscous preconditioning matrix, and proper implementation of boundary conditions. The novel algorithm provides uniformly efficient convergence at all aspect ratios for both Euler and Navier-Stokes computations for a variety of test problems.

Buelow, P. E. O.↗
NASA NTRS1991

Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model

The system of equations consisting of the full Navier-Stokes equations and two turbulence equations was solved for in the steady state using a multigrid strategy on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multistage Runge-Kutta time stepping scheme with a stability bound local time step, while the turbulence equations are advanced in a point-implicit scheme with a time step which guarantees stability and positively. Low Reynolds number modifications to the original two equation model are incorporated in a manner which results in well behaved equations for arbitrarily small wall distances. A variety of aerodynamic flows are solved for, initializing all quantities with uniform freestream values, and resulting in rapid and uniform convergence rates for the flow and turbulence equations.

Mavriplis, D. J.↗