Local Time-stepping Algorithm Research Articles

Advancements in enhancing flood evolution and urban inundation predictions: A study of local time stepping algorithm and GPU-accelerated hydrodynamic model

In recent years, floods occur frequently affected by extreme rainstorms. Timely flood prediction and rapid simulation are crucial in mitigating their impact. However, the current models often utilize the global time step method for updates and solving, resulting in high computational costs. To further improve the efficiency while ensuring calculational accuracy, this study proposes the local time stepping and GPU-accelerated hydrodynamic model based on non-uniform grid. The proposed model utilizes the HLLC approximate Riemannian solution to estimate interface flux and employs the local time step algorithm, enabling each unit to update variables using the locally permitted maximum time step. It relies on non-uniform grid technology, the local time step algorithm, and GPU acceleration to optimize computational efficiency, facilitating swift simulation of flood evolution processes across large-scale terrains. The results show that the hydrodynamic model has good stability. The results show that the model has good accuracy and stability. It can accurately predict the flood evolution process, flood arrival time, flood inundation range, and urban inundation process in complex terrains. Researchers can identify effective measures for reducing flood risks and improving the area’s resilience against flooding events. Due to its significantly improved efficiency in flood prediction, it is more suitable for actual large-scale flood processes.

Improved local time-stepping schemes for storm surge modeling on unstructured grids

This paper presents improved explicit local time-stepping (LTS) schemes of both first and second order accuracy for storm surge modeling. The two-dimensional shallow water equations are numerically solved on unstructured triangular meshes using finite volume method with Roe’s approximate Riemann solver. The LTS algorithms are designed based on explicit Euler and strong stability preserving Runge–Kutta time integration methods. A single-layer interface prediction–correction scheme is adopted to combine coarse and fine time discretization, further enhancing the stability of the LTS schemes, particularly at higher LTS levels and during long time simulations. An ideal numerical test validates the efficiency of the improved LTS models, revealing their capability to improve computational speed while preserving conservation properties and reducing accuracy loss as LTS levels increase. We further apply the LTS models to cross-scale simulations of storm surges in the Northwest Pacific. Results show that compared to the global time-stepping (GTS) models, the LTS models significantly boost computational speed by up to 37%, all while delivering equally reliable computational outcomes. With expanding high-resolution coastal data and the need for high-resolution modeling, the improved LTS models show great potential for cross-scale storm surge modeling.

Performance Analysis of Speculative Parallel Adaptive Local Timestepping for Conservation Laws

Stable simulation of conservation laws, such as those used to model fluid dynamics and plasma physics applications, requires the satisfaction of the so-called Courant-Friedrichs-Lewy condition. By allowing regions of the mesh to advance with different timesteps that locally satisfy this stability constraint, significant work reduction can be attained when compared to a time integration scheme using a single timestep size. However, parallelizing this algorithm presents considerable difficulty. Since the stability condition depends on the state of the system, dependencies become dynamic and potentially non-local. In this article, we present an adaptive local timestepping algorithm using an optimistic (Timewarp-based) parallel discrete event simulation. We introduce waiting heuristics to limit misspeculation and a semi-static load balancing scheme to eliminate load imbalance as parts of the mesh require finer or coarser timesteps. Last, we outline an interface for separating the physics of the specific conservation law from the temporal integration allowing for productive adoption of our proposed algorithm. We present a misspeculation study for three conservation laws, demonstrating both the productivity of the local timestepping API, for which 74% of the lines of code are reused across different conservation laws, and the robustness of the waiting heuristics—at most 1.5% of element updates are rolled back. Our performance studies demonstrate up to a 2.8× speedup versus a baseline unoptimized local timestepping approach, a 4x improvement in per-node throughput compared to an MPI parallelization of synchronous timestepping, and scalability up to 3,072 cores on NERSC’s Cori Haswell partition.

Local time stepping for the shallow water equations in MPAS

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.

A multiresolution local-timestepping scheme for particle–laden multiphase flow simulations using a level-set and point-particle approach

We present a novel storage-and-access approach for Lagrangian particles in a multiresolution local-timestepping framework for multiphase flow simulations. The proposed method applies a block-particle mapping strategy for efficient access of all particles on a specific refinement level while traversing through the multiresolution tree. This allows for extending the local timestepping algorithm with its refinement level-dependent timestep sizes to the evolution of particles, which results in significant speed-up in comparison to standard timestepping. For multi-scale multiphase flow simulations, the particle model is combined with a level-set based multiphase model on a Cartesian grid. To maintain robustness of fluid-state interpolation for particles near the level set-based fluid–fluid interface, WENO-based interpolation is applied which includes both real- and ghost-fluid cells. This enforces the sharp interface property also for interpolated fluid states, and suppresses spurious oscillations in the event of discontinuities.We validate the particle model with a one-dimensional simulation of a single particle in an air–helium shock tube for one-way coupling, and with two-dimensional simulations of a particle injected in a quiescent domain for the feedback force. Simulations of two-dimensional aerodynamic fragmentation in shear-induced entrainment and Rayleigh–Taylor piercing regimes use Lagrangian particles as sub-grid scale representation of small droplets post-breakup. Finally, three-dimensional massively-parallel simulations of single- and triple-bubble collapse near a wall coated with free-floating particles are presented. The particle cleaning radius of the single-bubble setup agrees reasonably well with experimental reference data. These simulations consider 106 particles and an Eulerian grid with effectively 1012 finite-volume cells at compression rates of more than 90% for the particulate phase. This underlines the advantageous effect of embedding the particles in the multiresolution tree with its spatial and temporal adaptivity, which is necessary for performing such large scale simulations efficiently.

Performance analysis and validation of an explicit local time-stepping algorithm for complex hypersonic flows

A local time-stepping (LTS) algorithm has been implemented and tested into the open source computational fluid dynamics code (CFD), Eilmer. The goal of this work is to investigate the performance of LTS for simulating complex hypersonic flows, in particular, flows that include strong inviscid/viscous interactions. Results from a range of laminar and turbulent test cases spanning a hypersonic flow range of Mach 4.5 to Mach 14.1 are presented. The relative efficiency of the LTS implementation, as compared to a global time-stepping (GTS) implementation, is reported. The LTS implementation is validated against experimental data for each test case to assess the accuracy and applicability of LTS for hypersonic flows. The results presented in this work show that LTS can accelerate the steady-state solution of complex hypersonic flows, in particular, speedup factors ranging from 17.5 to 174 are reported for the test cases presented. The accuracy of the solutions obtained using LTS match favourably to those obtained using GTS, in addition to comparing well to experimental data. Performance is further validated against complex scramjet cavity combustor flow fields with shear layer, shock interactions and recirculation regions. Overall, the results presented in this paper demonstrate that LTS is an easy to implement technique that can provide fast and accurate results for a gamut of high-speed flow problems.

Implementation of a Local Time Stepping Algorithm and Its Acceleration Effect on Two-Dimensional Hydrodynamic Models

The engineering applications of two-dimensional (2D) hydrodynamic models are restricted by the enormous number of meshes needed and the overheads of simulation time. The aim of this study is to improve computational efficiency and optimize the balance between the quantity of grids used in and the simulation accuracy of 2D hydrodynamic models. Local mesh refinement and a local time stepping (LTS) strategy were used to address this aim. The implementation of the LTS algorithm on a 2D shallow-water dynamic model was investigated using the finite volume method on unstructured meshes. The model performance was evaluated using three canonical test cases, which discussed the influential factors and the adaptive conditions of the algorithm. The results of the numerical tests show that the LTS method improved the computational efficiency and fulfilled mass conservation and solution accuracy constraints. Speedup ratios of between 1.3 and 2.1 were obtained. The LTS scheme was used for navigable flow simulation of the river reach between the Three Gorges and Gezhouba Dams. This showed that the LTS scheme is effective for real complex applications and long simulations and can meet the required accuracy. An analysis of the influence of the mesh refinement on the speedup was conducted. Coarse and refined mesh proportions and mesh scales observably affected the acceleration effect of the LTS algorithm. Smaller proportions of refined mesh resulted in higher speedup ratios. Acceleration was the most obvious when mesh scale differences were large. These results provide technical guidelines for reducing computational time for 2D hydrodynamic models on non-uniform unstructured grids.

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 (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. 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 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.

A discontinuous Galerkin immersed boundary solver for compressible flows: Adaptive local time stepping for artificial viscosity–based shock‐capturing on cut cells

SummaryWe present a higher‐order cut cell immersed boundary method (IBM) for the simulation of high Mach number flows. As a novelty on a cut cell grid, we evaluate an adaptive local time stepping (LTS) scheme in combination with an artificial viscosity–based shock‐capturing approach. The cut cell grid is optimized by a nonintrusive cell agglomeration strategy in order to avoid problems with small or ill‐shaped cut cells. Our approach is based on a discontinuous Galerkin discretization of the compressible Euler equations, where the immersed boundary is implicitly defined by the zero isocontour of a level set function. In flow configurations with high Mach numbers, a numerical shock‐capturing mechanism is crucial in order to prevent unphysical oscillations of the polynomial approximation in the vicinity of shocks. We achieve this by means of a viscous smoothing where the artificial viscosity follows from a modal decay sensor that has been adapted to the IBM. The problem of the severe time step restriction caused by the additional second‐order diffusive term and small nonagglomerated cut cells is addressed by using an adaptive LTS algorithm. The robustness, stability, and accuracy of our approach are verified for several common test cases. Moreover, the results show that our approach lowers the computational costs drastically, especially for unsteady IBM problems with complex geometries.

Improved local time‐stepping algorithm for leap‐frog discontinuous Galerkin time‐domain method

The discontinuous Galerkin time-domain (DGTD) method is now very popular for the solution of Maxwell's equations. Owing to the ability to deal with various types and shapes, and possibly locally refined meshes, it handles easily complicated geometries and remains globally explicit with easy parallelisation and extension to high orders of accuracy. However, the limitation of the stability of the method, related to the smallest elements in the mesh, requires the construction of local time-stepping algorithm. An improved local time-stepping algorithm of the explicit leap-frog DGTD method is proposed. The monostatic and bistatic radar cross section of several typical targets (a dielectric sphere, a perfect electric conductor almond, and a simplified missile) is calculated by the proposed algorithm. The simulated results show that although the entire solution relative error calculated by the proposed method has a slight increase compared with that of the previous scheme, the computational efficiency of the explicit leap-frog DGTD method is greatly improved. Meanwhile, the proposed algorithm does not require any additional storage. Compared to the improvement of the computational efficiency, this slight increase of the relative error can be absolutely acceptable.

A conservative finite volume scheme with time-accurate local time stepping for scalar transport on unstructured grids

In this article we propose a new conservative high resolution TVD (total variation diminishing) finite volume scheme with time-accurate local time stepping (LTS) on unstructured grids for the solution of scalar transport problems, which are typical in the context of water quality simulations. To keep the presentation of the new method as simple as possible, the algorithm is only derived in two space dimensions and for purely convective transport problems, hence neglecting diffusion and reaction terms. The new numerical method for the solution of the scalar transport is directly coupled to the hydrodynamic model of Casulli and Walters (2000) that provides the dynamics of the free surface and the velocity vector field based on a semi-implicit discretization of the shallow water equations. Wetting and drying is handled rigorously by the nonlinear algorithm proposed by Casulli (2009). The new time-accurate LTS algorithm allows a different time step size for each element of the unstructured grid, based on an element-local Courant-Friedrichs-Lewy (CFL) stability condition. The proposed method does not need any synchronization between different time steps of different elements and is by construction locally and globally conservative. The LTS scheme is based on a piecewise linear polynomial reconstruction in space–time using the MUSCL–Hancock method, to obtain second order of accuracy in both space and time.The new algorithm is first validated on some classical test cases for pure advection problems, for which exact solutions are known. In all cases we obtain a very good level of accuracy, showing also numerical convergence results; we furthermore confirm mass conservation up to machine precision and observe an improved computational efficiency compared to a standard second order TVD scheme for scalar transport with global time stepping (GTS). Then, the new LTS method is applied to some more complex problems, where the new scalar transport scheme has also been coupled to a semi-implicit model for the simulation of the free surface hydrodynamics, including nonlinear wetting and drying. The last application shown in this paper is carried out on a real geometry, for which we have taken available DTM data of the lake Guaiba in Brazil. Comparisons have been made in all cases with a second order TVD scheme based on GTS. For the new LTS algorithm we report a significant reduction in computational effort, with a savings of CPU time of the order of up to 95%.

A novel Local Time Stepping algorithm for shallow water flow simulation in the discontinuous Galerkin framework

This paper introduces a generic and efficient Local Time Stepping (LTS) algorithm called Local Flux Flagged-Local Time Stepping (LFF-LTS). The applicability of LFF-LTS algorithm to the solution of shallow water flow equations, utilizing the discontinuous Galerkin finite element framework, is investigated for uniform and non-uniform meshes. The proposed algorithm can be incorporated in any numerical scheme using an explicit time integration scheme. The new scheme is developed by combining and modifying the two existing LTS algorithms, where the domain time step (maximum allowable time step within the domain based on the Courant–Friedrichs–Lewy condition) and the local time step for an element is determined such that the stability criterion is satisfied for all elements. Frozen flux approximation is used during the intermediate time step as the solution for each element is advanced to the domain time step. The algorithm is applied to idealized and practical shallow water flow problems with wet/dry and partially wet bed conditions, and its accuracy and efficiency are compared to the traditional Global Time Stepping (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 47–73% CPU time reduction when compared to the GTS method. The accuracy of the LFF-LTS and GTS algorithms are compared with analytical/measured data and found to be in good agreement. The LFF-LTS algorithm is compared with the two existing LTS schemes and found to be more efficient.

High order cell-centered Lagrangian-type finite volume schemes with time-accurate local time stepping on unstructured triangular meshes

We present a novel cell-centered direct Arbitrary-Lagrangian–Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). It extends our previous investigations on high order Lagrangian finite volume schemes with LTS carried out in [46] in one space dimension. The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space–time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space–time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space–time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm.We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gas dynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder, the Saltzman and the Sedov problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as ∼4.7. Finally, we have also shown the improvement in terms of computational efficiency in a representative test for the special relativistic magnetohydrodynamics (RMHD) equations.

RKDG2 shallow-water solver on non-uniform grids with local time steps: Application to 1D and 2D hydrodynamics

This paper investigates local time stepping (LTS) with the RKDG2 (second-order Runge–Kutta Discontinuous Galerkin) non-uniform solutions of the inhomogeneous SWEs (shallow water equations) with source terms. A LTS algorithm – recently designed for homogenous hyperbolic PDE(s) – is herein reconsidered and improved in combination with the RKDG2 shallow-flow solver (LTS-RKDG2) including topography and friction source terms as well as wetting and drying. Two LTS-RKDG2 schemes that adapt 3 and 4 levels of LTSs are configured on 1D and/or 2D (quadrilateral) non-uniform meshes that, respectively, adopt 3 and 4 scales of spatial discretization. Selected shallow water benchmark tests are used to verify, assess and compare the LTS-RKDG2 schemes relative to their conventional Global Time Step RKDG2 alternatives (GTS-RKDG2) considering several issues of practical relevance to hydraulic modelling. Results show that the LTS-RKDG2 models could offer (depending on both the mesh setting and the features of the flow) comparable accuracy to the associated GTS-RKDG2 models with a savings in runtime of up to a factor of 2.5 in 1D simulations and 1.6 in 2D simulations.

DGTD for a Class of Low-Observable Targets: A Comparison With MoM and (2, 2) FDTD

The simulation of low-observable targets requires high accuracy, both in the geometrical discretization as well as in the numerical solution of the electromagnetic problem. In this letter, we employ the well-known NASA almond to illustrate the accuracy of the leapfrog discontinuous Galerkin method, combined with a local time-stepping algorithm, comparing it to the method of moments (MoM) and the (2, 2) finite-difference time-domain (FDTD) methods.

An efficient local time-stepping scheme for solution of nonlinear conservation laws

We develop an efficient local time-stepping algorithm for the method of lines approach to numerical solution of transient partial differential equations. The need for local time-stepping arises when adaptive mesh refinement results in a mesh containing cells of greatly different sizes. The global CFL number and, hence, the global time step, are defined by the smallest cell size. This can be inefficient as a few small cells may impose a restrictive time step on the whole mesh. A local time-stepping scheme allows us to use the local CFL number which reduces the total number of function evaluations. The algorithm is based on a second order Runge–Kutta time integration. Its important features are a small stencil and the second order accuracy in the L 2 and L ∞ norms.

An arbitrary high-order Discontinuous Galerkin method for elastic waves on unstructured meshes - V. Local time stepping andp-adaptivity

SUMMARY This article describes the extension of the arbitrary high-order Discontinuous Galerkin (ADER-DG) method to treat locally varying polynomial degress of the basis functions, so-called p-adaptivity, as well as locally varying time steps that may be different from one element to another. The p-adaptive version of the scheme is useful in complex 3-D models with small-scale features which have to be meshed with reasonably small elements to capture the necessary geometrical details of interest. Using a constant high polynomial degree of the basis functions in the whole computational domain can lead to an unreasonably high CPU effort since good spatial resolution at the surface may be already obtained by the fine mesh. Therefore, it can be more adequate in some cases to use a lower order method in the small elements to reduce the CPU effort without loosing much accuracy. To further increase computational efficiency, we present a new local time stepping (LTS) algorithm. For usual explicit time stepping schemes the element with the smallest time step resulting from the stability criterion of the method will dictate its time step to all the other elements of the computational domain. In contrast, by using local time stepping, each element can use its optimal time step given by the local stability condition. Our proposed LTS algorithm for ADER-DG is very general and does not need any temporal synchronization between the elements. Due to the ADER approach, accurate time interpolation is automatically provided at the element interfaces such that the computational overhead is very small and such that the method maintains the uniform high order of accuracy in space and time as in the usual ADER-DG schemes with a globally constant time step. However, the LTS ADER-DG method is computationally much more efficient for problems with strongly varying element size or material parameters since it allows to reduce the total number of element updates considerably. This holds especially for unstructured tetrahedral meshes that contain strongly degenerate elements, so-called slivers. We show numerical convergence results and CPU times for LTS ADER-DG schemes up to sixth order in space and time on irregular tetrahedral meshes containing elements of very different size and also on tetrahedral meshes containing slivers. Further validation of the algorithm is provided by results obtained for the layer over half-space (LOH.1) benchmark problem proposed by the Pacific Earthquake Engineering Research Center. Finally, we present a realistic application on earthquake modelling and ground motion prediction for the alpine valley of Grenoble.

A consistent hybrid PDF method: implementation details and application to the simulation of a bluff-body stabilised flame

PDF simulations of a bluff-body stabilised flame studied experimentally by Dally et al. have been performed using a hybrid Finite-Volume /particle method in combination with a local time-stepping algorithm. Using mixture fraction as only independent scalar and a flamelet chemistry model, satisfactory results for flow field and mean temperature are obtained. In the proposed hybrid method, the Finite-Volume submodel plays a leading role since it provides the mean velocity, mean velocity gradients, mean pressure gradient, mean dissipation and the gradients of Reynolds stresses used in the particle submodel, resulting in a low bias error. Details on the implementation are given, correction algorithms are presented and a new iteration averaging method is proposed. Despite an inconsistency in the treatment of triple correlation terms, the difference between the Reynolds stresses resolved in the Finite-Volume submodel and extracted from the ensemble of particles appears to be very small, demonstrating that the correction algorithms ensure consistency.

Local Time-Stepping Algorithm for Solving Probability Density Function Turbulence Model Equations

A local time-stepping algorithm has been developed to improve the numerical efe ciency of Lagrangian particlebased Monte Carlo methods for obtaining the steady-state solutions of the modeled probability density function equations of turbulent reacting e ows. On each step in the pseudo-time-marching algorithm, the properties of each particle are advanced by a time step, the magnitude of which depends on the particle’ s spatial location. This algorithm has been incorporated into the consistent hybrid e nite volume/particle method. The performance of the localtime-steppingmethodisevaluated intermsofnumericalefe ciencyandaccuracy throughapplication toa nonreacting bluff-body e ow. Forthistest case,itisfound that localtime stepping can acceleratetheglobal convergence of the hybrid method by as much as an order of magnitude, depending on the grid stretching. Additionally, local time stepping is found to improve signie cantly the robustness of the hybrid method mainly due to the accelerated convection of error waves out of thecomputational domain. The method is very simpleto implement, and the small increase in CPU time per step (typically 3%) is a negligible penalty compared to the substantial reduction in the number of time steps required to reach convergence.

Local time-stepping algorithm for solving probability density function turbulence model equations

Local time-stepping algorithm for solving probability density function turbulence model equations