Local Time-stepping Methods Research Articles

Improvements for matrix-type implicit temporal scheme regarding entropy fix and local time step

The matrix version of Symmetric Successive Over Relaxation (matrix-SSOR) scheme has been proved to be more efficient than the standard Lower-Upper Symmetric Gauss-Seidel (LU-SGS), but less robust for high-speed flows. In order to ulteriorly improve the convergence rate as well as numerical stability of matrix-SSOR, two improvements regarding entropy fix and local time step have been proposed and validated. Firstly, an augmented entropy fix method is imposed on the inviscid Jacobian matrix and proved to be effective in two high-speed flows, in which the key parameter in entropy fix is discussed and found to be insensitive within appropriate range of values. Since the time step also has great effects on the numerical stability and convergence rate, a modified cell residual adapted local time step method with consideration of the residual history is developed, which is found to be effective for increasing the convergence rate when the matrix-SSOR is applied, but invalid when the LU-SGS is used. The proposed modified local time step method is also insensitive to the key parameter within appropriate range of values. The two modifications can be conveniently implanted into analogous matrix-type implicit schemes to improve the numerical performance.

Cross-Scale Modeling of Shallow Water Flows in Coastal Areas with an Improved Local Time-Stepping Method

A shallow water equations-based model with an improved local time-stepping (LTS) scheme is developed for modeling coastal hydrodynamics across multiple scales, from large areas to detailed local regions. To enhance the stability of the shallow water model for long-duration simulations and at larger LTS gradings, a prediction-correction method using a single-layer interface that couples coarse and fine time discretizations is adopted. The proposed scheme improves computational efficiency with an acceptable additional computational burden and ensures accurate conservation of time truncation errors in a discrete sense. The model performance is verified with respect to conservation and computational efficiency through two idealized tests: the spreading of a drop of shallow water and a tidal flat/channel system. The results of both tests demonstrate that the improved LTS scheme maintains precision as the LTS grading increases, preserves conservation properties, and significantly improves computational efficiency with a speedup ratio of up to 2.615. Furthermore, we applied the LTS scheme to simulate tides at grid scales of 40,000 m to 200 m for a portion of the Northwest Pacific. The proposed model shows promise for modeling cross-scale hydrodynamics in complex coastal and ocean engineering problems.

Numerical study of shock wave boundary layer interaction flows using a novel turbulence model

This study proposes a novel partial average fluctuation velocity (PAFV) turbulence based on the PAFV and the average strain rate tensor, which is used to calculate the shock wave boundary layer interaction flows, including cascade flow and transonic compressor flow. In order to illustrate the adequacy, the calculation results of several other turbulence models were also compared with PAFV, including k-ω, shear stress transport, SA-neg (negative Spalart-Allmaras), and stress-BSL (ω-based Reynolds stress model). The suitability of the PAFV turbulence model was initially tested by calculating the transonic flow in the L030-4 cascade. Subsequently, numerical computations were performed for the 3D (three-dimensional) National Aeronautics and Space Administration Rotor 67 transonic compressor rotor. In the numerical simulation, the flow control equations were transformed in a general curvilinear coordinate system for precision enhancement, and the spatial discretization of control equations was executed using the finite difference method. The convection term was processed using the Steger–Warming flux vector splitting method, the diffusion term was discretized using a second-order central difference scheme, and the time derivative term was discretized using the third-order Runge–Kutta method. A parallel computing strategy with multi-block partitioning was employed to speed up the calculations and facilitate faster convergence alongside a local time step method. All numerical computation procedures were written in Python, revealing the following: (1) The PAFV turbulence model aptly simulates the transonic flow within the cascade channel. The shock wave pattern in the cascade channel aligns well with the experimental schlieren images. (2) During the Rotor 67 transonic compressor rotor calculation, the flow–pressure ratio, flow efficiency, and inlet and outlet total temperatures and pressures align well with the experimental data. Furthermore, the resolution of the flow field structure related to the interaction between the tip leakage vortex and channel shock wave is high. (3) The turbulence model uses PAFV as the turbulence velocity scale, thereby suppressing the fluctuation velocity and turbulence viscosity near shock wave areas, preventing excessive turbulence viscosity. Moreover, PAFV inherently exhibits transport properties and anisotropic characteristics, making it well-suited for handling transonic compressor flows.

Efficient Improved Local Time Stepping with the Leapfrog Scheme for Transient 3-D Electromagnetic Analyses

An alternate boundary local time stepping (ABLTS) method is proposed for the discontinuous Galerkin time domain method for transient electromagnetic simulations to reduce the computation complexity of the local time stepping (LTS) method. The proposed method exhibites lower storage and time complexity than the conventional LF-LTS method . The stability, accuracy, and effectiveness of the ABLTS method are verified by applying it to the simulation of a resonator cavity and multi-layer microstrip antenna. The numerical results revealed that the developed method is effective for the transient electromagnetic simulation of antennas.

A Global-Local Error-Driven -Adaptive Scheme for Discontinuous Galerkin Time-Domain Method With Local Time-Stepping Strategy

In this paper, a highly efficient <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive scheme has been developed for discontinuous Galerkin time domain (DGTD) method with local time stepping (LTS) strategy. In the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive procedure, the dual-fold error driven scheme has been developed. Compared to the global error range corresponding to the preset base order range, the local maximal error in each discretized element is used to determine the allowable local range of the base order. The local error in each element is compared with the local error range to determine which operation, i.e., the base order increasing or decreasing, is implemented. The resulting two error thresholds are insensitive in the dynamic <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive procedure. When applying the proposed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -adaptive scheme into the LTS method, a simple grouping approach in terms of the time steps related to the base orders has been developed to rapidly perform the marching-on-in time procedure. Several numerical examples including scattering from a missile, radiation of a ridge gap waveguide (RGW) antenna and transmission of a cylindrical cavity filter are given to demonstrate good accuracy and high efficiency of the proposed method.

Analysis and evaluation of steady-state and non-steady-state preserving operator splitting schemes for reaction-diffusion(-advection) problems

Reaction-diffusion(-advection) problems are well-known in chemical engineering and computational fluid dynamics. A common feature of these systems is a linear (or non-stiff) transport sub-system and a non-linear (or stiff), highly coupled chemistry sub-system. The expected numerical effort often prohibits a fully coupled solution of the system. Therefore, the system is split into a transport and a reaction sub-system, and each sub-system is solved using specialized solvers. Operator splitting schemes are required to reconstruct the solution of the initial system from the sub-system solutions. Steady-state preserving splitting schemes are particularly essential for steady-state calculations since local time stepping (LTS) or other methods based on fictional time rely on large time steps to be efficient. This work formally analyzes common splitting schemes for reaction-diffusion problems by stability analysis and checking the steady-state preservation of a representative linear scalar problem. Balanced and Simpler Balanced splitting are the only steady-state conservative schemes analyzed. Three new steady-state preserving splitting schemes are proposed based on the findings of the formal analysis. To achieve steady-state preservation, the new schemes use splitting constants based on either the mixing derivative or the chemistry derivative. The formal analysis is accompanied by a dimensionless perfectly stirred reactor (PSR) case and a hydrogen combustion case, both known to be challenging for operator splitting schemes. The test case results are in line with the theoretical results but indicate that the scalar linear analysis is nonviable to capture the full effects of the chemical sub-system. The Simpler and the newly proposed Consistent Staggered splitting schemes give significantly better results than the remaining ones while being second-order and first-order accurate, respectively. If temporal accuracy is irrelevant, e.g., for steady-state solvers, the proposed Consistent Adaptive splitting scheme is promising since it preserves the steady-state solution first-order time accurate with less function evaluations.

A Minimal Round-Trip Strategy Based on Graph Matching for Parallel DGTD Method With Local Time-Stepping

The load distribution in the local time stepping (LTS) method significantly impacts its computing efficiency. This letter proposes a minimal round-trip (MRT) strategy of the LTS method to balance the communication load of the discontinuous Galerkin time-domain (DGTD) method. By discovering the matching of the connected graph of computing nodes, independent communication with a similar load can be done in the same round trip to minimize waiting between nodes in nonblocking communication, thereby decreasing the communication time of the DGTD-LTS technique. The numerical results indicate that the MRT strategy reduces the communication time between processors by 50% and improves the parallel performance when the LTS method is implemented in the DGTD method. The parallel scale of the MRT approach may be increased to 16 000 nodes (1 040 000 cores) on the supercomputer, and the parallel efficiency is greater than 73.8%.

Operator Splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media

Operator Splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media

Novel Parallelization of Discontinuous Galerkin Method for Transient Electromagnetics Simulation Based on Sunway Supercomputers

A novel parallelization of discontinuous Galerkin time-domain (DGTD) method hybrid with the local time step (LTS) method on Sunway supercomputers for electromagnetic simulation is proposed. The proposed method includes a minimum number of roundtrip (MNR) strategy for processor-level parallelism and a double buffer strategy based on the remote memory access (RMA) of the Sunway processor. The MNR strategy optimizes the communication topology between nodes by recursively establishing the minimum spanning tree and the double buffer strategy is designed to make communication overlapped computation when RMA transmission. Combining the two methods, the proposed method achieves an unprecedented massively parallelism of the DGTD method. Several examples of radiation and scattering are used as cases to study the accuracy and validity of the proposed method. The numerical results show that the proposed method can effectively support 16,000 nodes (1,040,000 cores) parallelism on the Sunway supercomputer, which enables the DGTD method to solve the transient electromagnetic field in a very short time.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

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.

Stabilized leapfrog based local time-stepping method for the wave equation

Local time-stepping methods permit to overcome the severe stability constraint on explicit methods caused by local mesh refinement without sacrificing explicitness. Diaz and Grote [SIAM J. Sci. Comput. 31 (2009), pp. 1985–2014] proposed a leapfrog based explicit local time-stepping (LF-LTS) method for the time integration of second-order wave equations. Recently, optimal convergence rates were proved for a conforming FEM discretization, albeit under a CFL stability condition where the global time-step, Δt, depends on the smallest elements in the mesh (see M. J. Grote, M. Mehlin, and S. A. Sauter [SIAM J. Numer. Anal. 56 (2018), pp. 994–1021]). In general one cannot improve upon that stability constraint, as the LF-LTS method may become unstable at certain discrete values of Δt. To remove those critical values of Δt, we apply a slight modification (as in recent work on LF-Chebyshev methods by Carle, Hochbruck, and Sturm [SIAM J. Numer. Anal. 58 (2020), pp. 2404–2433]) to the original LF-LTS method which nonetheless preserves its desirable properties: it is fully explicit, second-order accurate, satisfies a three-term (leapfrog like) recurrence relation, and conserves the energy. The new stabilized LF-LTS method also yields optimal convergence rates for a standard conforming FE discretization, yet under a CFL condition where Δt no longer depends on the mesh size inside the locally refined region.

GPU-Accelerated Hybrid Discontinuous Galerkin Time Domain Algorithm With Universal Matrices and Local Time Stepping Method

In this article, a graphic processing unit (GPU)-based acceleration implementation of a hybrid discontinuous Galerkin time domain method based on Maxwell’s equations and Helmholtz vector wave equation (HDGTD) has been developed. The computational domain is discretized by tetrahedrons and the resultant meshes are categorized into two regions solved by Maxwell’s equations and Helmholtz vector wave equation, respectively. The hierarchical vector basis functions are used to expand the unknowns in the HDGTD method, and a universal matrix technique is proposed to decompose the geometry-dependent matrices in each tetrahedron into the summation of universal matrices defined in barycentric coordinates, thus giving rise to a great decrease of the memory usage. A local time stepping (LTS) method based on a simple interpolation is introduced in the proposed HDGTD method to achieve highly efficient solution of multiscale problems. Two kinds of compute unified device architecture (CUDA)-based mapping techniques, i.e., 1-D and 2-D blocks, are implemented to achieve a tradeoff between the parallel speedup and the memory usage. With the 1-D block mapping, over 590 times speedup can be achieved, and in the case of the 2-D block mapping, over 150 times acceleration and 13 times memory reduction are obtained. Some practical complex examples are given to demonstrate a good performance of the proposed parallel method.

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.

An implicit local time-stepping method based on cell reordering for multiphase flow in porous media

We discuss how to introduce local time-step refinements in a sequential implicit method for multiphase flow in porous media. Our approach relies heavily on causality-based optimal ordering, which implies that cells can be ordered according to total fluxes after the pressure field has been computed, leaving the transport problem as a sequence of ordinary differential equations, which can be solved cell-by-cell or block-by-block. The method is suitable for arbitrary local time steps and grids, is mass-conservative, and reduces to the standard implicit upwind finite-volume method in the case of equal time steps in adjacent cells. The method is validated by a series of numerical simulations. We discuss various strategies for selecting local time steps and demonstrate the efficiency of the method and several of these strategies by through a series of numerical examples.

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.

Local time-stepping for adaptive multiresolution using natural extension of Runge–Kutta methods

A space–time fully adaptive multiresolution method for evolutionary non-linear partial differential equations is presented introducing an improved local time-stepping method. The space discretisation is based on classical finite volumes, endowed with cell average multiresolution analysis for triggering the dynamical grid adaptation. The explicit time scheme features a natural extension of Runge–Kutta methods which allow local time-stepping while guaranteeing accuracy. The use of a compact Runge–Kutta formulation permits further memory reduction. The precision and computational efficiency of the scheme regarding CPU time and memory compression are assessed for problems in one, two and three space dimensions. As application Burgers equation, reaction–diffusion equations and the compressible Euler equations are considered. The numerical results illustrate the efficiency and superiority of the proposed local time-stepping method with respect to the reference computations.

Preconditioned WENO finite-difference lattice Boltzmann method for simulation of incompressible turbulent flows

In this work, a preconditioned high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (WENO-LBM) is applied to deal with the incompressible turbulent flows. Two different turbulence models namely, the Spalart–Allmaras (SA) and k−ωSST models are used and applied in the solution method for this aim. The spatial derivatives of the two-dimensional (2D) preconditioned LB equation in the generalized curvilinear coordinates are discretized by using the fifth-order WENO finite-difference scheme and an implicit–explicit Runge–Kutta scheme is adopted for the time discretization. For the convective and diffusive terms of the turbulence transport equations, the third-order WENO and second-order central finite-difference schemes are used, respectively. The preconditioning technique along with the local time-stepping method are applied to the WENO-LBM to further accelerate the solution to the converged steady-state condition, and therefore, an accurate and efficient incompressible LB solver is provided for simulating turbulent flows. The accuracy and robustness of the proposed solution method are assessed by computing two test cases: the 2D turbulent flow over a flat plate at Re=1.0×107 and the 2D turbulent flow over a NACA0012 airfoil at Re=6.0×106 and different angles of attack. The present results are compared with the available numerical and experimental results which show excellent agreement. It is demonstrated that the present solution methodology based on the WENO-LBM provides more accurate results of the incompressible turbulent flows and it requires lower number of grid points compared to the traditional (low-order accurate) LB and Navier–Stokes solvers.

Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time step eve...

Second order accurate asynchronous scheme for modeling linear partial differential equations

We propose an asynchronous method for the explicit integration of multi-scale partial differential equations. This method is restricted by a local CFL (Courant Friedrichs Lewy) condition rather than the traditional global CFL condition. Moreover, contrary to other local time-stepping (LTS) methods, the asynchronous algorithm permits the selection of independent time steps in each mesh element. We derived an asynchronous Runge–Kutta 2 (ARK2) scheme from a standard explicit Runge–Kutta method and we proved that the ARK2 scheme is second order convergent. Comparing with the classical integration, the asynchronous scheme is effective in terms of computation time.