Local Time Stepping Research Articles

Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

Space–time adaptive ADER-DG schemes for dissipative flows: Compressible Navier–Stokes and resistive MHD equations

This paper presents an arbitrary high-order accurate ADER Discontinuous Galerkin (DG) method on space–time adaptive meshes (AMR) for the solution of two important families of non-linear time dependent partial differential equations for compressible dissipative flows : the compressible Navier–Stokes equations and the equations of viscous and resistive magnetohydrodynamics in two and three space-dimensions.The work continues a recent series of papers concerning the development and application of a proper a posteriori subcell finite volume limiting procedure suitable for discontinuous Galerkin methods (Dumbser et al., 2014, Zanotti et al., 2015 [40,41]). It is a well known fact that a major weakness of high order DG methods lies in the difficulty of limiting discontinuous solutions, which generate spurious oscillations, namely the so-called ‘Gibbs phenomenon’. In the present work, a nonlinear stabilization of the scheme is sequentially and locally introduced only for troubled cells on the basis of a novel a posteriori detection criterion, i.e. the MOOD approach. The main benefits of the MOOD paradigm, i.e. the computational robustness even in the presence of strong shocks, are preserved and the numerical diffusion is considerably reduced also for the limited cells by resorting to a proper sub-grid. In practice the method first produces a so-called candidate solution by using a high order accurate unlimited DG scheme. Then, a set of numerical and physical detection criteria is applied to the candidate solution, namely: positivity of pressure and density, absence of floating point errors and satisfaction of a discrete maximum principle in the sense of polynomials. Furthermore, in those cells where at least one of these criteria is violated the computed candidate solution is detected as troubled and is locally rejected. Subsequently, a more reliable numerical solution is recomputed a posteriori by employing a more robust but still very accurate ADER-WENO finite volume scheme on the subgrid averages within that troubled cell. Finally, a high order DG polynomial is reconstructed back from the evolved subcell averages.We apply the whole approach for the first time to the equations of compressible gas dynamics and magnetohydrodynamics in the presence of viscosity, thermal conductivity and magnetic resistivity, therefore extending our family of adaptive ADER-DG schemes to cases for which the numerical fluxes also depend on the gradient of the state vector. The distinguished high-resolution properties of the presented numerical scheme standout against a wide number of non-trivial test cases both for the compressible Navier–Stokes and the viscous and resistive magnetohydrodynamics equations. The present results show clearly that the shock-capturing capability of the news schemes is significantly enhanced within a cell-by-cell Adaptive Mesh Refinement (AMR) implementation together with time accurate local time stepping (LTS).

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.

Semi-implicit discontinuous Galerkin methods for the incompressible Navier–Stokes equations on adaptive staggered Cartesian grids

In this paper a new high order semi-implicit discontinuous Galerkin method (SI-DG) is presented for the solution of the incompressible Navier–Stokes equations on staggered space-time adaptive Cartesian grids (AMR) in two and three space dimensions. The pressure is written in the form of piecewise polynomials on the main grid, which is dynamically adapted within a cell-by-cell AMR framework. According to the time dependent main grid, different face-based spatially staggered dual grids are defined for the piece-wise polynomials of the respective velocity components. Although the resulting adaptive staggered grids are more complex than classical uniform Cartesian meshes, the numerical scheme can still be written in a rather compact form. Thanks to the use of a tensor-product formulation for the definition of the nodal basis in the d-dimensional space (d=2,3), all the discrete operators can be efficiently written as a combination of linear one-dimensional operators acting in the d space directions separately.Arbitrary high order of accuracy is achieved in space, while a very simple semi-implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms, and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation. The real advantages of the staggered grid arise in the solution of the Schur complement associated with the saddle point problem of the discretized incompressible Navier–Stokes equations, i.e. after substituting the discrete momentum equations into the discrete continuity equation. This leads to a linear system for only one unknown, the scalar pressure. Indeed, the resulting linear pressure system is shown to be symmetric and positive-definite. In order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor, an implicit discretization is also used for the diffusive terms in the momentum equation. A particular feature of our staggered DG approach is that the viscous stress tensor is discretized on the dual mesh. This corresponds to the use of a lifting operator, but on the staggered grid. Both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix-free conjugate gradient method, for which fast convergence is observed. Moreover, it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner. Due to the explicit discretization of the nonlinear convective terms, the final algorithm is stable within the classical CFL-type time step restriction for explicit DG methods based on the fluid velocity. Moreover, by using a suitable local time stepping technique based on the ADER-DG framework, each element can use its optimal local time step also within the AMR context.The new space-time adaptive staggered DG scheme has been thoroughly verified for polynomial degrees up to N=9 for a large set of non-trivial test problems in two and three space dimensions, for which analytical, numerical or experimental reference solutions exist. The high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution.To the knowledge of the authors, this is the first staggered semi-implicit DG scheme for the incompressible Navier–Stokes equations on space-time adaptive meshes in two and three space dimensions.

An h-p Version of the Continuous Petrov–Galerkin Method for Nonlinear Delay Differential Equations

We investigate an h-p version of the continuous Petrov–Galerkin time stepping method for nonlinear delay differential equations with vanishing delays. We derive a priori error estimates in the $$L^{2}$$ -, $$H^{1}$$ - and $$L^\infty $$ -norm that are completely explicit with respect to the local time steps, the local polynomial degrees, and the local regularity of the exact solution. Moreover, we show that the h-p version continuous Petrov–Galerkin scheme based on geometrically refined time steps and on linearly increasing approximation orders achieves exponential rates of convergence for solutions with start-up singularities. The theoretical results are illustrated by some numerical experiments.

An h-p version of the continuous Petrov-Galerkin method for Volterra delay-integro-differential equations

We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L2-, H1- and L∞-norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.

MultiLevel Local Time-Stepping Methods of Runge--Kutta-type for Wave Equations

Local mesh refinement significantly influences the performance of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods improve the efficiency by using smaller time steps precisely where the smallest mesh elements are located, thus permitting a larger time step in the coarser regions of the mesh without violating the stability condition. However, when the mesh contains nested patches of refinement, any local time step will be unnecessarily small in some regions. To allow for an appropriate time step at each level of mesh refinement, multilevel local time-stepping (MLTS) methods have been proposed. Starting from the Runge--Kutta-based LTS methods derived by Grote, Mehlin, and Mitkova [SIAM J. Sci. Comput., 37 (2015), pp. A747--A775], we propose explicit MLTS methods of arbitrarily high accuracy. Numerical experiments with finite difference and continuous finite element spatial discretizations illustrate the usefulness of the novel MLTS methods and show that they r...

Local Time-Step TLM Unstructured Block Meshing for Electromagnetic and Bio-Thermal Applications

Many practical problems are of multiscale nature. This issue appears when relatively small regions inside the computational domain have a large impact on the overall behavior of the structure. This requires using small mesh size in these regions of interest for adequately representing their geometrical details. Using uniform fine grid everywhere is computationally very exhaustive. The use of irregular structured grids may lead to some memory cost reduction as compared to uniform fine mesh. However, it imposes a smaller time step and yields higher dispersion as cell aspect ratio deviates from unity. The alternate approach is to use block meshing which allows fine discretization in regions with fine details and larger mesh size in quasi homogenous regions. The main disadvantage of this technique is that a global time step corresponding to the Courant–Friedrichs–Lewy (CFL) limit in the smallest cells should be used to avoid long-term instability. In this paper, we present a time-domain transmission-line matrix (TLM) algorithm for interfacing block regions that allows the use of local time steps. Hence, cubic cells can be optimally used in all regions with substantial coarseness error reduction. Applied to both electromagnetic and heat transfer TLM models, simulation experiments are presented to show the behavior and performances of the algorithm.

Newmark local time stepping on high-performance computing architectures

In multi-scale complex media, finite element meshes often require areas of local refinement, creating small elements that can dramatically reduce the global time-step for wave-propagation problems due to the CFL condition. Local time stepping (LTS) algorithms allow an explicit time-stepping scheme to adapt the time-step to the element size, allowing near-optimal time-steps everywhere in the mesh. We develop an efficient multilevel LTS-Newmark scheme and implement it in a widely used continuous finite element seismic wave-propagation package. In particular, we extend the standard LTS formulation with adaptations to continuous finite element methods that can be implemented very efficiently with very strong element-size contrasts (more than 100x). Capable of running on large CPU and GPU clusters, we present both synthetic validation examples and large scale, realistic application examples to demonstrate the performance and applicability of the method and implementation on thousands of CPU cores and hundreds of GPUs.

A Novel Explicit-Implicit Coupled Solution Method of SWE for Long-term River Meandering Process Induced by Dambreak

Large amount of sediment deposits in the reservoir area can cause dam break, which not only leads to an immeasurable loss to the society, but also the sediments from the reservoir can be transported to generate further problems in the downstream catchment. This study aims to investigate the short-to-long term sediment transport and channel meandering process under such a situation. A coupled explicit-implicit technique based on the Euler-Lagrangian method (ELM) is used to solve the hydrodynamic equations, in which both the small and large time steps are used separately for the fluid and sediment marching. The main feature of the model is the use of the Characteristic-Based Split (CBS) method for the local time step iteration to correct the ELM traced lines. Based on the solved flow field, a standard Total Variation Diminishing (TVD) finite volume scheme is applied to solve the sediment transportation equation. The proposed model is first validated by a benchmark dambreak water flow experiment to validate the efficiency and accuracy of ELM modelling capability. Then an idealized engineering dambreak flow is used to investigate the long-term downstream channel meandering process with non-uniform sediment transport. The results showed that both the hydrodynamic and morphologic features have been well predicted by the proposed coupled model.

A comparative study of fractional step method in its quasi-implicit, semi-implicit and fully-explicit forms for incompressible flows

Purpose – The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared. Design/methodology/approach – This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split. Findings – In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations. Originality/value – A comprehensive comparison between three versions of the CBS method is provided for the first time.

Efficient construction of mock catalogs for baryon acoustic oscillation surveys

Precision measurements of the large scale structure of the Universe require large numbers of high fidelity mock catalogs to accurately assess, and account for, the presence of systematic effects. We introduce and test a scheme for generating mock catalogs rapidly using suitably derated N-body simulations. Our aim is to reproduce the large scale structure and the gross properties of dark matter halos with high accuracy, while sacrificing the details of the halo's internal structure. By adjusting global and local time-steps in an N-body code, we demonstrate that we recover halo masses to better than 0.5% and the power spectrum to better than 1% both in real and redshift space for k=1hMpc−1, while requiring a factor of 4 less CPU time. We also calibrate the redshift spacing of outputs required to generate simulated light cones. We find that outputs separated by Δ z=0.05 allow us to interpolate particle positions and velocities to reproduce the real and redshift space power spectra to better than 1% (out to k=1hMpc−1). We apply these ideas to generate a suite of simulations spanning a range of cosmologies, motivated by the Baryon Oscillation Spectroscopic Survey (BOSS) but broadly applicable to future large scale structure surveys including eBOSS and DESI. As an initial demonstration of the utility of such simulations, we calibrate the shift in the baryonic acoustic oscillation peak position as a function of galaxy bias with higher precision than has been possible so far. This paper also serves to document the simulations, which we make publicly available.

A fully coupled turbulent low-speed preconditioner for harmonic balance applications

Aeromechanical design of wind turbines requires a high-fidelity analysis toolbox that can cover a wide spectrum of flow speeds from incompressible to compressible flow regimes. As it is known, the convergence of the compressible density-based flow solvers is deteriorated when the flow lies in an essentially incompressible regime due to the disparity between the eigenvalues of the flux Jacobian matrices. This problem is usually resolved using a low-speed preconditioner that can balance the eigenvalues, thus alleviating the convergence issues in the density-based flow solver. In this paper, a low-speed preconditioner that fully couples the Reynolds-averaged Navier–Stokes equations with the one equation Spalart–Allmaras turbulence model is developed for steady as well as unsteady flows. The preconditioner incorporates the working variable of the turbulence model into the formulation, thus resulting in balanced artificial dissipation terms and modified local time-steps to ensure convergence acceleration for the compressible flow solver. The preconditioner is first applied to a set of two-dimensional steady flow cases to assess the convergence acceleration. This new preconditioner is further enhanced with an unsteady limiter that improves the convergence of the harmonic balance solver used for the simulation of unsteady periodic flows such as those that arise in wind turbine applications. Although the CPU time per iteration is increased by about 9%–11%, the present preconditioner is capable of reducing the number of required iterations by about 50%–60% in order to achieve the same level of accuracy as with the non-preconditioned solver.

Implicit unified gas-kinetic scheme for steady state solutions in all flow regimes

This paper presents an implicit unified gas-kinetic scheme (UGKS) for non-equilibrium steady state flow computation. The UGKS is a direct modeling method for flow simulation in all regimes with the updates of both macroscopic flow variables and microscopic gas distribution function. By solving the macroscopic equations implicitly, a predicted equilibrium state can be obtained first through iterations. With the newly predicted equilibrium state, the evolution equation of the gas distribution function and the corresponding collision term can be discretized in a fully implicit way for fast convergence through iterations as well. The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization method is implemented to solve both macroscopic and microscopic equations, which improves the efficiency of the scheme. Since the UGKS is a direct modeling method and its physical solution depends on the mesh resolution and the local time step, a physical time step needs to be fixed before using an implicit iterative technique with a pseudo-time marching step. Therefore, the physical time step in the current implicit scheme is determined by the same way as that in the explicit UGKS for capturing the physical solution in all flow regimes, but the convergence to a steady state speeds up through the adoption of a numerical time step with large CFL number. Many numerical test cases in different flow regimes from low speed to hypersonic ones, such as the Couette flow, cavity flow, and the flow passing over a cylinder, are computed to validate the current implicit method. The overall efficiency of the implicit UGKS can be improved by one or two orders of magnitude in comparison with the explicit one.

Sprinkler Irrigation Droplet Dynamics. II: Numerical Solution and Model Evaluation

AbstractA system of equations, with strong physical basis, was derived for sprinkler irrigation droplet dynamics in the companion paper. Numerical solution of these equations and model evaluation is discussed in this paper. With the aim of enhancing computational efficiency and robustness, the droplet dynamics equations were scaled using four characteristic variables: characteristic time, length, velocity, and density. The characteristic time, length, and velocity are derived based on consideration of the motion of a droplet falling freely, starting from rest, through a quiescent ambient air, to an eventual steady-state condition. The characteristic density was set to the density of water at standard conditions. The dimensionless system of equations was then solved numerically with a fourth-fifth order pair Runge-Kutta method capable of local error estimation and time-step size control. The numerical model was first evaluated through successful comparisons with simplified solutions derived based on more l...

Enabling local time stepping in the parallel implicit solution of reaction–diffusion equations via space-time finite elements on shallow tree meshes

For many applications, local time stepping offers an interesting and worthwhile alternative to the by now well established global time step control. In fact, local time stepping can allow for a highly detailed resolution of localized features of the solution with strongly reduced computational cost, when compared to global time step control. However local time stepping is not applicable in a straight-forward manner in the context of fully implicit time-discretizations.Here, we present a method for the efficient parallel adaptive solution of (non-linear) partial differential equations, in particular reaction–diffusion equations, using spatially adapted time step sizes in the context of a fully implicit solution strategy. Our proposed method uses a discontinuous Galerkin method in-time approach within a full space-time approach. Moreover, it is designed from scratch for efficient parallel computation. We employ shallow tree-based mesh data structures in order to ensure a low memory footprint of the adaptive meshes. By solving the time-dependent partial differential equation on a (d+1)-dimensional non-conforming mesh, space-time adaptivity is naturally achieved. In combination with a discontinuous Galerkin method in-time the size of the arising systems can be precisely controlled.We additionally introduce and discuss a stabilization scheme for space-time mortar element methods that also has a highly positive impact on the efficiency of preconditioning techniques for the arising systems of equations. We present results from extensive numerical experiments that address the question of convergence and efficiency, linear and non-linear solver performance, parallel scalability up to 2048 cores as well as accuracy for the linear heat equation and a real world, non-linear reaction–diffusion equation from the field of computational electrocardiology.

A high-order local time stepping finite volume solver for one-dimensional blood flow simulations: application to the ADAN model.

In recent years, the complexity of vessel networks for one-dimensional blood flow models has significantly increased, because of enhanced anatomical detail or automatic peripheral vasculature generation, for example. This fact, along with the application of these models in uncertainty quantification and parameter estimation poses the need for extremely efficient numerical solvers. The aim of this work is to present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on (i) a high-order finite volume type numerical scheme, (ii) a high-order treatment of the numerical solution at internal vertexes of the network, often called junctions, and (iii) an accurate LTS strategy. The accuracy of the proposed methodology is verified by empirical convergence tests. Then, the resulting LTS scheme is applied to arterial networks of increasing complexity and spatial scale heterogeneity, with a number of one-dimensional segments ranging from a few tens up to several thousands and vessel lengths ranging from less than a millimeter up to tens of centimeters, in order to evaluate its computational cost efficiency. The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant. Copyright © 2016 John Wiley & Sons, Ltd.

Comparison of Adaptive Multiresolution and Adaptive Mesh Refinement Applied to Simulations of the Compressible Euler Equations

We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in space with second order shock-capturing, and explicit time integration either with or without local time-stepping. The two methods are benchmarked for the compressible Euler equations in Cartesian geometry. As test cases a 2D Riemann problem, Lax-Liu 6, and a 3D ellipsoidally expanding shock wave have been chosen. We compare and assess their computational efficiency in terms of CPU time and memory requirements. We evaluate the accuracy by comparing the results of the adaptive computations with those obtained with the corresponding FV scheme using a regular fine mesh. We find that both approaches yield similar trends for CPU time compression for increasing number of refinement levels. MR exhibits more efficient memory compression than AMR and shows slightly enhanced convergence; however, a larger absolute overhead is measured for the tested codes.

A hybrid time stepping scheme combining explicit Runge–Kutta with implicit LU‐SGS for Navier–Stokes equations

SummaryA hybrid time stepping scheme is developed and implemented by a combination of explicit Runge–Kutta with implicit LU‐SGS scheme at the level of system matrix. In this method, the explicit scheme is applied to those grid cells of blocks that have large local time steps; meanwhile, the implicit scheme is applied to other grid cells of blocks that have smaller allowable local time steps in the same flow field. As a result, the discretized governing equations can be expressed as a compound of explicit and implicit matrix operator. The proposed method has been used to compute the steady transonic turbulent flow over the RAE 2822 airfoil. The numerical results are found to be in excellent agreement with the experimental data. In the validation case, the present scheme saved at least 50% of the memory resources compared with the fully implicit LU‐SGS. Copyright © 2015 John Wiley & Sons, Ltd.

Optimized Schwarz waveform relaxation for advection reaction diffusion equations in two dimensions

Optimized Schwarz waveform relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to mathematically hard best approximation problems of homographic functions. We study in this paper in detail the best approximation problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel (higher order) type, which can also be used in the various limits for example for the heat equation, since we include in our analysis a positive low frequency limiter both in space and time. We give for each case closed form asymptotic values for the parameters which can directly be used in implementations of these algorithms, and which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, including cases not covered by our analysis, for example decompositions with cross points. We use Q1 finite element discretizations in space and Forward and Backward Euler discretizations in time (other discretization could also have been considered, since all our analysis is at the continuous level), and in all cases, we measure performance corresponding to our analysis.