Local Time Stepping Research Articles

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.

Implicit fully mesh-less method for compressible viscous flow calculations

A dual-time implicit mesh-less scheme is developed for solution of governing viscous flow equations. The computational efficiency of the method is enhanced by adopting accelerating techniques such as local time stepping and residual smoothing. The Taylor series least square method is used for approximation of derivatives at each node which leads to a central difference spatial discretization. The capabilities of the method are demonstrated by flow computations about standard cases at subsonic and transonic laminar flow conditions. Results are presented which indicate good agreements with the alternative numerical and experimental data. The computational time is considerably reduced when using the proposed mesh-less method compared with the explicit mesh-less and finite-volume counterparts using the same distribution of points.

High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions

We present a space–time adaptive solver for single- and multi-phase compressible flows that couples average interpolating wavelets with high-order finite volume schemes. The solver introduces the concept of wavelet blocks, handles large jumps in resolution and employs local time-stepping for efficient time integration. We demonstrate that the inherently sequential wavelet-based adaptivity can be implemented efficiently in multicore computer architectures using task-based parallelism and introducing the concept of wavelet blocks. We validate our computational method on a number of benchmark problems and we present simulations of shock-bubble interaction at different Mach numbers, demonstrating the accuracy and computational performance of the method.

Local Timestepping Techniques Using Taylor Expansion for Modeling Electromagnetic Wave Propagation With Discontinuous Galerkin-FEM

A time integration scheme based on Taylor expansion using the higher order discontinuous Galerkin finite element method (DG-FEM) is presented. The special properties of the scheme allow for using explicit time integration. Local timestepping techniques are proposed, which are efficient whenever large ratios in element size affect the performance of global explicit time integration schemes. The algorithm is introduced, its characteristics are analyzed, and benchmarks are presented.

Application and comparison of SST model in numerical simulation of the axial compressors

The shear-stress transport (SST) turbulence model is incorporated into Navier-Stokes equations to simulate a turbomachinery flowfield. A staggered finite volume method is used to make the mean flow equations and turbulence model equations strongly coupled and enhance the stability of the numerical computation. The implicit treatment of the source terms is applied to the SST model. A steady state solution is obtained using five-stage Runge-Kutta time-stepping scheme with local time stepping and residual smoothing to accelerate convergence. The wall distance d, a key parameter in the SST model, is solved by a partial differential equation. The validations of the code are conducted on rotor 37, wp11 at design and off-design conditions by comparison with measurements and the Spalart-Allmaras (SA) turbulence model. The flow within the tip is calculated with a multi-block grid.

The momentum interpolation method based on the time-marching algorithm for All-Speed flows

The time-marching approach has clear physical meaning and strict mathematical nature and has been applied in computation of compressible flows widely and extended to many uniform algorithms for All-Speed flows. Remedy for its weakness in the problem of checkerboard decoupling of pressure field for incompressible flows is envisaged with the time-marching momentum interpolation method (MIM) taken into account in this paper. Existing preconditioning methods for suppressing decoupling and time-marching MIM are analyzed for this purpose, and algorithms of time-marching MIM are proposed for steady and unsteady flows and for All-Speed flows. Asymptotic analysis shows that the supposed time-marching MIM has at least a third-order accuracy, better than the existing time-marching coupling methods, which only have an accuracy of the same order as the adopted scheme has. Effects of the time step sizes on the ability of the time-marching MIM to suppress the checkerboard pressure decoupling are particularly discussed in terms of the dual-time stepping approach, and it is revealed how the decreased sizes of either the pseudo- or physical-time step increases the possibility of decoupling and how Choi’s modification, in which the history of the interface velocity is decided by itself instead of the arithmetic average of the velocities on its adjacent nodes, removes the unphysical pressure oscillation with small size of the physical time step but leads to divergence with the pseudo-time step as well. As a remedy for the pseudo-time step, such methods are recommended as implicit methods and the local-time step method with a proposed modification of the time-marching MIM preventing accuracy loss due to very large time step size. Numerical experiments support the theoretical analyses and show the validity of the time-marching MIM proposed.

Improving the time-splitting errors of one-dimensional advection schemes in multidimensional applications

A method is presented to reduce the time-splitting errors of one-dimensional flux form advection schemes. In multidimensional applications deformational correction terms are included in each one-dimensional advection step whereby all correction terms add up to zero. With the correction terms a consistent form of a given one-dimensional advection scheme is achieved when time-splitting is performed. Positive definiteness and shape preservation of the scheme are obtained by nonlinear limitation of the advective fluxes and the deformational correction terms. Large Courant numbers are treated by applying local timestep reduction. The resulting scheme is strictly one-dimensional, mass conservative and positive definite. Shape preservation of the method is nearly achieved. Two- and three-dimensional test runs are presented demonstrating that the method yields satisfactory results. The horizontal flow in a terrain with a steep mountain ridge is investigated by utilizing a terrain following vertical coordinate. It is shown that even in steep orographical terrain the method works quite well, however, small errors occur in the shape preservation of the scheme. These errors increase with increasing spatial variation of the vertical grid mesh.

Keeping Time with Maxwell's Equations

Introducing a commercial FETD solver breaks new ground in EM field simulation. Based on the DGTD method, it allows unstructured geometry-conforming meshes to be used for the first time in transient EM field simulation. Since the underlying method doesn't require the solution of a large matrix equation, its computer memory usage is modest. Simulation speed is optimized without compromising accuracy or stability by introducing an innovative local timestepping procedure. In this procedure, small time steps are taken only where needed in small mesh elements while appropriately larger time steps are used in larger mesh elements. Furthermore, a local implicit time-stepping algorithm is employed with selected elements to further improve simulation speed. DGTD is a competitive alternative to traditional FDTD-based methods to solving Maxwell's equations in the time domain. The applications presented here include the electromagnetic pulse susceptibility of the differential lines in a laptop computer, the radar signature of a landmine under undulating ground,the TDR of a bent flex circuit, and the return loss of a connector. All of these examples involve complicated/ curved geometries where the flexibility of the unstructured meshes used in DGTD provides powerful advantages over simulation by conventional brickshaped FDTD and FIT meshes.

Pricing American options using a space-time adaptive finite difference method

American options are priced numerically using a space- and time-adaptive finite difference method. The generalized Black–Scholes operator is discretized on a Cartesian structured but non-equidistant grid in space. The space- and time-discretizations are adjusted such that a predefined tolerance level on the local discretization error is met. An operator splitting technique is used to separately handle the early exercise constraint and the solution of linear systems of equations from the finite difference discretization of the linear complementarity problem. In numerical experiments three variants of the adaptive time-stepping algorithm with and without local time-stepping are compared.

Globalized matrix-explicit Newton-GMRES for the high-order accurate solution of the Euler equations

Implicit methods for finite-volume schemes on unstructured grids typically rely on a matrix-free implementation of GMRES and an explicit first-order accurate Jacobian for preconditioning. Globalization is typically achieved using a global timestep or a CFL based local timestep. We show that robustness of the globalization can be improved by supplementing the pseudo-timestepping method commonly used with a line search method. The number of timesteps required for convergence can be reduced by using a timestep that scales with the local residual. We also show that it is possible to form the high-order Jacobian explicitly at a reasonable computational cost. This is demonstrated for cases using both limited and unlimited reconstruction. This exact Jacobian can be used for preconditioning and directly in the GMRES method. The benefits of improvements in preconditioning and the elimination of residual evaluations in the inner iterations of the matrix-free GMRES method are substantial. Computational results focus on second- and fourth-order accurate schemes with some results for the third-order scheme. Overall computational cost for the matrix-explicit method is lower than the matrix-free method for all cases. The fourth-order matrix-explicit scheme is a factor of 1.6–3 faster than the matrix-free scheme while requiring about 50–100% more memory.

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.

An asynchronous and parallel time-marching method: Application to three-dimensional MHD simulation of solar wind

An asynchronous and parallel time-marching method for three-dimensional (3D) time-dependent magnetohydrodynamic (MHD) simulation is used for large-scale solar wind simulation. It uses different local time steps in the corona and the heliosphere according to the local Courant-Friedrichs-Levy (CFL) conditions. The solar wind background with observed solar photospheric magnetic field as input is first presented. The simulation time for the background solar wind by using the asynchronous method is <1/6 of that by using the normal synchronous time-marching method with the same computation precision. Then, we choose the coronal mass ejection (CME) event of 13 November, 2003 as a test case. The time-dependent variations of the pressure and the velocity configured from a CME model at the inner boundary are applied to generate transient structures in order to study the dynamical interaction of a CME with the background solar wind flow between 1 and 230 Rs. This time-marching method is very effective in terms of computation time for large-scale 3D time-dependent numerical MHD problem. In this validation study, we find that this 3D MHD model, with the asynchronous and parallel time-marching method, provides a relatively satisfactory comparison with the ACE spacecraft observations at L1 point.

Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems

We present a new non-conforming space–time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method.

A visual incompressible magneto‐hydrodynamics solver with radiation, mass, and heat transfer

AbstractA visual two‐dimensional (2D) nonlinear magneto‐hydrodynamics (MHD) code that is able to solve steady state or transient charged or neutral convection problems under the radiation, mass, and heat transfer effects is presented. The flows considered are incompressible and the divergence conditions on the velocity and magnetic fields are handled by similar relaxation schemes in the form of pseudo‐iterations between the real time levels. The numerical method utilizes a matrix distribution scheme that runs on structured or unstructured triangular meshes. The time‐dependent algorithm developed here utilizes a semi‐implicit dual time stepping technique with multistage Runge‐Kutta (RK) algorithm. It is possible for the user to choose different normalizations (natural, forced, Boussinesq, Prandtl, double‐diffusive and radiation convection) automatically. The code is visual and runs interactively with the user. The graphics algorithms work multithreaded and allow the user to follow certain flow features (color graphs, vector graphs, one‐dimensional profiles) during runs, see (Comput. Fluids 2007; 36:961–973) for details. With the code presented here nonlinear steady or time‐dependent evolution of heated and stratified neutral and charged liquids, convection of mixture of neutral and charged gases, double‐diffusive and salinity natural convection flows with internal heat generation/absorption and radiative heat transfer flows can be investigated. In addition, the numerical method (combining concentration, radiation, heat transfer, and MHD effects) takes the advantage of local time stepping and employs simplified residual jacobian matrix to increase pseudo‐convergence rate. This code is currently being improved to simulate three‐dimensional problems with parallel processing. Copyright © 2009 John Wiley &amp; Sons, Ltd.

A local time-stepping Discontinuous Galerkin algorithm for the MHD system

This work is devoted to the simulation of the Magneto-Hydro-Dynamics (MHD) equations on unstructured meshes. The starting point is the Discontinuous Galerkin (DG) method, semi-discrete in space. For the time integration, we choose the Adams-Bashforth scheme. This scheme allows very easily local time-stepping on the smallest cells of the mesh. Finally, we present several numerical experiments.

Space–time adaptive multiresolution methods for hyperbolic conservation laws: Applications to compressible Euler equations

Adaptive strategies in space and time allow considerable speed-up of finite volume schemes for conservation laws, while controlling the accuracy of the discretization. In this paper, a multiresolution technique for finite volume schemes with explicit time discretization is presented. An adaptive grid is introduced by suitable thresholding of the wavelet coefficients, which maintains the accuracy of the finite volume scheme of the regular grid. Further speed-up is obtained by local scale-dependent time stepping, i.e., on large scales larger time steps can be used without violating the stability condition of the explicit scheme. Furthermore, an estimation of the truncation error in time, using embedded Runge–Kutta type schemes, guarantees a control of the time step for a given precision. The accuracy and efficiency of the fully adaptive method is illustrated with applications for compressible Euler equations in one and two space dimensions.

Trajectory Optimization for DDE Models of Supercavitating Underwater Vehicles

Abstract In this study we first develop a flight mechanics model for supercavitating vehicles, which is formulated to account for the dependence of the cavity shape from the past history of the system. This mathematical model is governed by a particular class of delay differential equations, featuring time delays on the states of the system. Next, flight trajectories and maneuvering strategies for supercavitating vehicles are obtained by solving an optimal control problem, whose solution, given a cost function and general constraints and bounds on states and controls, yields the control time histories that maneuver the vehicle according to a desired strategy, together with the associated flight path. The optimal control problem is solved using a novel direct multiple shooting approach, which is formulated to properly handle conditions dictated by the delay differential equation formulation governing the dynamic behavior of the vehicle. Specifically, the new formulation enforces the state continuity line conditions in a least-squares sense using local interpolations, which supports local time stepping and drastically reduces the number of optimization unknowns. Examples of maneuvers and resulting trajectories demonstrate the effectiveness of the proposed methodology and the generality of the formulation. The results are also compared with those obtained from a previously developed model governed by ordinary differential equations to highlight the differences and demonstrate the need for the current formulation.

Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction–diffusion systems

Spatially two-dimensional, possibly degenerate reaction–diffusion systems, with a focus on models of combustion, pattern formation and chemotaxis, are solved by a fully adaptive multiresolution scheme. Solutions of these equations exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort on zones of strong variation. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree (“quadtree”), whose leaves are the non-uniform finite volumes on whose borders the numerical divergence is evaluated. By a thresholding procedure, namely the elimination of leaves with solution values that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen such that the total error of the adaptive scheme is of the same order as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. For scalar equations, this strategy has the advantage that consistence with a CFL condition is always enforced. Numerical experiments with five different scenarios, in part with local time stepping, illustrate the effectiveness of the adaptive multiresolution method. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.

Acceleration of steady-state lattice Boltzmann simulations for exterior flows

The simulation of a stationary fluid flow past an obstacle by the lattice Boltzmann method (LBM) in two dimensions is discussed. The combination of second-order expressions for far-field boundary conditions and a suitable treatment of the no-slip boundary condition at the obstacle surface with the nested grid-refinement technique can be applied to the LBM, resulting in a highly efficient method for the treatment of exterior flows. Furthermore, via replacing the nested time stepping by local time stepping, the resolution process can be substantially accelerated. A highly accurate drag coefficient was used to make an error assessment for various no-slip boundary conditions imposed on the obstacle surface. The analysis showed that the equilibrium method for treating the no-slip conditions is superior to halfway bounce-back and full-way bounce-back no-slip conditions when the relaxation time tau=1 . Also a tau -dependence test was made to evaluate the performance of different methods in the treatment of the no-slip boundary conditions.

A high‐order discontinuous Galerkin method with time‐accurate local time stepping for the Maxwell equations

AbstractWe present an explicit numerical method to solve the time‐dependent Maxwell equations with arbitrary high order of accuracy in space and time on three‐dimensional unstructured tetrahedral meshes. The method is based on the discontinuous Galerkin finite element approach, which allows for discontinuities at grid cell interfaces. The computation of the flux between the grid cells is based on the solution of generalized Riemann problems, which provides simultaneously a high‐order accurate approximation in space and time. Within our approach, we expand the solution in a Taylor series in time, where subsequently the Cauchy–Kovalevskaya procedure is used to replace the time derivatives in this series by space derivatives. The numerical solution can thus be advanced in time in one single step with high order and does not need any intermediate stages, as needed, e.g. in classical Runge–Kutta‐type schemes. This locality in space and time allows the introduction of time‐accurate local time stepping (LTS) for unsteady wave propagation. Each grid cell is updated with its individual and optimal time step, as given by the local Courant stability criterion. On the basis of a numerical convergence study we show that the proposed LTS scheme provides high order of accuracy in space and time on unstructured tetrahedral meshes. The application to a well‐acknowledged test case and comparisons with analytical reference solutions confirm the performance of the proposed method. Copyright © 2008 John Wiley &amp; Sons, Ltd.