Time Step Restriction Research Articles

Level set jet schemes for stiff advection equations: The semijet method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided.

Sequential and Simultaneous Model Predictive Control of a Drainage Canal Network Using an Implicit Diffusive Wave Model

AbstractModel predictive control (MPC) is an efficient approach to regulate water systems, for both water quantity and quality. It generates optimal control trajectories based on model predictions over a finite horizon. In this research, the focus is on nonlinear MPC with a nonlinear internal model and the comparison of a sequential and simultaneous optimization setup, referred to as sequential and simultaneous nonlinear model predictive control (SeNMPC, SiNMPC), for the solution of the optimum control problem. The representation of the water system in the internal model is based on the diffusive wave model. The model is integrated in time by an unconditionally stable Backward Euler scheme to avoid model instabilities and time step restrictions. This numerical robustness is essential in real-time control applications where large control time steps should not be jeopardized by local grid refinements owing to the system topology. In order to speed up the optimization, an adjoint model is set up to calculate...

A GPU accelerated level set reinitialization for an adaptive discontinuous Galerkin method

GPU accelerated high order reconstruction of signed distance function of the level set method is studied. The flow based reinitialization equation is discretized in space by using a nodal discontinuous Galerkin method on adaptive unstructured grids. Artificial diffusion with a modal decay rate based regularity estimator is used to damp out high frequency solution components near kinks, where mesh adaptivity is applied. A two rate Adams-Bashforth time integrator is developed to avoid time step restrictions resulting from artificial diffusion stabilization and local mesh refinement. Platform independence of the solver is achieved by using an extensible multi-threading programming API that allows runtime selection of different computing devices (GPU and CPU) and threading interfaces (CUDA, OpenCL and OpenMP). Overall, a highly scalable numerical scheme that preserves the simplicity of the original level set method is obtained. Performance and accuracy of the method to construct signed distance function on highly disturbed initial data with smooth and non-smooth interfaces are tested through distinct two- and three-dimensional problems.

Parallel level-set methods on adaptive tree-based grids

We present scalable algorithms for the level-set method on dynamic, adaptive Quadtree and Octree Cartesian grids. The algorithms are fully parallelized and implemented using the MPI standard and the open-source p4est library. We solve the level set equation with a semi-Lagrangian method which, similar to its serial implementation, is free of any time-step restrictions. This is achieved by introducing a scalable global interpolation scheme on adaptive tree-based grids. Moreover, we present a simple parallel reinitialization scheme using the pseudo-time transient formulation. Both parallel algorithms scale on the Stampede supercomputer, where we are currently using up to 4096 CPU cores, the limit of our current account. Finally, a relevant application of the algorithms is presented in modeling a crystallization phenomenon by solving a Stefan problem, illustrating a level of detail that would be impossible to achieve without a parallel adaptive strategy. We believe that the algorithms presented in this article will be of interest and useful to researchers working with the level-set framework and modeling multi-scale physics in general.

An unconditionally monotone numerical scheme for the two-factor uncertain volatility model

Under the assumption that two asset prices follow an uncertain volatility model, the maximal and 2 minimal solution values of an option contract are given by a two dimensional Hamilton-Jacobi-Bellman 3 (HJB) PDE. A fully implicit, unconditionally monotone nite dierence numerical scheme is developed 4 in this paper. Consequently, there are no time step restrictions due to stability considerations. The 5 discretized algebraic equations are solved using policy iteration. Our discretization method results in a 6 local objective function which is a discontinuous function of the control. Hence some care must be taken 7 when applying policy iteration. The main diculty in designing a discretization scheme is development 8 of a monotonicity preserving approximation of the cross derivative term in the PDE. We derive a hybrid 9 numerical scheme which combines use of a xed point stencil and a wide stencil based on a local coordinate 10 rotation. The algorithm uses the xed point stencil as much as possible to take advantage of its accuracy 11 and computational eciency. The analysis shows that our numerical scheme is l1 stable, consistent in 12 the viscosity sense, and monotone. Thus, our numerical scheme guarantees convergence to the viscosity 13 solution. 14

Voigt regularization for the explicit time stepping of the Hall effect term

ABSTRACTThis article studies the stabilization of the explicit time stepping methods for Hall Maxwell and incompressible Hall MHD equations. Since the Hall effect term contains two spatial derivatives, classical time-stepping schemes result in very restrictive CFL condition. We consider weakening the restrictive time step condition via Voigt regularization. We establish the energy stability result and validate our approach numerically. Numerical experiments demonstrate that we are able to take larger time steps, compared to classical schemes, at additional, negligible computational cost.

High Order Local Discontinuous Galerkin Methods for the Allen-Cahn Equation: Analysis and Simulation

In this paper, we present a local discontinuous Galerkin (LDG) method for the AllenCahn equation. We prove the energy stability, analyze the optimal convergence rate of k + 1 in L 2 norm and present the (2k + 1)-th order negative-norm estimate of the semidiscrete LDG method for the Allen-Cahn equation with smooth solution. To relax the severe time step restriction of explicit time marching methods, we construct a first order semi-implicit scheme based on the convex splitting principle of the discrete Allen-Cahn energy and prove the corresponding unconditional energy stability. To achieve high order temporal accuracy, we employ the semi-implicit spectral deferred correction (SDC) method. Combining with the unconditionally stable convex splitting scheme, the SDC method can be high order accurate and stable in our numerical tests. To enhance the efficiency of the proposed methods, the multigrid solver is adapted to solve the resulting nonlinear algebraic systems. Numerical studies are presented to confirm that we can achieve optimal accuracy of O(h k+1 ) in L 2 norm and improve the LDG solution from O(h k+1 ) to O(h 2k+1 ) with the

Numerical study of an inextensible, finite swimmer in Stokesian viscoelastic flow

A numerical investigation of an Immersed Boundary (IB) model of an effectively inextensible, finite swimmer in a Stokesian Oldroyd-B flow is presented. The swimmer model is a two-dimensional sheet of finite extent and its gait is generated by an elastic force which penalizes deviations from a target shape. A non-stiff IB method is employed to remove the impeding time step limitation induced by strong tangential forces on the swimmer. It is found that for a swimmer with a prescribed gait its mean propulsion speed decreases with increasing Deborah number De toward an apparent asymptotic minimal value. However, as the swimmer is allowed to deviate more from the target shape, the monotonic locomotion behavior with De is broken. For a sufficiently flexible swimmer, viscoelasticity can enhance locomotion but the swimmer in the viscoelastic fluid always remains slower than when it is propelling in a Newtonian fluid. Remarkably, the addition of viscoelastic stress diffusion dramatically alters the swimmer propulsion and can lead to a speed-up over the swimmer in the Newtonian fluid.

Accelerated trinomial trees applied to American basket options and American options under the Bates model

The accelerated trinomial tree (ATT) is a derivatives pricing lattice method that circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs), in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps, each of which contains an inner lattice/trinomial tree with N nonuniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, is constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk-neutral extended probabilities: extended in the sense that values can lie outside the range [0, 1] on the substep scale but aggregate to probabilities within the range [0,1] on the superstep scale. Hence, it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient or have comparable efficiency to competing implicit methods that use L time steps in pricing Black–Scholes American put options and two-dimensional American basket options. Crucially, this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs, which may be relatively complex in their implementation. We also extend ATTs to the pricing of American options under the Heston model and the Bates model in order to demonstrate the general applicability of the approach.

A space-time lower–upper symmetric Gauss–Seidel scheme for the time-spectral method

ABSTRACTThe time-spectral method (TSM) offers the advantage of increased order of accuracy compared to methods using finite-difference in time for periodic unsteady flow problems. Explicit Runge–Kutta pseudo-time marching and implicit schemes have been developed to solve iteratively the space-time coupled nonlinear equations resulting from TSM. Convergence of the explicit schemes is slow because of the stringent time-step limit. Many implicit methods have been developed for TSM. Their computational efficiency is, however, still limited in practice because of delayed implicit temporal coupling, multiple iterative loops, costly matrix operations, or lack of strong diagonal dominance of the implicit operator matrix. To overcome these shortcomings, an efficient space-time lower–upper symmetric Gauss–Seidel (ST-LU-SGS) implicit scheme with multigrid acceleration is presented. In this scheme, the implicit temporal coupling term is split as one additional dimension of space in the LU-SGS sweeps. To improve numerical stability for periodic flows with high frequency, a modification to the ST-LU-SGS scheme is proposed. Numerical results show that fast convergence is achieved using large or even infinite Courant–Friedrichs–Lewy (CFL) numbers for unsteady flow problems with moderately high frequency and with the use of moderately high numbers of time intervals. The ST-LU-SGS implicit scheme is also found to work well in calculating periodic flow problems where the frequency is not known a priori and needed to be determined by using a combined Fourier analysis and gradient-based search algorithm.

Comparison of Terrain-Following and Cut-Cell Grids Using a Nonhydrostatic Model

Abstract Terrain-following coordinates are widely used in operational models but the cut-cell method has been proposed as an alternative that can more accurately represent atmospheric dynamics over steep orography. Because the type of grid is usually chosen during model implementation, it becomes necessary to use different models to compare the accuracy of different grids. In contrast, here a C-grid finite-volume model enables a like-for-like comparison of terrain-following and cut-cell grids. A series of standard two-dimensional tests using idealized terrain are performed: tracer advection in a prescribed horizontal velocity field, a test starting from resting initial conditions, and orographically induced gravity waves described by nonhydrostatic dynamics. In addition, three new tests are formulated: a more challenging resting atmosphere case, and two new advection tests having a velocity field that is everywhere tangential to the terrain-following coordinate surfaces. These new tests present a challenge on cut-cell grids. The results of the advection tests demonstrate that accuracy depends primarily upon alignment of the flow with the grid rather than grid orthogonality. A resting atmosphere is well maintained on all grids. In the gravity waves test, results on all grids are in good agreement with existing results from the literature, although terrain-following velocity fields lead to errors on cut-cell grids. Because of semi-implicit time stepping and an upwind-biased, explicit advection scheme, there are no time step restrictions associated with small cut cells. In contradiction to other studies, no significant advantages of cut cells or smoothed coordinates are found.

High fidelity anisotropic adaptive variational multiscale method for multiphase flows with surface tension

We propose in this work an adaptive variational multiscale method for two-fluid flows with surface tension. A level set function is used to provide a precise position of the interfaces. The implementation of the surface tension in the context of the Continuum Surface Force is proposed to circumvent the capillary time step restriction. The obtained system is then solved using a new derived Variational Multiscale stabilized finite element method designed to handle the abrupt changes at the interface and large density and viscosity ratios. Combined with an a posteriori error estimator, we show that anisotropic mesh adaptation provides highly stretched elements at the interfaces and thus yield an accurate modeling framework for two-phase incompressible isothermal flows. Stable and accurate results are obtained for all two- and three-dimensional numerical examples. To the best of our knowledge, these are the first simulation results for representative time-dependent three-dimensional two-fluid flow problems using an implicit treatment of the surface tension and a dynamic unstructured anisotropic mesh adaptation.

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss–Legendre and Gauss–Legendre–Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

The eagle simulations of galaxy formation: Public release of halo and galaxy catalogues

We present the public data release of halo and galaxy catalogues extracted from the EAGLE suite of cosmological hydrodynamical simulations of galaxy formation. These simulations were performed with an enhanced version of the GADGET code that includes a modified hydrodynamics solver, time-step limiter and subgrid treatments of baryonic physics, such as stellar mass loss, element-by-element radiative cooling, star formation and feedback from star formation and black hole accretion. The simulation suite includes runs performed in volumes ranging from 25 to 100 comoving megaparsecs per side, with numerical resolution chosen to marginally resolve the Jeans mass of the gas at the star formation threshold. The free parameters of the subgrid models for feedback are calibrated to the redshift z=0 galaxy stellar mass function, galaxy sizes and black hole mass - stellar mass relation. The simulations have been shown to match a wide range of observations for present-day and higher-redshift galaxies. The raw particle data have been used to link galaxies across redshifts by creating merger trees. The indexing of the tree produces a simple way to connect a galaxy at one redshift to its progenitors at higher redshift and to identify its descendants at lower redshift. In this paper we present a relational database which we are making available for general use. A large number of properties of haloes and galaxies and their merger trees are stored in the database, including stellar masses, star formation rates, metallicities, photometric measurements and mock gri images. Complex queries can be created to explore the evolution of more than 10^5 galaxies, examples of which are provided in appendix. (abridged)

Ransom test results from various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?

The basic elliptic ill-posedness of physical models and numerical schemes for two-fluid flows is a recurring issue that has motivated the introduction of numerous possible correction strategies. In practical applications physical terms are generally present and regularize the models (viscosity, drag, surface tension, etc.). Yet, many numerical schemes were developed with the stringent and self-imposed constraint that the convective part of the models to be solved had to be hyperbolic, regardless of the type and magnitude of the particular physical regularizing terms. This leads to consider the simplest possible two-fluid “backbone” models corrected with the simplest “universal” terms to ensure hyperbolicity.Among the proposed corrections is the introduction of an interfacial pressure, either closed by algebraic relations or by supplementary evolution equations. Concurrently with the shift to hyperbolic behavior, these techniques also affect other features of systems: Kelvin–Helmholtz type instabilities are notably quenched at all scales, a highly undesirable effect in many practical situations. Less commonly recognized are also distortions in the transfers between kinetic, reversible, and irreversible energies, sometimes up to thermodynamic inconsistency.The present work aims at comparing on the standard Ransom-faucet test the results from various available hyperbolic and elliptic schemes and models against an explicit double Lagrange-plus-remap discretization of the basic elliptic, one-pressure, compressible, six-equations system (i.e. with energy equations). Four features are examined on this test: the entropy preservation, the stretched stream profile, the volume fraction discontinuity, and the unstable character of the analytical solution for the simplest backbone model.The paper highlights the fact that the convective part of two-fluid models might not be necessarily hyperbolic provided that it is physically consistent and numerically robust. Observation of published results for Ransom’s test shows that by enforcing hyperbolicity regardless of thermodynamical consistency, numerical models remove instabilities at the volume fraction discontinuity, but at the expense of distorted profiles of the stretched stream due to excessive numerical diffusion and to spurious forces in the momentum equation. The present approach provides a form of neutral starting point before including dissipative terms: robust but not excessively diffusive, with accurate capture of the stretched stream and volume fraction discontinuity for any practical mesh refinement. Moreover, and consistently with the chosen elliptic model, this numerical scheme eventually generates the elliptic instabilities for late times or fine meshes (but remains robust under the appropriate time step restrictions). It can be supplemented by any kind of small-scale regularization term in order to introduce a cut-off under which physical or numerical stability may be necessary.

Efficient High Order Semi-implicit Time Discretization and Local Discontinuous Galerkin Methods for Highly Nonlinear PDEs

In this paper, we develop a high order semi-implicit time discretization method for highly nonlinear PDEs, which consist of the surface diffusion and Willmore flow of graphs, the Cahn---Hilliard equation and the Allen---Cahn/Cahn---Hilliard system. These PDEs are high order in spatial derivatives, which motivates us to develop implicit or semi-implicit time marching methods to relax the severe time step restriction for stability of explicit methods. In addition, these PDEs are also highly nonlinear, fully implicit methods will incredibly increase the difficulty of implementation. In particular, we can not well separate the stiff and non-stiff components for these problems, which leads to traditional implicit-explicit methods nearly meaningless. In this paper, a high order semi-implicit time marching method and the local discontinuous Galerkin (LDG) spatial method are coupled together to achieve high order accuracy in both space and time, and to enhance the efficiency of the proposed approaches, the resulting linear or nonlinear algebraic systems are solved by multigrid solver. Specially, we develop a first order fully discrete LDG scheme for the Allen---Cahn/Cahn---Hilliard system and prove the unconditional energy stability. Numerical simulation results in one and two dimensions are presented to illustrate that the combination of the LDG method for spatial approximation, semi-implicit temporal integration with the multigrid solver provides a practical and efficient approach when solving this family of problems.

Unconditional long-time stability of a velocity–vorticity method for the 2D Navier–Stokes equations

We prove unconditional long-time stability for a particular velocity---vorticity discretization of the 2D Navier---Stokes equations. The scheme begins with a formulation that uses the Lamb vector to couple the usual velocity---pressure system to the vorticity dynamics equation, and then discretizes with the finite element method in space and implicit---explicit BDF2 in time, with the vorticity equation decoupling at each time step. We prove the method's vorticity and velocity are both long-time stable in the $$L^2$$L2 and $$H^1$$H1 norms, without any timestep restriction. Moreover, our analysis avoids the use of Gronwall-type estimates, which leads us to stability bounds with only polynomial (instead of exponential) dependence on the Reynolds number. Numerical experiments are given that demonstrate the effectiveness of the method.

An adaptive discretization of compressible flow using a multitude of moving Cartesian grids

We present a novel method for simulating compressible flow on a multitude of Cartesian grids that can rotate and translate. Following previous work, we split the time integration into an explicit step for advection followed by an implicit solve for the pressure. A second order accurate flux based scheme is devised to handle advection on each moving Cartesian grid using an effective characteristic velocity that accounts for the grid motion. In order to avoid the stringent time step restriction imposed by very fine grids, we propose strategies that allow for a fluid velocity CFL number larger than 1. The stringent time step restriction related to the sound speed is alleviated by formulating an implicit linear system in order to find a pressure consistent with the equation of state. This implicit linear system crosses overlapping Cartesian grid boundaries by utilizing local Voronoi meshes to connect the various degrees of freedom obtaining a symmetric positive–definite system. Since a straightforward application of this technique contains an inherent central differencing which can result in spurious oscillations, we introduce a new high order diffusion term similar in spirit to ENO-LLF but solved for implicitly in order to avoid any associated time step restrictions. The method is conservative on each grid, as well as globally conservative on the background grid that contains all other grids. Moreover, a conservative interpolation operator is devised for conservatively remapping values in order to keep them consistent across different overlapping grids. Additionally, the method is extended to handle two-way solid fluid coupling in a monolithic fashion including cases (in the appendix) where solids in close proximity do not properly allow for grid based degrees of freedom in between them.

A High-Order Asymptotic-Preserving Scheme for Kinetic Equations Using Projective Integration

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp., parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge--Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyze stability and consistency, and illustrate with numerical results.

Local Discontinuous Galerkin Method and High Order Semi-Implicit Scheme for the Phase Field Crystal Equation

In this paper, we present a local discontinuous Galerkin (LDG) method and two unconditionally energy stable schemes for the phase field crystal (PFC) equation. The semidiscrete energy stability of the LDG method is proved first. The PFC equation is a sixth order nonlinear partial differential equation (PDE), which leads to the severe time step restriction ($\Delta t=\mathcal{O}(\Delta x^6)$) of explicit time discretization methods to maintain stability. Due to this, we introduce semi-implicit first order and second order time discretization methods which are based on the convex splitting principle of a discrete energy and prove the corresponding unconditional energy stabilities. To improve the temporal accuracy, the spectral deferred correction (SDC) method and a high order semi-implicit Runge--Kutta method combining with the first--order convex splitting method are adopted for the PFC equation with constant and degenerate mobility, respectively. The equations at the implicit time level are nonlinear and ...