Time Step Restriction Research Articles

Improved direct integration auxiliary differential equation FDTD scheme for modeling graphene drude dispersion

In this communication, an improved direct-integration auxiliary differential equation (IM-DI-ADE) scheme is introduced for incorporating graphene's Drude dispersion into the finite difference time domain (FDTD) simulations in the GHz and THz frequency ranges. Stability analysis is performed and it is shown that the time-step stability limit of the presented IM-DI-ADE scheme coincides with the conventional non-dispersive FDTD Courant–Friedrichs–Lewy (CFL) limit and removes the additional stability stringent of the classical DI-ADE counterpart. In addition, the presented scheme does not increase the memory storage overhead. Numerical dispersion analysis is also addressed and it is shown that the presented scheme provides high accuracy. The formulation is validated by numerical tests that investigate the tunneling of electromagnetic wave through a graphene sheet and the existence of the surface plasmon polaritons (SPPs) waves created at the interface between the graphene sheet and a dielectric material.

A macroelement stabilization for mixed finite element/finite volume discretizations of multiphase poromechanics

Strong coupling between geomechanical deformation and multiphase fluid flow appears in a variety of geoscience applications. A common discretization strategy for these problems is a continuous Galerkin finite element scheme for the momentum balance equation and a finite volume scheme for the mass balance equations. When applied within a fully implicit solution strategy, however, this discretization is not intrinsically stable. In the limit of small time steps or low permeabilities, spurious oscillations in the piecewise-constant pressure field, i.e., checkerboarding, may be observed. Further, eigenvalues associated with the spurious modes will control the conditioning of the matrices and can dramatically degrade the convergence rate of iterative linear solvers. Here, we propose a stabilization technique in which the mass balance equations are supplemented with stabilizing flux terms on a macroelement basis. The additional stabilization terms are dependent on a stabilization parameter. We identify an optimal value for this parameter using an analysis of the eigenvalue distribution of the macroelement Schur complement matrix. The resulting method is simple to implement and preserves the underlying sparsity pattern of the original discretization. Another appealing feature of the method is that mass is exactly conserved on macroelements, despite the addition of artificial fluxes. The efficacy of the proposed technique is demonstrated with several numerical examples.

An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields

We present a new implicit asymptotic preserving time integration scheme for charged-particle orbit computation in arbitrary electromagnetic fields. The scheme is built on the Crank-Nicolson integrator and continues to recover full-orbit motion in the small time-step limit, but also recovers all the first-order guiding center drifts as well as the correct gyroradius when stepping over the gyration time-scale. In contrast to previous efforts in this direction, the new scheme also features exact energy conservation. In the derivation of the scheme, we find that a new numerical time-scale is introduced. This scale is analyzed and the resulting restrictions on time-step are derived. Based on this analysis, we develop an adaptive time-stepping strategy the respects these constraints while stepping over the gyration scale when physically justified. It is shown through numerical tests on single-particle motion that the scheme's energy conservation property results in tremendous improvements in accuracy, and that the scheme is able to transition smoothly between magnetized and unmagnetized regimes as a result of the adaptive time-stepping.

Efficient linear energy dissipative difference schemes for the coupled nonlinear damped space fractional wave equations

In this paper, several efficient energy dissipative linear difference schemes are presented and analyzed for solving the coupled nonlinear damped fractional wave equations. First, the weighted shifted Grünwald difference formula is used to approach the considered fractional system in space direction. Then, we apply second-order centered difference scheme and invariant energy quadratization Crank-Nicolson scheme to discrete the resulting system in time direction, respectively. Subsequently, the convergence and stability of the proposed schemes are discussed. By using the discrete energy method and a ‘cut-off’ function technique, it is proven that the suggested schemes attain the convergence orders of O(Δt2+h2) in the discrete L2- and L∞- norms, without any time step restrictions dependent on the spatial mesh size. Finally, some numerical results are provided to elucidate the physical behavior and unconditional energy stability of the proposed schemes, and confirm the correctness of theoretical results.

Is it necessary to resolve the Debye length in standard or δ f PIC codes?

Numerical instabilities of standard Particle-In-Cell (PIC) codes were observed and explained very early in their development. In the zero-time step limit, these instabilities arise from the interaction between the spatial grid and the (artificial) particle shape functions. δf PIC codes, which have recently been especially popular in gyrokinetic simulations, suffer from similar instabilities. In the zero time step limit, the numerical stabilities of standard and δf methods are equivalent. Numerical instabilities arise when the simulation grid does not “resolve the Debye length,” but many modern PIC codes use relatively high order shape functions, and as a result, the worst-case numerical growth rates are undetectably small; in addition, some codes use energy-conserving methods which usually prevent this numerical instability from arising. Similarly, a numerical instability was found in a gyrokinetic δf code using a first-order shape function; we show that this is related to the usual PIC numerical instability. In the gyrokinetic case, where waves have acoustic dispersion at a large wavenumber, increasing the grid resolution actually increases the growth rate of the numerical instability, and the prescription of resolving an effective Debye length is not applicable. However, using higher order shape functions is still an effective remedy.

A second order all Mach number IMEX finite volume solver for the three dimensional Euler equations

This article deals with the development of a numerical method for the compressible Euler system valid for all Mach numbers: from extremely low to high regimes. In classical fluid dynamic problems, one faces both situations in which the flow is subsonic, and consequently acoustic waves are very fast compared to the velocity of the fluid, and situations in which the fluid moves at high speed and compressibility may generate shock waves. Standard explicit fluid solvers such as Godunov method fail in the description of both flows due to time step restrictions caused by the stiffness of the equations which leads to prohibitive computational costs. In this work, we develop a new method for the full Euler system of gas dynamics based on partitioning the equations into a fast and a low scale. Such a method employs a time step which is independent of the speed of the pressure waves and works uniformly for all Mach numbers. Cell centered discretization on Cartesian meshes is proposed. Numerical results up to the three dimensional case show the accuracy, the robustness and the effectiveness of the proposed approach.

An ALE method for simulations of axisymmetric elastic surfaces in flow

SummaryThe dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in‐plane stretching and out‐of‐plane bending, possibly accompanied by a surface tension force. In this work, we present a novel arbitrary Lagrangian‐Eulerian (ALE) method to simulate such elastic surfaces immersed in Navier‐Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can, therefore, be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions, which are present in a wide range of biophysical problems. We formulate axisymmetric elastic surface forces and propose a discretization with surface finite‐differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. We show in several numerical test cases that accurate results can be achieved at computational times on the order of minutes on a single core CPU. We demonstrate two state‐of‐the‐art applications which to our knowledge cannot be simulated with any other numerical method so far: we present first simulations of the observed shape oscillations of novel microswimming shells and the uniaxial compression of the cortex of a biological cell during an AFM experiment.

A high-order accurate scheme for Maxwell's equations with a Generalized Dispersive Material (GDM) model and material interfaces

A high-order accurate scheme for solving the time-domain dispersive Maxwell's equations and material interfaces is described. Maxwell's equations are solved in second-order form for the electric field. A generalized dispersive material (GDM) model is used to represent a general class of linear dispersive materials and this model is implemented in the time-domain with the auxiliary differential equation (ADE) approach. The interior updates use our recently developed second-order and fourth-order accurate single-stage three-level space-time finite-difference schemes, and this paper extends these schemes to treat interfaces between different dispersive materials. Composite overlapping grids are used to treat complex geometry, with Cartesian grids generally covering most of the domain and local conforming grids representing curved boundaries and interfaces. Compatibility conditions derived from the interface jump conditions and governing equations are used to derive accurate numerical interface conditions that define values at ghost points. Although some compatibility conditions couple the equations for the ghost points in tangential directions due to mixed-derivatives, it is shown how to decouple the equations to avoid solving a larger system of equations for all ghost points on the interface. The stability of the interface approximations is studied with mode analysis and it is shown that the schemes retain close to a CFL-one time-step restriction. Numerical results are presented in two and three space dimensions to confirm the accuracy and stability of the schemes. The schemes are verified using exact solutions for a planar interface, a disk in two dimensions, and a solid sphere in three dimensions.

Stability Analysis of the Explicit Difference Scheme for Richards Equation.

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

Nonconforming finite element method for a generalized nonlinear Schrödinger equation

In this paper, an efficient nonconforming finite element method (FEM) is studied with EQ1rot element for a generalized nonlinear Schrödinger equation. First, we prove a novel result of the consistency error estimate with order O(h2) for EQ1rot element which leads to the superconvergent error estimate in broken H1-norm for semi-discrete scheme, while previous work only derive convergent results for this element. Second, a linearized backward Euler scheme is established and a time-discrete system is introduced to split the error into two parts, the temporal error and the spatial error. By using a rigorous analysis for the regularity of the time-discrete system and the proved characters of EQ1rot element, the optimal L2-error estimate is obtained without any time-step restrictions, which leads to the numerical solution can be bounded in L∞-norm by an inverse inequality unconditionally. Then, the supercloseness estimate is arrived at with the above achievements. Third, global superconvergence results are deduced through interpolated postprocessing technique. At last, numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter, and τ is the time step.

Stability analysis of hierarchical tensor methods for time-dependent PDEs

We address the question of whether it is possible to stably integrate time-dependent high-dimensional linear PDEs with hierarchical tensor methods and explicit time stepping schemes. To this end, we develop sufficient conditions for stability and convergence of tensor solutions evolving on tensor manifolds with constant rank. We also argue that the applicability of PDE solvers with explicit time-stepping may be limited by time-step restriction dependent on the dimension of the problem. Numerical applications are presented and discussed for variable coefficients linear hyperbolic and parabolic PDEs.

Multigrid dual-time-stepping lattice Boltzmann method.

The lattice Boltzmann method often involves small numerical time steps due to the acoustic scaling (i.e., scaling between time step and grid size) inherent to the method. In this work, a second-order dual-time-stepping lattice Boltzmann method is proposed in order to avoid any time-step restriction. The implementation of the dual time stepping is based on an external source in the lattice Boltzmann equation, related to the time derivatives of the macroscopic flow quantities. Each time step is treated as a pseudosteady problem. The convergence rate of the steady lattice Boltzmann solver is improved by implementing a multigrid method. The developed solver is based on a two-relaxation time model coupled to an immersed-boundary method. The reliability of the method is demonstrated for steady and unsteady laminar flows past a circular cylinder, either fixed or towed in the computational domain. In the steady-flow case, the multigrid method drastically increases the convergence rate of the lattice Boltzmann method. The dual-time-stepping method is able to accurately reproduce the unsteady flows. The physical time step can be freely adjusted; its effect on the simulation cost is linear, while its impact on the accuracy follows a second-order trend. Two major advantages arise from this feature. (i) Simulation speed-up can be achieved by increasing the time step while conserving a reasonable accuracy. A speed-up of 4 is achieved for the unsteady flow past a fixed cylinder, and higher speed-ups are expected for configurations involving slower flow variations. Significant additional speed-up can also be achieved by accelerating transients. (ii) The choice of the time step allows us to alter the range of simulated timescales. In particular, increasing the time step results in the filtering of undesired pressure waves induced by sharp geometries or rapid temporal variations, without altering the main flow dynamics. These features may be critical to improve the efficiency and range of applicability of the lattice Boltzmann method.

A novel three-level time-split MacCormack scheme for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term

ABSTRACT This study presents a three-level explicit time-split MacCormack method to compute approximate solutions of two-dimensional time-dependent linear convection-diffusion-reaction equations with source term. The difference operators split the two-dimensional problem into two pieces so that each subproblem is easily solvable using the original MacCormack approach. Second order accuracy in time and fourth-order convergence in space are achieved by the application of the Taylor series expansion. The proposed algorithm minimizes the computational time, computer memory requirement and is easy to implement. Under a suitable time-step restriction, both stability and error estimates of the numerical scheme are deeply analysed in -norm. Numerical evidences which confirm the theoretical analysis are considered and discussed.

An immersed boundary projection method for simulating the inextensible vesicle dynamics

We develop an immersed boundary projection method (IBPM) based on an unconditionally energy stable scheme to simulate the vesicle dynamics in a viscous fluid. Utilizing the block LU decomposition of the algebraic system, a novel fractional step algorithm is introduced by decoupling all solution variables, including the fluid velocity, fluid pressure, and the elastic tension. In contrast to previous works, the present method preserves both the fluid incompressibility and the interface inextensibility at a discrete level simultaneously. In conjunction with an implicit discretization of the bending force, the present method alleviates the time-step restriction, so the numerical stability is assured by non-increasing total discrete energy during the simulation. The numerical algorithm takes a linearithmic complexity by using preconditioned GMRES and FFT-based solvers. The grid convergence studies confirm the solution variables exhibit first-order convergence rate in L2-norm. We demonstrate the numerical results of the vesicle dynamics in a quiescent fluid, Poiseuille flow, and shear flow, which are congruent with the results in the literature.

On Energy Stable, Maximum-Principle Preserving, Second-Order BDF Scheme with Variable Steps for the Allen--Cahn Equation

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:=\tau_k/\tau_{k-1}<(3+\sqrt{17})/2\approx3.561.$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

Chebyshev pseudospectral approximation of two dimensional fractional Schrodinger equation on a convex and rectangular domain

In this article, the authors report the Chebyshev pseudospectral method for solving twodimensional nonlinear Schrodinger equation with fractional order derivative in time and space both. The modified Riemann-Liouville fractional derivatives are used to define the new fractional derivatives matrix at CGL points. Using the Chebyshev fractional derivatives matrices, the given problem is reduced to a diagonally block system of nonlinear algebraic equations, which will be solved using Newton's Raphson method. The proposed methods have shown error analysis without any dependency on time and space step restrictions. Some model examples of the equations, defined on a convex and rectangular domain, have tested with various values of fractional order <i>α</i> and <i>β</i>. Moreover, numerical solutions are demonstrated to justify the theoretical results.

Real-time hybrid simulation using analogue electronic computer technology

In the field of structural dynamics, real-time hybrid simulation (RTHS) continues to receive increased interest from researchers conducting component testing incorporating system-level behaviour. Increased digital computing power has allowed advances in RTHS. However, limitations in RTHS will persist due to the effects of discrete errors caused by quantisation and corresponding numerical integration time step limitations, which can limit the bandwidth and nonlinearities of the system studied. In this paper, an analogue electronic computer is constructed to conduct RTHS of a modelled base-isolated structure with a physically tested viscous damping device. The analogue computer models a two-degree-of-freedom (2DOF) structure and solves the equations of motion required in RTHS testing. Results of the RTHS tests using the analogue computer are compared to RTHS tests implemented with a digital computer in order to validate the proposed analogue RTHS method. Further applications of RTHS testing with analogue computing including high-frequency dynamic testing are discussed.

Real-time hybrid simulation using analogue electronic computer technology

In the field of structural dynamics, real-time hybrid simulation (RTHS) continues to receive increased interest from researchers conducting component testing incorporating system-level behaviour. Increased digital computing power has allowed advances in RTHS. However, limitations in RTHS will persist due to the effects of discrete errors caused by quantisation and corresponding numerical integration time step limitations, which can limit the bandwidth and nonlinearities of the system studied. In this paper, an analogue electronic computer is constructed to conduct RTHS of a modelled base-isolated structure with a physically tested viscous damping device. The analogue computer models a two-degree-of-freedom (2DOF) structure and solves the equations of motion required in RTHS testing. Results of the RTHS tests using the analogue computer are compared to RTHS tests implemented with a digital computer in order to validate the proposed analogue RTHS method. Further applications of RTHS testing with analogue computing including high-frequency dynamic testing are discussed.

Semi-implicit operator splitting for the simulation of Herschel–Bulkley flows with smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) has become a popular numerical framework of choice for simulating free-surface flows, mainly for Newtonian fluids. The topic regarding the simulation of non-Newtonian free-surface flows, however, remains relatively untouched due to difficulties regarding the computation of viscous forces. In previous approaches, the viscous forces acting on each SPH particle were computed explicitly. Non-Newtonian fluids such as Herschel–Bulkley fluids, the effective viscosity between yielded and unyielded regions can differ by several orders of magnitudes; imposing severe time step restrictions for the simulation for explicit methods. Numerically, this can be seen as a stiff problem. We propose a semi-implicit time-stepping approach where the viscous forces are computed implicitly, within the context of SPH. We demonstrate the convergence of the method via a simple 2D test case.

Unconditionally optimal error estimates of a new mixed FEM for nonlinear Schrödinger equations

In this paper, a new mixed finite element scheme in space and a linearized backward Euler scheme in time are presented and investigated for the nonlinear Schrodinger equations. By introducing a suitable time-discrete system, both the errors in L2- and H1-norms for the original variable and L2-norm for the flux variable are derived without any time-step restriction, while previous works always required certain conditions between time step and space size. Finally, some numerical results are provided to verify the theoretical analysis.