Sub-iteration leads to accuracy and stability enhancements of semi-implicit schemes for the Navier–Stokes equations

We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes.     

Mass,momentum and energy conserving (MaMEC) discretizations on general grids for the compressible Euler and shallow water equations

The paper explains a method by which discretizations of the continuity and momentum equations can be designed, such that they can be combined with an equation of state into a discrete energy equation. The resulting ‘MaMEC’ discretizations conserve mass, momentum as well as energy, although no explicit conservation law for the total energy is present. Essential ingredients are (i) discrete convection that leaves the discrete energy invariant, and (ii) discrete consistency between the thermodynamic terms. Of particular relevance is the way in which finite volume fluxes are related to nodal values. The method is an extension of existing methods based on skew-symmetry of discrete operators, because it allows arbitrary equations of state and a larger class of grids than earlier methods.The method is first illustrated with a one-dimensional example on a highly stretched staggered grid, in which the MaMEC method calculates qualitatively correct results and a non-skew-symmetric finite volume method becomes unstable. A further example is a two-dimensional shallow water calculation on a rectilinear grid as well as on an unstructured grid. The conservation of mass, momentum and energy is checked, and losses are found negligible up to machine accuracy.     

Conservative, unconditionally stable discretization methods for Hamiltonian equations, applied to wave motion in lattice equations modeling protein molecules

A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics.Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives.This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.     

On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations

Finite element and finite difference discretizations for evolutionary convection–diffusion–reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank–Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge–Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.     

A Finite Variable Difference Relaxation Scheme for hyperbolic–parabolic equations

Using the framework of a new relaxation system, which converts a nonlinear viscous conservation law into a system of linear convection–diffusion equations with nonlinear source terms, a finite variable difference method is developed for nonlinear hyperbolic–parabolic equations. The basic idea is to formulate a finite volume method with an optimum spatial difference, using the Locally Exact Numerical Scheme (LENS), leading to a Finite Variable Difference Method as introduced by Sakai [Katsuhiro Sakai, A new finite variable difference method with application to locally exact numerical scheme, Journal of Computational Physics, 124 (1996) pp. 301–308.], for the linear convection–diffusion equations obtained by using a relaxation system. Source terms are treated with the well-balanced scheme of Jin [Shi Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, Mathematical Modeling Numerical Analysis, 35 (4) (2001) pp. 631–645]. Bench-mark test problems for scalar and vector conservation laws in one and two dimensions are solved using this new algorithm and the results demonstrate the efficiency of the scheme in capturing the flow features accurately.     

多元高斯声束衍射修正下液体非线性系数检测方法

有效检测材料的非线性系数β是非线性超声评价材料力学性能及早期疲劳损伤等的前提和关键,针对当前的有限幅值法仅适用于有限孔径探头近场测量的现状,本论文研究了一种不受检测距离影响的测量方法。为抑制实际检测过程中声能的损失和声场扩散对测量结果的影响,对基波和二次谐波检测值进行衍射和衰减修正,其中利用多元高斯声束精确计算二次谐波衍射系数,在此基础上计算非线性系数β以消除其与理想的平面波推导结果间的差异,提高不同距离下测量值的精度。针对水的非线性系数β进行了仿真分析和实验验证,结果均显示本文方法相比于传统有限幅值法具有明显的精度优势,且表明该方法测量材料的β不受检测距离的影响,为放宽非线性超声检测的应用条件提供了理论依据。      

Klein-Gordon方程初边值问题的一种新的差分方法

对非线性Kiein-Gordon方程的初边值问题提出了一种能量守恒差分格式。证明了该格式的收敛性和稳定性。并给出数值计算结果。     

高功率微波土壤击穿的数值验证研究

高功率微波在土壤中传播时, 会引起土壤击穿电离而导致土壤电阻率的非线性变化, 土壤电阻率的变化又将反作用于传播过程, 加剧高功率微波衰减, 影响其能量传输效率. 通过对土壤动态电离过程的分析, 结合Maxwell方程组构建了高功率微波土壤传播模型, 采用时域有限差分法对该模型进行数值验证. 数值结果显示了高功率微波在土壤中传播、衰减等物理图像, 以及土壤电阻率的非线性变化过程. 理论分析验证了这些数值结果.     

A local energy-preserving scheme for Zakharov system

In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local timespace region which is independent of the boundary condition and more essential than the global energy conservation law.Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.     

非线性波动方程的新数值迭代方法

提出了一种新的求解非线性波动方程的数值迭代法,它是一种半解析的方法.与完全的数值计算方法扰法相比,它能够考虑各阶谐波的相互作用,且能够满足能量守恒定律.用它研究了非线性声波在液体中的传播性质,结果表明,在微扰法适用的声强范围内迭代法也适用,在微扰法不适用的一个较宽的声强范围内迭代法依然适用.     

An efficient flexible-order model for 3D nonlinear water waves

The flexible-order, finite difference based fully nonlinear potential flow model described in [H.B. Bingham, H. Zhang, On the accuracy of finite difference solutions for nonlinear water waves, J. Eng. Math. 58 (2007) 211–228] is extended to three dimensions (3D). In order to obtain an optimal scaling of the solution effort multigrid is employed to precondition a GMRES iterative solution of the discretized Laplace problem. A robust multigrid method based on Gauss–Seidel smoothing is found to require special treatment of the boundary conditions along solid boundaries, and in particular on the sea bottom. A new discretization scheme using one layer of grid points outside the fluid domain is presented and shown to provide convergent solutions over the full physical and discrete parameter space of interest. Linear analysis of the fundamental properties of the scheme with respect to accuracy, robustness and energy conservation are presented together with demonstrations of grid independent iteration count and optimal scaling of the solution effort. Calculations are made for 3D nonlinear wave problems for steep nonlinear waves and a shoaling problem which show good agreement with experimental measurements and other calculations from the literature.     

Toward textbook multigrid efficiency for fully implicit resistive magnetohydrodynamics

Multigrid methods can solve some classes of elliptic and parabolic equations to accuracy below the truncation error with a work-cost equivalent to a few residual calculations – so-called “textbook” multigrid efficiency. We investigate methods to solve the system of equations that arise in time dependent magnetohydrodynamics (MHD) simulations with textbook multigrid efficiency. We apply multigrid techniques such as geometric interpolation, full approximate storage, Gauss–Seidel smoothers, and defect correction for fully implicit, nonlinear, second-order finite volume discretizations of MHD. We apply these methods to a standard resistive MHD benchmark problem, the GEM reconnection problem, and add a strong magnetic guide field, which is a critical characteristic of magnetically confined fusion plasmas. We show that our multigrid methods can achieve near textbook efficiency on fully implicit resistive MHD simulations.     

Statistical mechanics of Arakawa’s discretizations

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa [Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1966) 119–143] which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.     

Nonlinear thermoelectric response due to energy-dependent transport properties of a quantum dot

Quantum dots are useful model systems for studying quantum thermoelectric behavior because of their highly energy-dependent electron transport properties, which are tunable by electrostatic gating. As a result of this strong energy dependence, the thermoelectric response of quantum dots is expected to be nonlinear with respect to an applied thermal bias. However, until now this effect has been challenging to observe because, first, it is experimentally difficult to apply a sufficiently large thermal bias at the nanoscale and, second, it is difficult to distinguish thermal bias effects from purely temperature-dependent effects due to overall heating of a device. Here we take advantage of a novel thermal biasing technique and demonstrate a nonlinear thermoelectric response in a quantum dot which is defined in a heterostructured semiconductor nanowire. We also show that a theoretical model based on the Master equations fully explains the observed nonlinear thermoelectric response given the energy-dependent transport properties of the quantum dot.     

Nonlinear optical properties of a Woods–Saxon quantum dot under an electric field

A theoretical study of the effect of the confining potential on the nonlinear optical properties of two dimensional quantum dots is performed. A three-parameter Woods–Saxon potential is used within the density matrix formalism. The control of confinement by three parameters and an applied electric field gives one quite an advantage in studying their effects on the nonlinear properties. The coefficients investigated include the optical rectification, second and third-harmonic generation and the change in the refractive index. Their dependence on the electric field values, dot size and the energy of the incoming photons is studied extensively.It is shown that the Woods–Saxon potential can be used to model the confinement in quantum dots with considerable success.     

Statistical mechanics of Arakawa’s discretizations

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical mechanical theories are constructed for the discrete dynamical systems arising from three discretizations due to Arakawa [Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part I. J. Comput. Phys. 1 (1966) 119–143] which conserve energy, enstrophy or both. Numerical experiments with conservative and projected time integrators show that the statistical theories accurately explain the differences observed in statistics derived from the discretizations.     

A retrieval method for nonlinear metamaterials

Metamaterials are engineered periodic structures for which it is possible to assign effective homogenized constitutive properties. In recent years, metamaterials in which the constituent elements are integrated with inherently nonlinear materials or electronic components have been considered for their potential impact on nonlinear wave propagation. As is the case with their linear counterparts, nonlinear metamaterials can also be assigned homogenized effective properties. The effective constitutive parameters of a metamaterial can be determined by a retrieval method applied to full-wave numerical simulations of a single layer of the structure. In this work, we present a transfer matrix approach that extends the retrieval of metamaterial properties to include the effective nonlinear susceptibilities. Comparisons with time-domain finite element simulations of continuous nonlinear slabs confirm the validity of this approach. The proposed approach is also applied to determine the nonlinear susceptibility of a simple nonlinear metamaterial.     

Gigantic stimulated Raman scattering in one-dimensional Mott-Hubbard insulators: A possible THz source

One-dimensional Mott-Hubbard insulators like Sr₂CuO₃, halogen-bridged Nickel chain compounds have orders-of-magnitude nonlinear optical properties compared to other one-dimensional organic or inorganic compounds. We show theoretically, that the stimulated Raman scattering susceptibility for such insulators could be order(s)-of-magnitude larger even compared to other nonlinear optical susceptibilities. The lowest two-photon state is at lower energy than the lowest one-photon state in some of these insulators. This leads to a potential for strong Stokes generation in the THz regime from these compounds. Our results and conclusions are based on exact numerical solution of finite size two-band extended Hubbard model.     

Lagrangian shock hydrodynamics on tetrahedral meshes: A stable and accurate variational multiscale approach

In the past, a number of attempts have failed to robustly compute highly transient shock hydrodynamics flows on tetrahedral meshes. To a certain degree, this is not a surprise, as prior attempts emphasized enhancing the structure of shock-capturing operators rather than focusing on issues of stability with respect to small, linear perturbations. In this work, a new method is devised to stabilize computations on piecewise-linear tetrahedral finite elements. Spurious linear modes are prevented by means of the variational multiscale approach. The resulting algorithm can be proven stable in the linearized limit of acoustic wave propagation. Starting from this solid base, the approach is generalized to fully nonlinear shock computations, by augmenting the discrete formulation with discontinuity-capturing artificial viscosities. Extensive tests in the case of Lagrangian shock dynamics of ideas gases on triangular and tetrahedral grids confirm the stability and accuracy properties of the method. Incidentally, the same tests also reveal the lack of stability of current compatible/mimetic/staggered discretizations: This is due to the presence of specific unstable modes which are theoretically analyzed and verified in computations.