THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS

This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.

Numerical Stability Analysis for a Stationary and Translating Droplet at Extremely Low Viscosity Values Using the Lattice Boltzmann Method Color-GradientMulti-Component Model with Central Moments Formulation

Multicomponent models based on the Lattice Boltzmann Method(LBM)have clear advantages with respect to other approaches,such as good parallel performances and scalability and the automatic resolution of breakup and coalescence events.Multicomponent flow simulations are useful for a wide range of applications,yet many multicomponent models for LBMare limited in their numerical stability and therefore do not allow exploration of physically relevant low viscosity regimes.Here we performa quantitative study and validations,varying parameters such as viscosity,droplet radius,domain size and acceleration for stationary and translating droplet simulations for the color-gradientmethod with centralmoments(CG-CM)formulation,as this method promises increased numerical stability with respect to the non-CMformulation.We focus on numerical stability and on the effect of decreasing grid-spacing,i.e.increasing resolution,in the extremely low viscosity regime for stationary droplet simulations.The effects of small-and large-scale anisotropy,due to grid-spacing and domain-size,respectively,are investigated for a stationary droplet.The effects on numerical stability of applying a uniform acceleration in one direction on the domain,i.e.on both the droplet and the ambient,is explored into the low viscosity regime,to probe the numerical stability of the method under dynamical conditions.

Research of the CICC Stability by the Numerical Code Gandalf

The basic approach to computer analysis of the CICC in superconducting Tokamak HT-7U is given and discussed. We will apply a 1-D mathematical model (Gandalf) to investigate the stability of CICC at real operating modes of Tokamak. 1-D model can be directly adopted to follow the evolution of the zone when the energy input is large enough and the coil quenches. In this report, we will analyze the stability of typical CICC (including pure copper) and discuss effect of copper on the stability of CICC.

A Long-Time-Step-Permitting Tracer Transport Model on the Regular Latitude–Longitude Grid

If an explicit time scheme is used in a numerical model, the size of the integration time step is typically limited by the spatial resolution. This study develops a regular latitude–longitude grid-based global three-dimensional tracer transport model that is computationally stable at large time-step sizes. The tracer model employs a finite-volume flux-form semiLagrangian transport scheme in the horizontal and an adaptively implicit algorithm in the vertical. The horizontal and vertical solvers are coupled via a straightforward operator-splitting technique. Both the finite-volume scheme's onedimensional slope-limiter and the adaptively implicit vertical solver's first-order upwind scheme enforce monotonicity. The tracer model permits a large time-step size and is inherently conservative and monotonic. Idealized advection test cases demonstrate that the three-dimensional transport model performs very well in terms of accuracy, stability, and efficiency. It is possible to use this robust transport model in a global atmospheric dynamical core.

Stability conditions of explicit integration algorithms when using 3D viscoelastic artificial boundaries

Viscoelastic artificial boundaries are widely adopted in numerical simulations of wave propagation problems.When explicit time-domain integration algorithms are used,the stability condition of the boundary domain is stricter than that of the internal region due to the influence of the damping and stiffness of an viscoelastic artificial boundary.The lack of a clear and practical stability criterion for this problem,however,affects the reasonable selection of an integral time step when using viscoelastic artificial boundaries.In this study,we investigate the stability conditions of explicit integration algorithms when using three-dimensional(3D)viscoelastic artificial boundaries through an analysis method based on a local subsystem.Several boundary subsystems that can represent localized characteristics of a complete numerical model are established,and their analytical stability conditions are derived from and further compared to one another.The stability of the complete model is controlled by the corner regions,and thus,the global stability criterion for the numerical model with viscoelastic artificial boundaries is obtained.Next,by analyzing the impact of different factors on stability conditions,we recommend a stability coefficient for practically estimating the maximum stable integral time step in the dynamic analysis when using 3D viscoelastic artificial boundaries.

Investigation of stability of a tunnel in a deep coal mine in China

Stability level of tunnels that exist in an underground mine has a great influence on the safety,production and economic performance of mines.Ensuring of stability for soft-rock tunnels is an important task for deep coal mines located in high in situ stress conditions.Using the available information on stratigraphy,geological structures,in situ stress measurements and geo-mechanical properties of intact rock and discontinuity interfaces,a three-dimensional numerical model was built by using 3DEC software to simulate the stress conditions around a tunnel located under high in situ stress conditions in a coal rock mass in China.Analyses were conducted for several tunnel shapes and rock support patterns.Results obtained for the distribution of failure zones,and stress and displacement felds around the tunnel were compared to select the best tunnel shape and support pattern to achieve the optimum stability conditions.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

A DIRECT DISCONTINUOUS GALERKIN METHOD FOR TIME FRACTIONAL DIFFUSION EQUATIONS WITH FRACTIONAL DYNAMIC BOUNDARY CONDITIONS

This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions.The well-posedness for the weak solutions is studied.A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes,together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh,and then the fully discrete scheme is constructed.Furthermore,the stability and the error estimate for the full scheme are analyzed in detail.Numerical experiments are also given to illustrate the effectiveness of the proposed method.

Stability Analysis and Order Improvement for Time Domain Differential Quadrature Method

The differential quadrature method has been widely used in scientific and engineering computation.However,for the basic characteristics of time domain differential quadrature method,such as numerical stability and calculation accuracy or order,it is still lack of systematic analysis conclusions.In this paper,according to the principle of differential quadrature method,it has been derived and proved that the weighting coefficients matrix of differential quadrature method meets the important V-transformation feature.Through the equivalence of the differential quadrature method and the implicit Runge-Kutta method,it has been proved that the differential quadrature method is A-stable and s-stage s-order method.On this basis,in order to further improve the accuracy of the time domain differential quadrature method,a class of improved differential quadrature method of s-stage 2s-order has been proposed by using undetermined coefficients method and Pad´e approximations.The numerical results show that the proposed differential quadrature method is more precise than the traditional differential quadrature method.

On Dissipation and Dispersion Errors Optimization,A-Stability and SSP Properties

In a recent paper(Du and Ekaterinaris,2016)optimization of dissipation and dispersion errors was investigated.A Diagonally Implicit Runge-Kutta(DIRK)scheme was developed by using the relative stability concept,i.e.the ratio of absolute numerical stability function to analytical one.They indicated that their new scheme has many similarities to one of the optimized Strong Stability Preserving(SSP)schemes.They concluded that,for steady state simulations,time integration schemes should have high dissipation and low dispersion.In this note,dissipation and dispersion errors for DIRK schemes are studied further.It is shown that relative stability is not an appropriate criterion for numerical stability analyses.Moreover,within absolute stability analysis,it is shown that there are two important concerns,accuracy and stability limits.It is proved that both A-stability and SSP properties aim at minimizing the dissipation and dispersion errors.While A-stability property attempts to increase the stability limit for large time step sizes and by bounding the error propagations via minimizing the numerical dispersion relation,SSP optimized method aims at increasing the accuracy limits by minimizing the difference between analytical and numerical dispersion relations.Hence,it can be concluded that A-stability property is necessary for calculations under large time-step sizes and,more specifically,for calculation of high diffusion terms.Furthermore,it is shown that the oscillatory behavior,reported by Du and Ekaterinaris(2016),is due to Newton method and the tolerances they set and it is not related to the employed temporal schemes.

An Energy Stable SPH Method for Incompressible Fluid Flow

In this paper,a novel unconditionally energy stable Smoothed Particle Hydrodynamics(SPH)method is proposed and implemented for incompressible fluid flows.In this method,we apply operator splitting to break the momentum equation into equations involving the non-pressure term and pressure term separately.The idea behind the splitting is to simplify the calculation while still maintaining energy stability,and the resulted algorithm,a type of improved pressure correction scheme,is both efficient and energy stable.We show in detail that energy stability is preserved at each full-time step,ensuring unconditionally numerical stability.Numerical examples are presented and compared to the analytical solutions,suggesting that the proposed method has better accuracy and stability.Moreover,we observe that if we are interested in steady-state solutions only,our method has good performance as it can achieve the steady-state solutions rapidly and accurately.

Stable and Accurate Second-Order Formulation of the Shifted Wave Equation

High order finite difference approximations are derived for a one-dimensional model of the shifted wave equation written in second-order form. Thedomain is discretized using fully compatible summation by parts operators and theboundary conditions are imposed using a penalty method, leading to fully explicittime integration. This discretization yields a strictly stable and efficient scheme. Theanalysis is verified by numerical simulations in one-dimension. The present study isthe first step towards a strictly stable simulation of the second-order formulation ofEinstein’s equations in three spatial dimensions.

Fourth Order Difference Approximations for Space Riemann-Liouville Derivatives Based on Weighted and Shifted Lubich Difference Operators

High order discretization schemes playmore important role in fractional operators than classical ones.This is because usually for classical derivatives the stencil for high order discretization schemes is wider than low order ones;but for fractional operators the stencils for high order schemes and low order ones are the same.Then using high order schemes to solve fractional equations leads to almost the same computational cost with first order schemes but the accuracy is greatly improved.Using the fractional linear multistep methods,Lubich obtains the n-th order(n≤6)approximations of the a-th derivative(a>0)or integral(a<0)[Lubich,SIAM J.Math.Anal.,17,704-719,1986],because of the stability issue the obtained scheme can not be directly applied to the space fractional operator with a∈(1,2)for time dependent problem.By weighting and shifting Lubich’s 2nd order discretization scheme,in[Chen&Deng,SINUM,arXiv:1304.7425]we derive a series of effective high order discretizations for space fractional derivative,called WSLD operators there.As the sequel of the previous work,we further provide new high order schemes for space fractional derivatives by weighting and shifting Lubich’s 3rd and 4th order discretizations.In particular,we prove that the obtained 4th order approximations are effective for space fractional derivatives.And the corresponding schemes are used to solve the space fractional diffusion equation with variable coefficients.