Stability of the mixed time partitioning methods in relation to the size of time step

In this paper we focus on stability of a mixed time partitioning methods in relation to time step size which is using in numerical modelling of two-component alloys solidification. We present the numerical integration methods to solve solidification problems in a fast and accurate way. Our approach exploits the fact that physical processes inside a mould are of different nature than those in a solidifying cast. As a result different time steps can be used to run computations within both sub-domains. Because processes that are modeled in the cast sub-domain are more dynamic they require very fine-grained time step. On the other hand a heat transfer within the mould sub-domain is less intense, and thus coarse-grained step is sufficient to guarantee desired precision of computations. We propose using a fixed time step in the cast and its integer multiple in other parts of mould. We use one-step explicit and implicit time integration Θ schemes. These time integration schemes are applied to equations obtained after spatial discretization. The implicit scheme is unconditionally stable, but stability of the explicit scheme depends on the size of time step. Critical time step size can be determined on the basis of eigenvalues of the amplification matrix that depend on the material properties, size and type of the finite element. In this work we present the manner of determining the critical time step and its affect on the course of numerical simulation of solidification. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant curves for variable step size integrators

The behaviour of one-step methods with variable step size applied to is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.The first author was partially supported by NSF Grant DMS87-19952 during his stay at UCLA.     

A discontinuous Galerkin method for time fractional diffusion equations with variable coefficients

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order μ ∈ (0, 1) with variable coefficients. For the spatial discretization, we apply the standard continuous Galerkin method of total degree ≤ 1 on each spatial mesh elements. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval (0, T) and a spatial domain Ω, our analysis suggest that the error in \(L^{2}\left ((0,T),L^{2}({\Omega })\right )\)-norm is \(O(k^{2-\frac {\mu }{2}}+h^{2})\) (that is, short by order \(\frac {\mu }{2}\) from being optimal in time) where k denotes the maximum time step, and h is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal O(k² + h²) error bound in the stronger \(L^{\infty }\left ((0,T),L^{2}({\Omega })\right )\)-norm. Variable time steps are used to compensate the singularity of the continuous solution near t = 0.     

Stability criteria for differential equations with variable time delays

Time delays are an important aspect of mathematical modelling, but often result in highly complicated equations which are difficult to treat analytically. In this paper it is shown how careful application of certain undergraduate tools such as the Method of Steps and the Principle of the Argument can yield significant results. Certain delay differential equations arising in population dynamics may serve as good teaching examples for these methods. The determination of linear stability properties for an ordinary differential equation with a varying time delay is carried out through discrete point analysis, either by seeking explicit solutions or leading to the consideration of a difference equation and the roots of a characteristic polynomial. Numerical simulations carried out using MATLAB Simulink are compared to the analytical solutions, and computation is also used to suggest extensions to some results.     

A modified variable time step method for solving ice melting problem

A modified numerical method was used by authors for solving 1D Stefan problem. In this paper a modified method is proposed with difference formulae and different methods of calculating the variable time step, which are deduced from Taylor series expansions of different conditions at the boundary. Also an extrapolation formula for the solution at the first point at the right of the computational domain is proposed. The numerical results are compared with those obtained from other methods.     

Solving the order reduction phenomenon in variable step size quasi-consistent Nordsieck methods

The phenomenon is studied of reducing the order of convergence by one in some classes of variable step size Nordsieck formulas as applied to the solution of the initial value problem for a first-order ordinary differential equation. This phenomenon is caused by the fact that the convergence of fixed step size Nordsieck methods requires weaker quasi-consistency than classical Runge-Kutta formulas, which require consistency up to a certain order. In other words, quasi-consistent Nordsieck methods on fixed step size meshes have a higher order of convergence than on variable step size ones. This fact creates certain difficulties in the automatic error control of these methods. It is shown how quasi-consistent methods can be modified so that the high order of convergence is preserved on variable step size meshes. The regular techniques proposed can be applied to any quasi-consistent Nordsieck methods. Specifically, it is shown how this technique performs for Nordsieck methods based on the multistep Adams-Moulton formulas, which are the most popular quasi-consistent methods. The theoretical conclusions of this paper are confirmed by the numerical results obtained for a test problem with a known solution.     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).     

Interval methods of Adams-Bashforth type with variable step sizes

Numerical Algorithms - In a number of our previous papers, we have proposed interval versions of multistep methods (explicit and implicit), including interval predictor-corrector methods, in which...     

Numerical stability analysis of an acceleration scheme for step size constrained time integrators

Time integration schemes with a fixed time step, much smaller than the dominant slow time scales of the dynamics of the system, arise in the context of stiff ordinary differential equations or in multiscale computations, where a microscopic time-stepper is used to compute macroscopic behaviour. We discuss a method to accelerate such a time integrator by using extrapolation. This method extends the scheme developed by Sommeijer [Increasing the real stability boundary of explicit methods, Comput. Math. Appl. 19(6) (1990) 37–49], and uses similar ideas as the projective integration method. We analyse the stability properties of the method, and we illustrate its performance for a convection–diffusion problem.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

Particles of variable size

A problem of all particle methods is that they produce large vacuum regions when they are applied to a free gas flow, for example. With the approach recently proposed by the author [Numer. Math. (1997) 76: 111–142], this difficulty can be avoided. One can let the particles adapt their size to the local state of the fluid. How, is described in the present article. The diameter as an additional degree of freedom strongly improves the performance of the numerical methods based on this particle model. Received November 22, 1996 / Revised version received March 30, 1998     

Computational inelasticity for loading conditions on multiple time scales by adaptive step size control

This contribution proposes an algorithm based on adaptive step size control for the simulation of inelastic solids and structures undergoing loading conditions at multiple time scales. Adaptivity in time integration of viscoelastic constitutive laws is directed by an refinement indicator which is constructed from integrators of different order, here a fourth-order Runge-Kutta (RK) method and linear Backward-Euler. The key novel aspect is that by virtue of an recently established consistency condition the higher order methods, p ≥ 2, can achieve their full nominal order without fulfilling the weak form of balance of linear momentum in the RK stages, but only at the end of the time interval. A representative numerical example illustrates the performance of the present adaptive method and underpins the computational savings compared with uniform time step sizes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A novel variable step size fractional order incremental conductance algorithm to maximize power tracking of fuel cells

The maximum power tracking of Proton Exchange Membrane Fuel Cells (PEMFC) is important for the optimization of fuel cell system design. It is necessary to operate a fuel cell at maximum power to ensure full efficiency. This study presents a novel Fractional Order Incremental Conductance Algorithm (FOINC) with variable step size control which can be used for maximum power point tracking in the design of fuel cells. The method has high maximum power point tracking speed and good steady-state response, and does not require extra sensing elements for different fuel cell equipment. When compared to the traditional Incremental Conductance (INC) and Perturbation and Observation (P&O) methods, the system simulation results show the method to be feasible and effective.     

Stability of impulsive control systems with variable times

In this paper, cone-valued Lyapunov functions are employed to study the impulsive control system with variable times. The stability criteria on the non-zero solution of the impulsive control system are given by the cone-valued Lyapunov functions and the results of the controllability on the control system are also obtained.     

On Stability of Symplectic Algorithms

1.FundamentalDeflnitionsLemma1.Thesolutionofalinearoofinarydtherentialequationwithcon8tantcoeffcientY=AYissta6leifalleigenvalue8ofAhaven0nP6sitivercalpartsandtheeigenvalueswithnullrealpartaresingleroots0ftheminimalp0lynomial.,/P\ThelinearHamiltoniansystemcanbeden0tedasZ=JSZwhereZ=(q),J=(ELs),andtheHamiltonianfuncti0nH(z)=ty.Lemma2.Thesolution80flinearHamiltoniansy8temsarecmticallysta6leifalleigenvaluesofJShavenullrsalpartandaresinglerootsojtheminitnalp0lyno?nial.Definiti0n1.Whenthemo…     

A stochastic approximation algorithm with multiplicative step size modification

An algorithm of searching a zero of an unknown function ϕ: ℝ → ℝ is considered: x _t = x _t−1 − γ _t−1 y _t , t = 1, 2, ..., where y _t = ϕ(x _t−1) + ξ _t is the value of ϕ measured at x _t−1 and ξ _t is the measurement error. The step sizes γ _t > 0 are modified in the course of the algorithm according to the rule: γ _t = min{uγ _t−1, } if y _t−1 y _t > 0, and γ _t = dγ _t−1, otherwise, where 0 < d < 1 < u, > 0. That is, at each iteration γ _t is multiplied either by u or by d, provided that the resulting value does not exceed the predetermined value . The function ϕ may have one or several zeros; the random values ξ _t are independent and identically distributed, with zero mean and finite variance. Under some additional assumptions on ϕ, ξ _t , and , the conditions on u and d guaranteeing a.s. convergence of the sequence {x _t }, as well as a.s. divergence, are determined. In particular, if P(ξ ₁ > 0) = P (ξ ₁ < 0) = 1/2 and P(ξ ₁ = x) = 0 for any x ∈ ℝ, one has convergence for ud < 1 and divergence for ud > 1. Due to the multiplicative updating rule for γ _t , the sequence {x _t } converges rapidly: like a geometric progression (if convergence takes place), but the limit value may not coincide with, but instead, approximate one of the zeros of ϕ. By adjusting the parameters u and d, one can reach arbitrarily high precision of the approximation; higher accuracy is obtained at the expense of lower convergence rate.      

Stability properties of variable stepsize variable formula methods

Summary When variable stepsize variable formula methods (VSVFM's) are used in the solution of systems of first order differential equations instability arises sometimes. Therefore it is important to find VSVFM's whose zerostability properties are not affected by the choice of both the stepsize and the formula. The Adams VSVFM's are such methods. In this work a more general class of methods which contains the Adams VSVFM's is discussed and it is proved that the zero-stability of the class is not affected by the choice of the stepsize and of the formula.