Error analysis of variable stepsize Runge–Kutta methods for a class of multiply-stiff singular perturbation problems

AbstractIn this paper, we present some results on the error behavior of variable stepsize stiffly-accurate Runge–Kutta methods applied to a class of multiply-stiff initial value problems of ordinary differential equations in singular perturbation form, under some weak assumptions on the coefficients of the considered methods. It is shown that the obtained convergence results hold for stiffly-accurate Runge–Kutta methods which are not algebraically stable or diagonally stable. Some results on the existence and uniqueness of the solution of Runge–Kutta equations are also presented

The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations

In this work, we consider subordinated processes controlled by a family of subordinators which consist of a power function of time variable and a negative power function of $\alpha-$stable random variable. The effect of parameters in the subordinators on the subordinated process is discussed. By suitable variable substitutions and Laplace transform technique, the corresponding fractional Fokker-Planck-type equations are derived. We also compute their mean square displacements in a free force field. By choosing suitable ranges of parameters, the resulting subordinated processes may be subdiffusive, normal diffusive or superdiffusive.Comment: 11 pages, accepted by J. Stat. Mech.: Theor. Ex

Order properties of symplectic runge-kutta-nyström methods

AbstractIn this paper, some characteristics and order properties of symplectic Runge-Kuttar-Nyström (RKN) methods are given. By using a transformation technique, a family of high-order implicit symplectic RKN methods of order 2s -1 or order 2s is constructed, and some available symplectic RKN methods including singly-implicit, multiply-implicit, and diagonally-implicit symplectic RKN methods are investigated. As examples, two-stage and three-stage symplectic RKN methods are derived in detail

Is the formal energy of the mid-point rule convergent?

AbstractWe obtain some formulae for calculation of the coefficients of four special types of terms in τ2k, k = 1, 2, … (1−1 corresponding to four type of (2k + 1)-vertex free unlabeled trees, k = 1, 2, …, respectively), for a fixed step size τ, in the tree-expansion of the formal energy of the mid-point rule. And, we give an estimate of the difference between the formal energy H and the standard Hamiltonian H in some domain Ω under the assumptions 1.(i)|H is smooth and bounded in Ω, and2.(ii)|the absolute values of the coefficients of the terms in τ2k are uniformly bounded by ησ2k for some constants η ≥ 1, σ > 0 and for any k ≥ 1

Generalized Young equation for a spherical droplet inside a smooth and homogeneous cone involved by quadratic parabola

We thermodynamically investigate the wetting characteristics of a spherical droplet in a smooth and homogeneous cone rotated by the quadratic parabola through the mechanisms of both Gibbs’s dividing surfaces and Rusanov’s dividing line. For the triple phase system including the solid, liquid and vapor phases, the derivation of a generalized Young equation containing the influences of the line tension is successfully carried out. Additionally, we as well analyze various approximate forms for this generalized Young equation by using the corresponding assumptions