Estimation of the Hurst index of the solutions of fractional SDE with locally Lipschitz drift

Strongly consistent and asymptotically normal estimates of the Hurst index H are obtained for stochastic differential equations (SDEs) that have a unique positive solution. A strongly convergent approximation of the considered SDE solution is constructed using the backward Euler scheme. Moreover, it is proved that the Hurst estimator preserves its properties, if we replace the solution with its approximation.

A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing

First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.

Factorized Schemes of Second-Order Accuracy for Numerically Solving Unsteady Problems

Schemes with the second-order approximation in time are considered for numerically solving the Cauchy problem for an evolutionary equation of first order with a self-adjoint operator. The implicit two-level scheme based on the Padé polynomial approximation is unconditionally stable. It demonstrates good asymptotic properties in time and provides an adequate evolution in time for individual harmonics of the solution (has spectral mimetic (SM) stability). In fact, the only drawback of this scheme is the necessity to solve an equation with an operator polynomial of second degree at each time level. We consider modifications of these schemes, which are based on solving equations with operator polynomials of first degree. Such computational implementations occur, for example, if we apply the fully implicit two-level scheme (the backward Euler scheme). A three-level modification of the SM-stable scheme is proposed. Its unconditional stability is established in the corresponding norms. The emphasis is on the scheme, where the numerical algorithm involves two stages, namely, the backward Euler scheme of first order at the first (prediction) stage and the following correction of the approximate solution using a factorized operator. The SM-stability is established for the proposed scheme. To illustrate the theoretical results of the work, a model problem is solved numerically.

A second order numerical method for a class of parameterized singular perturbation problems on adaptive grid

A nonlinear singularly perturbed boundary value problem depending on a parameter is considered. First, we solve the problem using the backward Euler finite difference scheme on an adaptive grid. The adaptive grid is a special nonuniform mesh generated through equidistribution principle by a positive monitor function depending on the solution. The behavior of the solution, the stability and the error estimates are discussed. Then, the Richardson extrapolation technique is applied to improve the accuracy of the computed solution associated to the backward Euler scheme. The proofs of the uniform convergence for the backward Euler scheme and the Richardson extrapolation are carried out. Numerical experiments validate the theoretical estimates and indicates that the estimates are sharp.

A linear backward Euler scheme for a class of degenerate advection–diffusion equations: A mathematical analysis of the convergence in L∞(0, T0;L2(Ω)) and in L2(0, T0;H1(Ω))

This paper concerns itself with establishing convergence estimates for a linearized scheme for solving numerically the saturation equation. In a previous paper, error estimates were obtained for the same scheme in L2(0, T0;L2(Ω)). In this work, we establish error estimates for the linear scheme in L∞(0, T0;L2(Ω)) and in L2(0, T0;H1(Ω)) (in the discrete norms). Under certain realistic conditions, we show that, if the regularization parameter β and the spatial discretization parameter h are carefully chosen in terms of the time-stepping parameter Δt, the convergence, in these spaces, is at least of order O((Δt)α) for some determined α > 0, function of a parameter μ > 0 defined in the problem. Examples of possible choices of β and h in terms of Δt are given.

Parareal in Time Simulation Of Morphological Transformation in Cubic Alloys with Spatially Dependent Composition

In this paper, a reduced morphological transformation model with spatially dependent composition and elastic modulus is considered. The parareal in time al-gorithm introduced by Lions et al. is developed for longer-time simulation. The fine solver is based on a second-order scheme in reciprocal space, and the coarse solver is based on a multi-model backward Euler scheme, which is fast and less expensive. Numerical simulations concerning the composition with a random noise and a discontinuous curve are performed. Some microstructure characteristics at very low temperature are obtained by a variable temperature technique.