Computational methods for delay parabolic and time‐fractional partial differential equations

This article is concerned with ?‐methods for delay parabolic partial differential equations. The methodology is extended to time‐fractional‐order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay‐independent asymptotic stability and the solution continuously depends on the time‐fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

The Uniform <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Behavior for Time Discretization of an Evolution Equation

The uniform L^2 stability and convergence properties for the time discretization of an evolution equation with a memory term are studied.The methods are based on the second-order backward difference methods.The memory term is approximated by the second-order convolution quadrature and interpolant quadrature.     

On the global stability of a temporal discretization scheme for the Navier-Stokes equations

The time discretization by a linear backward Euler scheme forthe non-stationary viscous incompressible Navier–Stokesequations with a non-zero external force in a bounded 2D domainwith no-slip boundary condition or periodic boundary conditionis studied. Improved global stability results are obtained. The boundedness of the solution sequence in V and D(A) normsuniform with respect to &t for t [0, ) is proved. A similarresult in the V norm was previously obtained by (Geveci, 1989Math. Comp., 53, 43–53) for the non-forced system. A differentapproach is used here. As a corollary, the global attractorfor the approximation scheme is proved to exist, which is boundedin both V and D(A) spaces, thus compact in both H and V spaces.Applying the same techniques developed here, we are able toimprove the main result of (Hill and Süli 2000 IMA J. Numer.Anal., 20, 633–667) by showing that besides the existenceof a global attractor, the whole solution sequence is uniformlybounded in V as well, which is of significance from the pointof view of computing. As a corollary of local convergence results,upper semi-continuity of the attractor with respect to the numericalperturbation induced by the linear scheme is also establishedin both H and V spaces. Finally, some preliminary estimates,which are to our knowledge the first of their kind, on the dimensionsof the attractors in H and V spaces are also obtained.     

Towards a compact high-order method for non-linear hyperbolic systems. I: The Hermite Least-Square Monotone (HLSM) reconstruction

A new Hermite Least-Square Monotone (HLSM) reconstruction to calculate accurately complex flows on non-uniform meshes is presented.     

Convergence of the backward Euler method for type-II superconductors

Our paper is devoted to the study of a nonlinear degenerate transient eddy current problem of the type t_∂(|E|^−1/pE)+∇×(∇×E)=0, p>1, along with appropriate initial and boundary conditions. We design a nonlinear time-discrete numerical scheme for the approximation in suitable function spaces. We show the well-posedness of the problem, prove the convergence of the approximation to a weak solution and finally derive the error estimates. In the proofs, the monotonicity methods and the Minty-Browder argument are employed.     

Convergence of Solutions of Discrete Reflected Backward SDE＇s and Simulations

The objective of this paper is to introduce elementary discrete reflected backward equations and to give a simple method to discretize in time a （continuous） reflected backward equation. A presentation of numerical simulations is also described.     

Convolution quadrature time discretization of fractional diffusion-wave equations

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

     

Solving the Charged-Particle Scattering Problem by Wave Packet Continuum Discretization

We generalize our previously developed packet continuum discretization method to take the long-range Coulomb repulsion in the charged-particle interaction into account. We derive an analytic finite-dimensional approximation for the exact Coulomb resolvent in the basis of stationary Coulomb wave packets. In the suggested approach, determining the so-called additional partial scattering phase shifts that appear because of the additional short-range interaction reduces to simple matrix algebra, and the related calculations can be performed using an arbitrary complete L₂ basis. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 3, pp. 393–410, December, 2005.     

A numerical solution of Burgers’ equation by finite element method constructed on the method of discretization in time

In this paper, the Galerkin finite element method constructed on the method of discretization in time was applied to solve the one-dimensional nonlinear Burgers’ equation. The system of nonlinear equations obtained for each time step was solved by using the Newton method. In order to show the efficiency of the presented method, the numerical solutions obtained for various values of viscosity were compared with the exact solutions. It was seen that they were in excellent agreement.     

Attaining exponential convergence for the flux error with second- and fourth-order accurate finite-difference equations. I. Presentation of the basic concept and application to a pure diffusion system

It is a well-known phenomenon called superconvergence in the mathematical literature that the error level of an integral quantity can be much smaller than the magnitude of the local errors involved in the computation of this quantity. When discretizing an integrated form of Fick's second law of diffusion the local errors reflect the accuracy of individual concentration points while the integral quantity has the physical meaning of the flux. This article demonstrates how an extraordinary fast exponential convergence towards zero can be achieved for the simulated flux error on the basis of finite-difference approximations that are only second-order (Box 2 method) or fourth-order (Box 4 method) accurate as far as the level of local errors is concerned.