This paper deals with numerical solving semilinear parabolic problems based on the method of upper and lower solutions. A monotone iterative method with quadratic convergence rate is constructed. The monotone iterative method combines an explicit construction of initial upper and lower solutions and the modified accelerated monotone iterative method. The monotone iterative method leads to the existence-uniqueness theorem. An analysis of convergence rates of the monotone iterative method, based on different stoping tests, is given. Results of numerical experiments are presented, where iteration counts are compared with a monotone iterative method, whose convergence rate is linear.

A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its firstand second-order derivatives are established in the L_1 norm by employing the parametrix metho'. The dependence of these bounds on the small perturbation parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem to derive a robust a posteriori error estimator in the maximum norm.

One way to solve complicated systems of differential equations is the application of operator splitting techniques. The original problem is split into several subsystems that are solved cyclically one after the other. Naturally, this procedure introduces an error, which is called splitting error, into the calculations. It is known that if the splitting procedure is applied to autonomous systems of ordinary differential equations, then the frequently used splitting procedures: the sequential, the Strang-Marchuk and the symmetrically weighted sequential splittings generally have splitting errors of order one and two, respectively. In this paper, we show that the order of the splitting procedures is preserved for non-autonomous problems. The theoretical results will be verified on numerical test problems.

In this paper, finite difference schemes based on asymptotic analysis and the augmented immersed interface method are proposed for potential problems with an inclusion whose characteristic width is much smaller than the characteristic length in one and two dimensions. We call such a problem as a crack problem for simplicity. In the proposed methods, we use asymptotic analysis to approximate the problem with a single sharp interface. The jump conditions for the interface problem are derived. For one-dimensional problem, or two-dimensional problems in which the center line of the crack is parallel to one of axis, we can simply modify the finite difference scheme with added correction terms at irregular grid points. The coefficient matrix of the finite difference equations is still an M-matrix. For problems with a general thin crack, an augmented variable along the center line of the crack is introduced so that we can apply the immersed interface method to get the discretization. The augmented equation is the asymptotic jump condition. Numerical experiments including the case with large jump discontinuity in the coefficient are presented.

The problem on propagation of long waves in a domain of arbitrary form with the sufficiently smooth boundary on a sphere is considered. The boundary consists of “solid” parts passing along the coastline and “liquid” parts passing through the water area. We assume, free surface level observation data on “liquid” boundary is known. In general case the boundary condition on “liquid” part of boundary contains unknown boundary function, which must be found together with component of velocity vector and free surface level. We put an assimilation observation data problem by Prof. V.I. Agoshkov methodology. To solve our ill-posed inverse problem an approach, based on optimal control methods and adjoint equations theory, is used. Numerical solution of direct and adjoint problems is based on finite elements method. Parallel software using MPI is discussed.

A numerical method for solving parabolic equations based on multigrid techniques is proposed. The stability, approximation and conservation properties of the method are investigated theoretically and numerically for several initial-boundary model problems for the heat conduction equation. The use of the method makes it possible to considerably reduce the computational work as compared to either implicit or explicit schemes. A parallel implementation of the method is presented.

The stability of a four-dimensional nonlinear model of a wastewater treatment process (proposed by Bernard, Hadj-Sadok, Dochain, Genovesi and Steyer in 2001) is studied. A feedback control law, depending only on on-line measurable quantities, is proposed. This feedback law stabilizes asymptotically the closed-loop system towards a previously chosen operating point. In order to prove that the closed-loop system is asymptotically stable, two Lyapunov-like functions are constructed explicitly. A numerical extremum seeking algorithm is then applied to stabilize the dynamics towards an equilibrium point corresponding to the maximum methane output flow rate. The robustness of the feedback is demonstrated by involving uncertainties in the growth rate model functions. Computer simulations are reported to illustrate the theoretical results.

Computational algorithm for the solution of 2D and 3D dynamic problems of the Cosserat elasticity theory using multiprocessor computer systems is worked out. The algorithm is based on the two-cyclic splitting method with respect to the spatial variables in combination with the explicit monotone finite-difference ENO-scheme for solving 1D problems adapted for the calculation of discontinuities. Boundary conditions of symmetry, permitting to reduce computations many times, are formulated. Computations of 3D Lamb's problem on the action of concentrated load on the surface of a homogeneous elastic half-space and the problem on the action of concentrated impulsive periodic load are performed. The shock waves of four types (longitudinal, transverse, torsional and rotational waves), characteristic of a couple-stressed elastic medium, are recognized numerically. The oscillations of rotational motion of particles on the wave fronts are also found.

In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the option price leads to a free boundary problem for a parabolic partial differential equation. Applying fixed domain transformation and transformation of variables we develop an effcient numerical algorithm based on a solution to a non-local parabolic partial differential equation for the transformed variable representing the synthesized portfolio. For various types of averaging methods we investigate the dependence of the early exercise boundary on model parameters.

We study localised attractors of the parametrically driven, damped nonlinear Schrödinger equation. Time-periodic solitons of this equation are obtained as solutions of the boundary-value problem on a two-dimensional domain. Stability and bifurcations of periodic solitons and their complexes is classified. We show that the bifurcation diagram can be reproduced using a few-mode approximation.

This paper is concerned with the spline interpolation of functions with a boundary layer component. It is shown that polynomial formulas of spline interpolation for such functions lead to significant errors. We proposed nonpolynomial spline interpolation formulas, fitted to a boundary layer component. Nonsmooth and smooth on whole interval interpolants are constructed. The accuracy of constructed formulas is estimated. Numerical results are discussed.