For a singularly perturbed parabolic convection–diffusion equation with a perturbation parameter ε , ε∈ (0,1] , multiplying the highest-order derivative in the equation, we construct an improved computer difference scheme (with approximation of the first-order spatial derivative in the convective term by the central difference operator) on uniform meshes and study the behavior of discrete solutions in the presence of perturbations in the problem data. When solving such a problem numerically, errors in the grid solution depend on the parameter ε , on the parameters of the difference scheme, and also on the value of perturbations introduced in the process of computations (computer perturbations). For small values of the parameter ε , such errors, in general, significantly exceed the solution itself. For the computer perturbations, the conditions imposed on these admissible perturbations are obtained, under which accuracy of the computer solution in order is the same as for the solution of the unperturbed improved difference scheme, namely, 𝒪(ε ^-2 N^-2+ N^-1_0) . As a result, we have been constructed the improved computer difference scheme suitable for practical use.
On the set G =G ∪ S , G=(0,d]× (0,T] with the boundary S=S_0 ∪ S^ ℓ , we consider an initial-boundary value problem for the singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative, ε∈ (0,1] . For small values of the perturbation parameter ε , the solution of such a problem has a singularity of the boundary layer type, which makes standard difference schemes unsuitable for practical computations. To solve this problem numerically, an approach to the development of a robust difference scheme is proposed, similar to that used for constructing special ε -uniformly convergent difference schemes for singularly perturbed elliptic and parabolic equations. In this paper, we give a technique for constructing a robust difference scheme and justifying its ε -uniform convergence, and we study numerically solutions of standard and special robust difference schemes for a model initial-boundary value problem for a singularly perturbed transport equation. The results of numerical experiments confirm theoretical results.
Initial value problems and two-point boundary value problems for nonlinear pantograph type differential equations are investigated by presenting a new iterative numerical method based on constructing a sequence of splines that converges to the solution. The convergence of the method was proved by providing an error estimate and is tested on some numerical experiments.
In this work, we consider a coupled system of singularly perturbed reaction-diffusion equations (SPRDEs) with distinct small positive parameters, exhibiting overlapping boundary layers at both ends of the domain. In [4], the authors designed an overlapping domain decomposition method that gives almost second order accurate approximations to the solution of coupled systems of SPRDEs. High order methods are of great significance to the numerical community. To this end, for numerically solving the coupled systems of SPRDEs, we designed a high order accurate overlapping domain decomposition method via. defining an appropriate decomposition of the original domain and then considering a hybrid difference scheme on a uniform mesh on each subdomain. More precisely, the method gives almost fourth order accurate approximations to the solution of the problem, as compared to almost second order accurate approximations in [4]. Numerical results are given to demonstrate the effectiveness of the proposed method.
Permafrost takes place approximately 35 million km\(^2\) of the globe land. In Russia, it is most widely distributed in Eastern Siberia and Baikal region with oil and gas fields. Exploitation of the fields promotes the permafrost melting because of different technical devices affect on the dynamics of thawing. The permafrost thawing due to various human-generated impacts will be accompanied by subsidence of the earth’s surface around engineering facilities and development of dangerous permafrost geological processes, called thermokarst, which can lead to accidents in oil and gas fields with great damage to the environment. Therefore, an important problem for computer simulation is prediction of dynamics of the permafrost boundaries changes under long-term thermal impact of technical systems operating. In the paper a model and an algorithm for solving the problem of propagation of thermal fields in frozen ground from horizontal flare systems operated under a special regime are proposed. The maximum number of climatic and technical parameters is taken into account in the simulations. The calculations allow to choose an optimal thermal insulation of the ground surface under the flare system.
