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.
The method of adaptive artificial viscosity is used to model the process of one-dimensional nonlinear convection-diffusion equation. For this purpose, a finite difference scheme (FDS) of the second order of time and space approximation has been developed. The scheme was tested using a numerical solution of the problem on formation of a gradient catastrophe. The process of two-phase filtration was analyzed with the help of constructed FDS. Numerical calculations showed that the proposed method, and in this case reliably tracks the discontinuities of the solution.
In this work we suggest an iterative process for coefficient inverse problem. A parabolic equation in a bounded area supplied with initial condition and monotonic nondecreasing on time Dirichlet condition on a boundary is considered. We state a problem to recover the lowest order coefficient that depends only on spatial variables under an additional information as the observation of a solution taken at the final point of time. For numerical recovering of the coefficient we build the iterative process, at each iteration we perform finite-element approximation in space and fully implicit two-level discretization in time. For capabilities of given iterative process we present computational test for a model problem.
In the paper, we propose an efficient method based on the use of a bicubic Hermite finite element coupled with collocation for the diffusion equation. This enables one to reduce the dimension of the system of equations in comparison with the standard finite element scheme. Numerical experiments confirm a theoretical convergence estimate and demonstrate the advantage of the proposed method.
The paper deals with periodic freezing and thawing of soil due to seasonal changes under the climatic conditions that affect the formation of thermal fields in the ground. A model and algorithm for simulation of seasonal cooling systems operating in certain temperature parameters and the effect on thermal stabilization of the soil is proposed. In modeling, the maximum number of climatic and technical parameters of devices are taken into account. The computations make it possible to estimate the efficiency of cooling devices using for keep the properties of the soil.
Efficient implementation of the Two-times Repeated Richardson Extrapolation is studied in this paper under the assumption that systems of ordinary differential equations (ODEs) are solved numerically by Explicit Runge-Kutta Methods (ERKMs). The combinations of the Two-times Repeated Richardson Extrapolation with the ERKMs are new numerical methods. The computational cost per step of these new numerical methods is higher than the computational cost per step of the underlying ERKMs. However, the order of accuracy of the combined methods becomes very high: if the order of accuracy of the underlying ERKM is p, then the order of accuracy of its combination with the Two-times Repeated Richardson Extrapolation is at least \(p+3\) when the right-hand-side function of the system of ODEs is sufficiently many times continuously differentiable. Moreover, the stability properties of the new methods are always better than those of the underlying numerical methods when \(p=m\) and \( m=1,2,3,4\) (where m is the number of stage vectors in the chosen ERKM). These two useful properties, higher accuracy and better stability, are often giving a very reasonable compensation for the increased computational cost per step, because the same degree of accuracy can be achieved by applying a large stepsize which leads to a considerable reduction of the number of steps when the Two-times Repeated Richardson Extrapolation is used. This fact is verified by several numerical experiments.
To perform an analytical and numerical investigation of optical bullets in a focusing bulk waveguide with quadratic nonlinearity we use the well-known quasi-optical approach. We give an approximate soliton solution representing a two-component light bullet. To investigate numerically the regimes of the formation and propagation of two-component optical bullets we construct a conservative difference scheme. To realize the multi-dimensional nonlinear difference scheme we propose a multi-step effective iterative solver. This method allows us to carry out an accurate and efficient modeling of the considered processes.
In this paper we consider a numerical solution of the non-classical Dirichlet problem and its modifications for the second-order two-dimensional hyperbolic equations. In order to determine the missing initial condition using an additional condition specified at the final time, an iterative method of conjugate gradient is used. A direct problem is numerically realized at each iteration. The efficiency of the proposed computational algorithm is confirmed by calculations for model two-dimensional problems.
Stochastic differential games have recently been key mathematical tools for resolution of environmental and ecological optimization problems. A finite difference scheme is proposed for solving a variational inequality arising in a stochastic differential game having several singular control variables: optimization of algae population management under model ambiguity. The present scheme employs fitted-exponential and upwind discretization methods to generate stable numerical solutions. Accuracy of the scheme is verified against an exact solution to a simplified problem and an asymptotic solution to a more complicated problem. The scheme is finally applied to numerical computation of the optimal algae population management policy against a range of an incurred cost. The computational results suggest that qualitatively different optimal policies are obtained depending on the magnitude of the incurred cost.
The paper is devoted to the problem of calculation of the spatial vibrations of elementary pipeline sections that appear under shock pulse. The equation of the motion and the boundary conditions describing the pipe deflections are presented as the forth-order partial differential equation with the Dirichlet and Neumann boundary conditions. To solve the problem, the numerical method based on finite-difference equations and a regularization technique is proposed. In order to evaluate the efficiency of the proposed method, the computational experiments were carried out. The results demonstrated sufficient accuracy of numerical solutions and confirm the sensitivity of method to changes in system.
A conservative semi-Lagrangian method is developed in order to solve three-dimensional linear advection equation. It based on balance equation in integral form. Main feature of proposed method consists in way of computation of integral at lower time level. To compute integral, we decompose a domain of integration into several tetrahedrons and approximate integrand by trilinear function.
In this paper, we consider a mixed dimensional discrete fracture model with highly conductive fractures. Mathematically the problem is described by a coupled system of equations consisting a d - dimensional equation for flow in porous matrix and a $$(d-1)$$ - dimensional equation for fracture networks with a specific exchange term for coupling them. For the numerical solution on the fine grid, we construct unstructured mesh that is conforming with fracture surface and use the finite element approximation. Fine grid approximation typically leads to very large systems of equations since it resolves the fracture networks, and therefore some multiscale methods or upscaling methods should be applied. The main contribution of this paper is that we propose a new upscaled model using Non-local multi-continuum (NLMC) method and construct an effective coarse grid approximation. The upscaled model has only one additional coarse degree of freedom (DOF) for each fracture network. We will present results of the numerical simulations using our proposed upscaling method to illustrate its performance.
We develop discrete variant of a theory of pseudo-differential operators and equations. For some canonical domains we obtain solvability results for such equations and use these results to construct approximate solutions.
Finding repeated zero for a nonlinear equation f(x)=0 , f:I⊆ R→ R , has always been of much interest and attention due to it’s wide applications in many fields of science and engineering. The modified Newton’s method is usually applied to solve this problem. Keeping in view that very few optimal higher order convergent methods exist for multiple roots, we present a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity involving multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from Life Science, Engineering and Physics are considered for the sake of comparison. The numerical experiments show that our proposed methods are efficient for determining multiple roots of non-linear equations.
The problem of spline interpolation of functions with large gradients in the boundary layer is studied. It is assumed that the function contains the known up to a factor boundary layer component responsible for the large gradients of this function in the boundary layer. A modification of the cubic spline, based on the fitting to the boundary layer component is proposed. The questions of existence, uniqueness and accuracy of such spline are investigated. Estimates of the interpolation error which are uniform with respect to a small parameter are obtained.
Dynamics of temperature distribution and interfacial front propagation in a generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. Finally, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters.
