A difference scheme of improved accuracy for a quasilinear singularly perturbed elliptic convection-diffusion equation.

A Dirichlet boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion equation on a strip is considered. For such a problem, a difference scheme based on classical approximations of the problem on piecewise uniform meshes condensing in the layer converges epsilon-uniformly with an order of accuracy not more than 1. We construct a linearized iterative scheme based on the nonlinear Richardson scheme, where the nonlinear term is computed using the sought function taken at the previous iteration; the solution of the iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of a geometry progression epsilon-uniformly with respect to the number of iterations. The use of lower and upper solutions of the linearized iterative Richardson scheme as a stopping criterion allows us during the computational process to define a current iteration under which the same epsilon-uniform accuracy of the solution is achieved as for the nonlinear Richardson schem

An a-posteriori adaptive mesh technique for singularly perturbed convection-diffusion problems with a moving interior layer

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not~converge $eps$-uniformly in the uniform norm, even under the condition $N^{-1}+N_0^{-1}approx eps$, where $eps$ is the perturbation parameter and $N$ and $N_0$ denote the number of mesh points with respect to $x$ and $t$. In the case of rectangular meshes which are ({it a~priori,} or {it a~posteriori,}) locally refined in the transition layer, there are no schemes that convergence uniformly in $eps$ even under the {it very restrictive,} condition $N^{-2}+N_0^{-2} approx eps$. However, the condition for convergence can be {it essentially weakened} if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an {it a~priori,}, or an {it a~posteriori,} adaptive mesh technique. Here we construct a scheme on {it a~posteriori,} adaptive meshes (based on the gradient of the solution), whose solution converges `almost $eps$-uniformly', viz., under the condition $N^{-1}=o(eps^{ u})$, where $u>0$ is an arbitrary number from the half-open interval $(0,1]$

Using the Kellogg-Tsan Solution Decomposition in NumericalMethods for Singularly Perturbed Convection-Diffusion Problems

The linear one-dimensional singularly perturbed convection-diffusion problem is solved numerically by a second-order method that is uniform in the perturbation parameter . The method uses the Kellogg-Tsan decomposition of the continuous solution. This increases the accuracy of the numerical results and simplifies the proof of their -uniformit

Nonpolynomial Spline for Numerical Solution of Singularly Perturbed Convection-Diffusion Equations with Discontinuous Source Term

This research addresses the numerical solution of singularly perturbed convection-diffusion kind boundary value problem of second-order with a discontinuity term. Due to the perturbation parameter and discontinuity term, the problem solution has a boundary layer and an interior layer. A nonpolynomial cubic spline method is utilized to solve the boundary value problem. A specific set of parameters associated with nonpolynomial spline is used to tailor the method. A comprehensive analysis of the stability and convergence of the recommended method is presented which gives second-order convergence results. The suggested method is implemented on two examples, and the obtained results are contrasted with an existing method, highlighting the precision and efficacy of the proposed method, which would enhance the method's novelty

Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension

In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used

The Investigation of Efficiency of Physical Phenomena Modelling Using Differential Equations on Distributed Systems

This work is dedicated to development of mathematical modelling software. In this dissertation numerical methods and algorithms are investigated in software making context. While applying a numerical method it is important to take into account the limited computer resources, the architecture of these resources and how do methods affect software robustness. Three main aspects of this investigation are that software implementation must be efficient, robust and be able to utilize specific hardware resources. The hardware specificity in this work is related to distributed computations of different types: single CPU with multiple cores, multiple CPUs with multiple cores and highly parallel multithreaded GPU device. The investigation is done in three directions: GPU usage for 3D FDTD calculations, FVM method usage to implement efficient calculations of a very specific heat transferring problem, and development of special techniques for software for specific bacteria self organization problem when the results are sensitive to numerical methods, initial data and even computer round-off errors. All these directions are dedicated to create correct technological components that make a software implementation robust and efficient. The time prediction model for 3D FDTD calculations is proposed, which lets to evaluate the efficiency of different GPUs. A reasonable speedup with GPU comparing to CPU is obtained. For FVM implementation the OpenFOAM open source software is selected as a basis for implementation of calculations and a few algorithms and their modifications to solve efficiency issues are proposed. The FVM parallel solver is implemented and analyzed, it is adapted to heterogeneous cluster Vilkas. To create robust software for simulation of bacteria self organization mathematically robust methods are applied and results are analyzed, the algorithm is modified for parallel computations