Robust computing technique for reaction diffusion 2D parabolic problems with shift

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

A combination of two approaches to numerically solving second-order ODEs with a small parameter and singularities, such as interior and boundary layers, is considered, namely, compact high-order approximation schemes and explicit generation of layer resolving grids. The generation of layer resolving grids, which is based on estimates of solution derivatives and formulations of coordinate transformations eliminating the solution singularities, is a generalization of a method for a first-order scheme developed earlier. This paper presents formulas of the coordinate transformations and numerical experiments for first-, second-, and third-order schemes on uniform and layer resolving grids for equations with boundary, interior, exponential, and power layers of various scales. Numerical experiments confirm the uniform convergence of the numerical solutions performed with the compact high-order schemes on the layer resolving grids. By using transfinite interpolation or numerical solutions to the Beltrami and diffusion equations in a control metric based on coordinate transformations eliminating the solution singularities, this technology can be generalized to the solution of multi-dimensional equations with boundary and interior layers.

This volume will contain selected papers from the lectures held at the BAIL 2010 Conference, which took place from July 5th to 9th, 2010 in Zaragoza (Spain). The papers present significant advances in the modeling, analysis and construction of efficient numerical methods to solve boundary and interior layers appearing in singular perturbation problems. Special emphasis is put on the mathematical foundations of such methods and their application to physical models. Topics in scientific fields such as fluid dynamics, quantum mechanics, semiconductor modeling, control theory, elasticity, chemical reactor theory, and porous media are examined in detail.

In this paper, we investigate the features of the numerical solution of Cauchy problems for nonlinear differential equations with contrast structures (interior layers). Similar problems arise in the modeling of certain problems of hydrodynamics, chemical kinetics, combustion theory, computational geometry. Analytical solution of problems with contrast structures can be obtained only in particular cases. The numerical solution is also difficult to obtain. This is due to the ill-conditionality of the equations in the neighborhood of the interior and boundary layers. To achieve an acceptable accuracy of the numerical solution, it is necessary to significantly reduce the step size, which leads to an increase of a computational complexity. The disadvantages of using the traditional explicit Euler method and fourth-order Runge-Kutta method, as well as the implicit Euler method with constant and variable step sizes are shown on the example of one test problem with two boundary and one interior layers. Two approaches have been proposed to eliminate the computational disadvantages of traditional methods. As the first method, the best parametrization is applied. This method consists in passing to a new argument measured in the tangent direction along the integral curve of the considered Cauchy problem. The best parametrization allows obtaining the best conditioned Cauchy problem and eliminating the computational difficulties arising in the neighborhood of the interior and boundary layers. The second approach for solving the Cauchy problem is a semi-analytical method developed in the works of Alexander N. Vasilyev and Dmitry A. Tarkhov their apprentice and followers. This method allows obtaining a multilayered functional solution, which can be considered as a type of nonlinear asymptotics. Even at high rigidity, a semi-analytical method allows obtaining acceptable accuracy solution of problems with contrast structures. The analysis of the methods used is carried out. The obtained results are compared with the analytical solution of the considered test problem, as well as with the results of other authors.

В статье исследуются особенности численного решения задач Коши для нелинейных дифференциальных уравнений с контрастными структурами (внутренними пограничными слоями). Подобные задачи возникают при моделировании некоторых задач гидродинамики, химической кинетики, теории горения, вычислительной геометрии. Аналитическое решение задач с контрастными структурами удается найти только в исключительных случаях. Численное решение также затруднительно, что связано с плохой обусловленностью уравнений в окрестностях внутренних и пограничных слоев. Для достижения приемлемой точности численного решения необходимо значительно уменьшать шаг интегрирования, что приводит к возрастанию вычислительной сложности. На примере одной тестовой задачи с двумя пограничными и одним внутренним слоями показаны недостатки использования традиционных явных методов Эйлера и Рунге-Кутты 4 порядка точности, а также неявного метода Эйлера с постоянным и переменным шагами интегрирования. Для устранения вычислительных недостатков традиционных методов предложено два подхода. В качестве первого подхода применяется метод наилучшей параметризации, смысл которого состоит в переходе к новому аргументу, отсчитываемому по касательной вдоль интегральной кривой рассматриваемой задачи Коши. Этот метод позволяет получить наилучшим образом обусловленную задачу Коши и устранить вычислительные трудности, возникающие в окрестности внутренних и пограничных слоев. Вторым подходом является полуаналитический способ решения задачи Коши, разрабатываемый в работах А. Н. Васильева, Д. А. Тархова, их учеников и последователей. Данный подход позволяет получить многослойное функциональное решение, которое можно рассматривать как своего рода нелинейную асимптотику. Применительно к решению задач с контрастными структурами полуаналитический метод позволяет получать решение приемлемой точности, даже при высокой жесткости. Проводится анализ используемых методов. Полученные результаты сравниваются с аналитическим решением выбранной тестовой задачи, а также с результатами, представленными в работах других авторов. In this paper, we investigate the features of the numerical solution of Cauchy problems for nonlinear differential equations with contrast structures (interior layers). Similar problems arise in the modeling of certain problems of hydrodynamics, chemical kinetics, combustion theory, computational geometry. Analytical solution of problems with contrast structures can be obtained only in particular cases. The numerical solution is also difficult to obtain. This is due to the ill conditionality of the equations in the neighborhood of the interior and boundary layers. To achieve an acceptable accuracy of the numerical solution, it is necessary to significantly reduce the step size, which leads to an increase of a computational complexity. The disadvantages of using the traditional explicit Euler method and fourth-order Runge-Kutta method, as well as the implicit Euler method with constant and variable step sizes are shown on the example of one test problem with two boundaries and one interior layers. Two approaches have been proposed to eliminate the computational disadvantages of traditional methods. As the first method, the best parametrization is applied. This method consists in passing to a new argument measured in the tangent direction along the integral curve of the considered Cauchy problem. The best parametrization allows obtaining the best conditioned Cauchy problem and eliminating the computational difficulties arising in the neighborhood of the interior and boundary layers. The second approach for solving the Cauchy problem is a semi-analytical method developed in the works of Alexander N. Vasilyev and Dmitry A. Tarkhov their apprentice and followers. This method allows obtaining a multilayered functional solution, which can be considered as a type of nonlinear asymptotic. Even at high rigidity, a semi-analytical method allows obtaining acceptable accuracy solution of problems with contrast structures. The analysis of the methods used is carried out. The obtained results are compared with the analytical solution of the considered test problem, as well as with the results of other authors.

This paper is concerned with the asymptotic analysis of the transient semiconductor device equations. By appropriately scaling the equations the semiconductor device problem is reformulated as a singular perturbation problem. This singular perturbation problem is analyzed by the method of matchedasymptotic expansions. A formal solution that solves the equations approximately is constructed. The existence of interior and boundary layers in the transient model is shown. As a new phenomenon the existence of an initial temporal layer is shown. The existence of a “fast” timescale is proved along which certain components of the solutions decay. Equations for these components are deduced and analyzed.

In this paper, we propose a tailored-finite-point method for a type of linear singular perturbation problem in two dimensions. Our finite point method has been tailored to some particular properties of the problem. Therefore, our new method can achieve very high accuracy with very coarse mesh even for very small ?, i.e. the boundary layers and interior layers do not need to be resolved numerically. In our numerical implementation, we study the classification of all the singular points for the corresponding degenerate first order linear dynamic system. We also study some cases with nonlinear coefficients. Our tailored finite point method is very efficient in both linear and nonlinear coefficients cases.

In this work, we study an important class of singular perturbation problems: singularly perturbed two-point turning point problems (TPP's) in ordinary differential equations (ODE's) exhibiting interior layer at the mid-point of their intervals. We consider this type of problems as two different singular perturbation problems which have a boundary layer at an end point of their intervals. This study is devoted to the Successive complementary expansion method (SCEM) to solve these two sub-problems. Two test problems are considered to check the efficiency and accuracy of the proposed method. Our numerical experiments show that SCEM approximations are in good agreement with exact and previously obtained solutions.

A system of \(k(\ge 2)\) linear singularly perturbed differential equations of reaction–diffusion type coupled through their reactive terms is considered with Robin type boundary conditions, and the system has discontinuous source terms. The highest order derivative term of each equation is multiplied by a small positive parameter and these parameters are assumed to be different in magnitude, due to which the overlapping and interacting interior and boundary layers may appear in the solution of the considered problem. A numerical scheme involving a central difference scheme for the differential equations and a cubic spline technique for the Robin boundary conditions is developed on an appropriate piecewise-uniform Shishkin mesh. Error analysis is done and the constructed scheme is proved to be almost second-order uniformly convergent with respect to each perturbation parameter. Numerical experiments are conducted to verify the theoretical findings.KeywordsCoupled systemSingular perturbationShishkin meshBakhvalov meshDiscontinuous source termRobin boundary conditionsBoundary layerParameter-uniform convergenceFinite difference schemeInterior layerCubicspline

We consider a class of singular Sturm‐Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction‐diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.

In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε‐uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

We integrate optimal quadratic and cubic spline collocation methods for second-order two-point boundary value problems with adaptive grid techniques, and grid size and error estimators. Some adaptive grid techniques are based on the construction of a mapping function that maps uniform to non-uniform points, placed appropriately to minimize a certain norm of the error. One adaptive grid technique for cubic spline collocation is mapping-free and resembles the technique used in COLSYS (COLNEW) [2], [4]. Numerical results on a variety of problems, including problems with boundary or interior layers, and singular perturbation problems indicate that, for most problems, the cubic spline collocation method requires less computational effort for the same error tolerance, and has equally reliable error estimators, when compared to Hermite piecewise cubic collocation. Comparison results with quadratic spline collocation are also presented.

Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers

Two singular perturbation problems were considered: One with multiple boundary layers and one with interior layers. Numerical schemes were developed from the improved a'priori bounds. With a constant number of layer adapted grid points, numerical accuracy was maintained at the same level for a family of singular perturbation problems.

A review on singularly perturbed differential equations with turning points and interior layers