On Numerical Methods for Constructing Benchmark Solutions to a Nonlinear Heat Equation with a Singularity

A. L. Kazakov; L. F. Spevak; E. L. Spevak

doi:10.17804/2410-9908.2020.5.026-044

A. L. Kazakov, L. F. Spevak, E. L. Spevak

ON NUMERICAL METHODS FOR CONSTRUCTING BENCHMARK SOLUTIONS TO A NONLINEAR HEAT EQUATION WITH A SINGULARITY

DOI: 10.17804/2410-9908.2020.5.026-044

The paper deals with the construction of exact solutions to a nonlinear heat equation with degeneration in the case of the zero value of the required function. Generically self-similar solutions and traveling wave solutions are considered, the construction of which reduces to solving Cauchy problems for a nonlinear second-order ordinary differential equation with a singularity before the higher derivative. Two approaches are proposed to solve the Cauchy problems: the analytical solution by the power series method and the numerical solution by the boundary element method on a specified segment. A complex computational experiment is carried out to compare the above two methods with each other and with the finite difference methods, namely the Euler method and the fourth-order Runge-Kutta method. Power series segments are used on the first step of the finite difference solutions in order to resolve the singularity. The comparison of the application domains, the accuracy of the solutions and their dependence on the parameters of a certain problem shows that the boundary element method is the most universal, although not the most accurate for some particular examples. The conclusions drawn allow us to construct benchmark solutions to verify the approximate solutions of the nonlinear heat equation by various methods in a wide range of parameter values.

Acknowledgements: The work was supported by the RFBR, project No. 20-07-00407.

Keywords: nonlinear heat equation, exact solution, generically self-similar solution, traveling wave, power se-ries, boundary element method

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А. Л. Казаков, Л. Ф. Спевак, Е. Л. Спевак

О ЧИСЛЕННЫХ МЕТОДАХ ПОСТРОЕНИЯ ЭТАЛОННЫХ РЕШЕНИЙДЛЯ НЕЛИНЕЙНОГО УРАВНЕНИЯ ТЕПЛОПРОВОДНОСТИ С ОСОБЕННОСТЬЮ

Работа посвящена построению точных решений нелинейного уравнения теплопроводности с вырождением в случае нулевого значения искомой функции. Рассмотрены обобщенно-автомодельные решения и решения типа бегущей волны, нахождение которых сводится к решению задач Коши для обыкновенного дифференциального уравнения второго порядка с особенностью при старшей производной. Для решения этих задач предложены два подхода: построение аналитического решения методом степенных рядов и построение решения на заданном отрезке методом граничных элементов. В проведенном объемном вычислительном эксперименте построенные двумя методами решения сравнивались между собой, а также с разностными решениями методами Эйлера и Рунге–Кутта, в которых для раскрытия особенностей на первом шаге используются отрезки степенных рядов. Сопоставление областей применимости методов, точности решений и их зависимости от параметров конкретной задачи показало набольшую универсальность метода граничных элементов, хотя для отдельных примеров лучшую точность имели решения другими методами. Сделанные выводы позволят с достаточной корректностью строить эталонные решения для верификации приближенных решений нелинейного уравнения теплопроводности различными методами в широком диапазоне значений параметров.

Благодарности: Работа выполнена при поддержке РФФИ, проект № 20-07-00407.

Ключевые слова: нелинейное уравнение теплопроводности, точное решение, обобщенно-автомодельное решение, бегущая волна, степенной ряд, метод граничных элементов

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31. Kazakov A. L., Spevak L. F. Numerical Study of Travelling Wave Type Solutions for the Nonlinear Heat Equation // AIP Conference Proceedings. – 2019. – Vol. 2176. – P. 030006. – DOI: 10.1063/1.5135130.

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Библиографическая ссылка на статью

Kazakov A. L., Spevak L. F., Spevak E. L. On Numerical Methods for Constructing Benchmark Solutions to a Nonlinear Heat Equation with a Singularity // Diagnostics, Resource and Mechanics of materials and structures. - 2020. - Iss. 5. - P. 26-44. -
DOI: 10.17804/2410-9908.2020.5.026-044. -
URL: http://dream-journal.org/issues/2020-5/2020-5_306.html
(accessed: 24.04.2024).