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On the numerical solution of advection-diffusion equations with singular reactions terms
Mälardalen University, School of Education, Culture and Communication.
2023 (English)Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

Advection-diffusion equations with singular reaction terms, involving Dirac delta functions, have applications in many fields within science and engineering. In general, solutions of problems involving differential equations with Dirac delta function cannot be found analytically. Consequently, numerical methods become indispensable, but the discretization process requires careful consideration due to the nature of the Dirac delta function. Furthermore, solutions to equations involving the Dirac delta function exhibit a lack of smoothness, requiring special attention in the selection and implementation of numerical methods where the order reduction is expected.

This thesis focuses on investigating the convergence order of numerical solutions for different problems in one and two dimensions. The method of lines is used, along with the three commonly used spatial discretization methods: finite difference, finite element, and finite volume. Time integration is carried out using a second-order trapezoidal method. Notably, spatial discretization by finite difference requires the regularization of the singular term, unlike finite element and finite volume approaches that utilize the integral form and exploit properties of the Dirac delta function to avoid regularization.

In experiments, numerical solutions for problems with smooth solutions exhibit second-order convergence, which is consistent with theory. Most remarkable is the fact that second-order convergence is also achieved for test problem with nonsmooth solution, exceeding initial expectations. Given the significance of this result, it is important to make additional investigations to determine the underlying reasons for this behavior and explore its implications and to understand whether this behavior is specific to the collocation of the reaction term or it can be generalized to a broader class of problems with nonsmooth solutions.

Place, publisher, year, edition, pages
2023. , p. 76
Keywords [en]
Advection-diffusion-reaction equations, Singular reaction terms, Regularization of Dirac delta function, Finite difference method, Finite element method, Finite volume method
National Category
Computational Mathematics Mathematical Analysis
Identifiers
URN: urn:nbn:se:mdh:diva-63063OAI: oai:DiVA.org:mdh-63063DiVA, id: diva2:1765325
Subject / course
Mathematics/Applied Mathematics
Supervisors
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Available from: 2023-06-14 Created: 2023-06-09 Last updated: 2023-06-14Bibliographically approved

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