On the numerical solution of diffusion–reaction equations with singular source terms

• Published in: Journal of computational and applied mathematics Vol. 216; no. 1; pp. 20 - 38
• Main Authors: Ashyraliyev, M., Blom, J.G., Verwer, J.G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.06.2008; Elsevier
• Subjects: Exact sciences and technology; Finite volume methods; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Reaction–diffusion equations; Sciences and techniques of general use; Singular source terms; 65L05; Reaction–diffusion equations; 65M20; Finite volume methods; Singular source terms; primary; Source terms; Grid pattern; Singular equation; Finite volume method; Reaction diffusion equation; Reaction-diffusion equations; Singular source terms; Finite volume methods; Discretization method; Surface; primary 65M12;65M20; 65L05; Numerical analysis; Applied mathematics; Numerical solution; Diffusion equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2007.04.017

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.