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On the numerical solution of diffusion-reaction equations with singular source terms. (English) Zbl 1138.65065

Summary: 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.
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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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References:

[1] Greenberg, M. D., Application of Green’s Functions in Science and Engineering (1971), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[2] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31, Springer, Berlin, 2002.; E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31, Springer, Berlin, 2002. · Zbl 0994.65135
[3] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, vol. 33, Springer, Berlin, 2003.; W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, vol. 33, Springer, Berlin, 2003. · Zbl 1030.65100
[4] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), New York: New York London · Zbl 0241.65046
[5] Saad, Y., Iterative Methods for Sparse Linear Systems (2003), SIAM: SIAM Philadelphia · Zbl 1002.65042
[6] Temme, N. M., Special Functions, an Introduction to the Classical Functions of Mathematical Physics (1996), Wiley: Wiley New York · Zbl 0863.33002
[7] N.M. Temme, Personal communication, 2005.; N.M. Temme, Personal communication, 2005.
[8] Tornberg, A.-K.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200, 462-488 (2004) · Zbl 1115.76392
[9] van der Vorst, H. A., BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13, 2, 631-644 (1992) · Zbl 0761.65023
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