Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes
https://doi.org/10.18255/1818-1015-2016-3-334-341
Abstract
The main objective of the paper is to present a new analytic-numerical approach to singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layers (fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numerical solution of such problems. It is based on a priori information about the moving front properties provided by the asymptotic analysis. In particular, for the mesh construction we take into account a priori asymptotic evaluation of the location and speed of the moving front, its width and structure. Our algorithms significantly reduce the CPU time and enhance the stability of the numerical process compared with classical approaches.
The article is published in the authors’ wording.
About the Authors
D. V. Lukyanenko
Lomonosov Moscow State University
Russian Federation
PhD, associate professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics
Russian Federation
PhD, associate professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics
V. T. Volkov
Lomonosov Moscow State University
Russian Federation
Russian Federation
PhD, associate professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics
N. N. Nefedov
Lomonosov Moscow State University
Russian Federation
Russian Federation
Professor, 119991, Moscow, Leninskie Gory, MSU, Faculty of Physics
L. Recke
HU Berlin, Institut fЁur Mathematik, Rudower Chaussee, Berlin, Germany
Russian Federation
Professor
Russian Federation
Professor
K. Schneider
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany
Russian Federation
Professor
Russian Federation
Professor
References
1. G. I. Shishkin, “Grid approximation of a singularly perturbed quasilinear equation in the presence of a transition layer”, Russian Acad. Sci. Dokl. Math., 47:1 (1993), 83–88.
2. E. O’Riordan, J. Quinn, “Numerical method for a nonlinear singularly perturbed interior layer problem”, Lectures Notes in Computational Science and Engeneering, 81 (2011), 187–195.
3. E. O’Riordan, J. Quinn, “Parameter-uniform numerical method for some linear and nonlinear singularly perturbed convection-diffusion boundary turning point problems”, BIT Numerical Mathematics, 51:2 (2011), 317–337.
4. N. Kopteva, M. Stynes, “Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction-diffusion problem”, Numerische Mathematik, 119:4 (2011), 787–810.
5. N. Kopteva, E. O’Riordan, “Shishkin meshes in the numerical solution of singularly perturbed differential equations”, International Journal of Numerical Analysis and Modeling, 7:3 (2010), 393–415.
6. N. Kopteva, “Numerical analysis of a 2D singularly perturbed semilinear reaction-diffusion problem”, Lecture Notes in Computer Science, 5434 (2009), 80–91.
7. P. A. Farrell, E. O’Riordan, G. I. Shishkin, “A class of singularly perturbed semilinear differential equations with interior layers”, Mathematics of Computation, 74:252 (2005), 1759–1776.
8. G. I. Shishkin, “Necessary conditions for ε-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers”, Computational Mathematics and Mathematical Physics, 47:10 (2007), 1636–1655.
9. G. I. Shishkin, L. P. Shishkina, P. W. Hemker, “A Class of Singularly Perturbed Convection- iffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique”, Comput. Meth. Appl. Math., 4:1 (2004), 105–127.
10. G. I. Shishkin, “Grid Approximation of a Singularly Perturbed Parabolic Equation on a Composite Domain in the case of a Concentrated Source on a Moving Interface”, Computational Mathematics and Mathematical Physics, 43:12 (2003), 1738–1755.
11. J. Quinn, “A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location”, Computational and Applied Mathematics, 290:15 (2015), 500–515.
12. V. T. Volkov, N. N. Nefedov, “Development of the Asymptotic Method of Differential Inequalities for Investigation of Periodic Contrast Structures in Reaction–Diffusion Equations”, Computational Mathematics and Mathematical Physics, 46:4 (2006), 585–593.
13. N. N. Nefedov and L. Recke and K. R. Schneider, “Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations”, Journal of Mathematical Analysis and Applications, 405 (2013), 90–103.
14. V. T. Volkov, N. N. Nefedov, “Asymptotic-numerical investigation of generation and motion of fronts in phase transition models”, Lecture Notes in Computer Science, 8236 (2013), 524–531.
Review
For citations:
Lukyanenko D.V., Volkov V.T., Nefedov N.N., Recke L., Schneider K. Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes. Modeling and Analysis of Information Systems. 2016;23(3):334-341. https://doi.org/10.18255/1818-1015-2016-3-334-341
Views:
1253
Email this article 