Abstract
In this article, we study a novel adaptive mesh strategy for singularly perturbed problems (SPPs) of the parabolic convection-diffusion type that exhibit regular boundary layers. Our central insight is that the introduction of an auxiliary inequality for an entropy-like variable also serves as a remarkably effective adaptation indicator. The primary novelty of this method is that, unlike extant methods used for layer adapted meshes (in which enough mesh points exist in the layer region for well resolved numerical solution), the current method requires no a priori knowledge of the location and width of the boundary layers. Further, the current method, [(which is an extension of the methodology from Kumar and Srinivasan (Appl Math Model 39:2081–2091, 2015)] is completely independent of the perturbation parameter and results in accurate solutions for a wide range of problems. We include some preliminary error estimates and also the results of several numerical experiments including the Black–Scholes equation. The results exhibit the promise of the proposed strategy to generate efficient adaptive meshes for time dependent convection-diffusion problems.
Similar content being viewed by others
References
Kumar, V., Srinivasan, B.: An adaptive mesh strategy for singularly perturbed convection diffusion problems. Appl. Math. Model. 39, 2081–2091 (2015)
Miller, J.J.H., ORiordan, E., Shishkin, I.G.: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore (1996)
Farrell, P.A., Hegarty, A., Miller, J.J.H., O’Riordan, E., Shishkin, G.I.: Robust Computational Techniques for Boundary Layers. Chapman and Hall, New York (2000)
Bakhvalov, N.S.: Towards optimization of methods for solving boundary value problems in the presence of boundary layers. Zh. Vychisl. Mater. Fiz. 9, 841–859 (1969). (In Russian)
Linb, Torsten: Layer-adapted meshes for convection—diffusion problems. Comput. Methods Appl. Mech. Eng. 192, 1061–1105 (2003)
Bobisud, L.: Second-order linear parabolic equation with a small parameter. Arch. Ration. Mech. Anal. 27, 385–397 (1967)
Kellogg, R.B., Tsan, A.: Analysis of some difference approximations for a singular perturbation problem without turning point. Math. Comput. 32, 1025–1039 (1978)
Kadalbajoo, M.K., Awasthi, A.: A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension. Appl. Math. Comput. 183, 42–60 (2006)
Srinivasan, B., Kumar, V.: The versatility of an entropy inequality for the robust computation of convection dominated problems. Procedia Comput. Sci. 108C, 887–896 (2017)
Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)
Kadalbajoo, M.K., Tripathi, L.P., Arora, P.: A robust nonuniform B-spline collocation method for solving the generalized Black–Scholes equation. IMA J. Numer. Anal. 34, 252–278 (2013)
Acknowledgements
We would like to thank the reviewers for their valuable and constructive comments and suggestions. First author would like to acknowledge the National board of higher mathematics (NBHM) for research Grant no. -Ref.No. \( 2/48(6)/2016/NBHM(R.P.)/R \& D II/15455\).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kumar, V., Srinivasan, B. A Novel Adaptive Mesh Strategy for Singularly Perturbed Parabolic Convection Diffusion Problems. Differ Equ Dyn Syst 27, 203–220 (2019). https://doi.org/10.1007/s12591-017-0394-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12591-017-0394-2