# Approximation of singularly perturbed parabolic equations in unbounded domains subject to piecewise smooth boundary conditions in the case of solutions that grow at infinity

- 47 Downloads

## Abstract

The initial-boundary value problem in a domain on a straight line that is unbounded in *x* is considered for a singularly perturbed reaction-diffusion parabolic equation. The higher order derivative in the equation is multiplied by a parameter ɛ^{2}, where ɛ ∈ (0, 1]. The right-hand side of the equation and the initial function grow unboundedly as *x* → ∞ at a rate of *O*(*x* ^{2}). This causes the unbounded growth of the solution at infinity at a rate of *O*(Ψ(*x*)), where Ψ(*x*) = *x* ^{2} + 1. The initialboundary function is piecewise smooth. When ɛ is small, a boundary and interior layers appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristics of the reduced equation passing through the discontinuity points of the initial function. In the problem under examination, the error of the grid solution grows unboundedly in the maximum norm as *x* → ∞ even for smooth solutions when ɛ is fixed. In this paper, the proximity of solutions of the initial-boundary value problem and its grid approximations is considered in the weighted maximum norm ∥·∥^{ w } with the weighting function Ψ^{−1}(*x*); in this norm, the solution of the initial-boundary value problem is ɛ-uniformly bounded. Using the method of special grids that condense in a neighborhood of the boundary layer or in neighborhoods of the boundary and interior layers, special finite difference schemes are constructed and studied that converge ɛ-uniformly in the weighted norm. It is shown that the convergence rate considerably depends on the type of nonsmoothness in the initial-boundary conditions. Grid approximations of the Cauchy problem with the right-hand side and the initial function growing as *O*(Ψ(*x*)) that converge ɛ-uniformly in the weighted norm are also considered.

## Key words

parabolic reaction-diffusion equation unbounded domain unbounded growth of the solution at infinity piecewise smooth initial-boundary function boundary and interior layers ɛ-uniform convergence weighted maximum norm Cauchy problem## Preview

Unable to display preview. Download preview PDF.

## References

- 1.N. S. Bakhvalov, “On the Optimization of Methods for Solving Boundary Value Problems in the Presence of a Boundary Layer,” Zh. Vychisl. Mat. Mat. Fiz.
**9**, 841–859 (1969).zbMATHGoogle Scholar - 2.A. M. Il’in, “Differencing Scheme for a Differential Equation with a Small Parameter Affecting the Highest Derivative,” Mat. Zametki
**6**(2), 237–248 (1969) [Math. Notes**6**, 596–602 (1969)].zbMATHMathSciNetGoogle Scholar - 3.G. I. Shishkin,
*Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations*(Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].Google Scholar - 4.P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin,
*Robust Computational Techniques for Boundary Layers*(Chapman & Hall/CRC Press, Boca Raton, FL, 2000).zbMATHGoogle Scholar - 5.J. J. H. Miller, E. O’Riordan, and G. I. Shishkin,
*Fitted Numerical Methods for Singular Perturbation Problems*(World Sci., Singapore, 1996).zbMATHGoogle Scholar - 6.H.-G. Roos, M. Stynes, and L. Tobiska,
*Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems*(Springer, Berlin, 1996).zbMATHGoogle Scholar - 7.G. I. Shishkin and L. P. Shishkina,
*Difference Methods for Singular Perturbation Problems*, Ser.*Monographs & Surveys in Pure & Applied Mathematics*(Chapman & Hall/CRC Press, Boca Raton, FL, 2009).Google Scholar - 8.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,
*Linear and Quasilinear Equations of Parabolic Type*(Nauka, Moscow, 1967; American Mathematical Society, Providence, 1968).zbMATHGoogle Scholar - 9.G. I. Shishkin, “Grid Approximation in a Half Plane for Singularly Perturbed Elliptic Equations with Convective Terms That Grow at Infinity,” Zh. Vychisl. Mat. Mat. Fiz.
**45**, 298–314 (2005) [Comput. Math. Math. Phys.**45**, 285–301 (2005)].zbMATHMathSciNetGoogle Scholar - 10.G. I. Shishkin, “Grid Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations on Large Domains with Respect to the Space and Time Variables,” Zh. Vychisl. Mat. Mat. Fiz.
**46**, 2045–2064 (2006) [Comput. Math. Math. Phys.**46**, 1953–1971 (2006)].MathSciNetGoogle Scholar - 11.S. Li, G. Shishkin, and L. Shishkina, “Approximation of the Solution and Its Derivative for the Singularly Perturbed Black-Scholes Equation with Nonsmooth Initial Data,” Comput. Math. Math. Phys
**47**, 442–462 (2007).CrossRefMathSciNetGoogle Scholar - 12.A. A. Samarskii,
*Theory of Finite Difference Schemes*(Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).Google Scholar - 13.G. I. Shishkin, “Grid Approximation of Singularly Perturbed Parabolic Convection-Diffusion Equations Subject to a Piecewise Smooth Initial Condition,” Zh. Vychisl. Mat. Mat. Fiz.
**46**, 52–76 (2006) [Comput. Math. Math. Phys.**46**, 49–72 (2006)].zbMATHMathSciNetGoogle Scholar - 14.G. I. Shishkin, “Grid Approximation of Singularly Perturbed Parabolic Reaction-Diffusion Equations with Piecewise Smooth Initial-Boundary Conditions,” Math. Modelling Analys.
**12**, 235–254 (2007).zbMATHCrossRefMathSciNetGoogle Scholar - 15.G. I. Shishkin, “Grid Approximation of Singularly Perturbed Parabolic Equations with Piecewise Continuous Initial-Boundary Conditions,” Proc. Steklov Inst. Math. Suppl.
**2**, S213–S230 (2007).CrossRefGoogle Scholar - 16.P. W. Hemker and G. I. Shishkin, “Discrete Approximation of Singularly Perturbed Parabolic PDEs with a Discontinuous Initial Condition,” Comput. Fluid Dynamics J.
**2**, 375–392 (1994).Google Scholar - 17.V. L. Kolmogorov and G. I. Shishkin, “Numerical Methods for Singularly Perturbed Boundary Value Problems Modeling Diffusion Processes,” in
*Singular Perturbation Problems in Chemical Physics, Ser. Advances in Chemical Physics*(Wiley, 1997), Vol. XCVII, pp. 281–362.Google Scholar - 18.G. I. Shishkin, “Singularly Perturbed Boundary Value Problems with Concentrated Sources and Discontinuous Initial Conditions,” Zh. Vychisl. Mat. Mat. Fiz.
**37**(4), 429–446 (1997) [Comput. Math. Math. Phys.**37**, 417–434 (1997)].zbMATHMathSciNetGoogle Scholar - 19.G. I. Shishkin, “Grid Approximation of a Singularly Perturbed Elliptic Convection-Diffusion Equation in an Unbounded Domain,” Rus. J. Numer. Analys. Math. Modelling
**21**(1), 67–94 (2006).zbMATHCrossRefMathSciNetGoogle Scholar - 20.G. I. Shishkin, Grid Approximation of Singularly Perturbed Boundary Value Problems for Quasi-Linear Parabolic Equations in Case of Complete Degeneracy in Spatial Variables,” Sov. J. Numer. Analys. Math. Modelling
**6**(3), 243–261 (1991).zbMATHMathSciNetCrossRefGoogle Scholar - 21.N. S. Bakhvalov,
*Numerical Methods*(Nauka, Moscow, 1973) [in Russian].Google Scholar