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

G. I. Shishkin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 10, pp. 1827-1843

Résumé

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 $\varepsilon^2$, where $\varepsilon\in(0,1]$. The right-hand side of the equation and the initial function grow unboundedly as $x\to\infty$ at a rate of $O(x^2)$. This causes the unbounded growth of the solution at infinity at a rate of $O(\Psi(x))$, where $\Psi(x)=x^2+1$. The initialboundary function is piecewise smooth. When $\varepsilon$ 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 $x\to\infty$ even for smooth solutions when $\varepsilon$ 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 $\|\cdot\|^w$ with the weighting function $\Psi^{-1}(x)$; in this norm, the solution of the initial-boundary value problem is $\varepsilon$-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 $\varepsilon$-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 $O(\Psi(x))$ that converge $\varepsilon$-uniformly in the weighted norm are also considered.

Bibliographie

[1] Bakhvalov N. S., “K optimizatsii metodov resheniya kraevykh zadach pri nalichii pogranichnogo sloya”, Zh. vychisl. matem. i matem. fiz., 9:4 (1969), 841–859 | Zbl

[2] Ilin A. M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Matem. zametki, 6:2 (1969), 237–248

[3] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, RAN, UrO, Ekaterinburg, 1992

[4] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., Robust computational techniques for boundary layers, Chapman and Hall/CRC, Boca Raton, 2000 | MR | Zbl

[5] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted numerical methods for singular perturbation problems, World Sci., Singapore, 1996 | MR

[6] Roos H.-H., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, Springer, Berlin, 1996 | MR

[7] Shishkin G. I., Shishkina L. P., Difference methods for singular perturbation problems., Sen Monographs Surveys in Pure Appl. Math., Chapman and Hall/CRC, Boca Raton, 2009 | MR | Zbl

[8] Ladyzhenskaya O. A., Solonnikov B. A., Uraltseva H. H., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

[9] Shishkin G. I., “Setochnaya approksimatsiya na poluploskosti singulyarno vozmuschennykh ellipticheskikh uravnenii s konvektivnymi chlenami, rastuschimi na beskonechnosti”, Zh. vychisl. matem. i matem. fiz., 45:2 (2005), 298–314 | MR | Zbl

[10] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennykh parabolicheskikh uravnenii reaktsii-diffuzii nabolshikh oblastyakh po prostranstvennoi i vremennoi peremennym”, Zh. vychisl. matem. i matem. fiz., 46:11 (2006), 2045–2064 | MR

[11] Li S., Shishkin G., Shishkina L., “Approximation of the solution and its derivative for the singularly perturbed Black–Scholes equation with nonsmooth initial data”, Zh. vychisl. matem. i matem. fiz., 47:3 (2007), 460–480 | MR | Zbl

[12] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[13] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennykh parabolicheskikh uravnenii konvektsii-diffuzii s kusochno-gladkim nachalnym usloviem”, Zh. vychisl. matem. i matem. fiz., 46:1 (2006), 52–76 | MR | Zbl

[14] Shishkin G. I., “Grid approximation of singularly perturbed parabolic reaction-diffusion equations with piecewise smooth initial-boundary conditions”, Math. Modelling and Analys, 12:2 (2007), 235–254 | DOI | MR | Zbl

[15] Shishkin G. I., “Grid approximation of singularly perturbed parabolic equations with piecewise continuous initialboundary conditions”, Proc. Steklov Inst. Math., Suppl. 2, 2007, S213–S230 | DOI

[16] Hemker P. W., Shishkin G. I., “Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition”, Comput. Fluid Dynamics J., 2:4 (1994), 375–392

[17] Kolmogorov V. L., Shishkin G. I., “Numerical methods for singularly perturbed boundary value problems modeling diffusion processes”, Singular Perturbation Problems in Chem. Phys., Advances in Chem. Phys. Ser., 457, J. Wiley Sons, 1997, 181–362

[18] Shishkin G. I., “Singulyarno vozmuschennye kraevye zadachi s sosredotochennymi istochnikami i razryvnymi nachalnymi usloviyami”, Zh. vychisl. matem. i matem. fiz., 37:4 (1997), 429–446 | MR | Zbl

[19] Shishkin G. I., “Grid approximation of a singularly perturbed elliptic convection-diffusion equation in an unbounded domain”, Rus. J. Numer. Analys. Math. Modelling, 21:1 (2006), 67–94 | DOI | MR | Zbl

[20] Shishkin G. I., “Grid approximation of singularly perturbed boundary value problems for quasi-linear parabolic equations in case of complete degeneracy in spatial variables”, Soviet J. Numer. Analys. Math. Modelling, 6:3 (1991), 234–261 | MR

[21] Bakhvalov H. C., Chislennye metody, Nauka, M., 1973 | Zbl