Geodesic
Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 218-233 
G. I. Shishkin. Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 13 (2007) no. 2, pp. 218-233. http://geodesic.mathdoc.fr/item/TIMM_2007_13_2_a17/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter $\varepsilon^2$, $\varepsilon\in (0,1]$. For small values of the parameter $\varepsilon$, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width $\varepsilon$), respectively, which have bounded smoothness for fixed values of the parameter $\varepsilon$. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge $\varepsilon$-uniformly with the second order of accuracy in $x$ and the first order of accuracy in $t$, up to logarithmic factors.

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

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

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

[4] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E. and Shishkin G. I., Robust Computational Techniques for Boundary Layers, FL: Chapman and Hall/CRC, Boca Raton, 2000 | MR | Zbl

[5] Miller J. J. H., O'Riordan E. and Shishkin G. I., Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996 | MR

[6] Roos H.-G., Stynes M., Tobiska L., Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems, Springer-Verlag, Berlin, 1996 | MR

[7] Kolmogorov V. L., Shishkin G. I., “Numerical methods for singularly perturbed boundary value problems modelling diffusion processes”, Singular Perturbation Problems in Chemical Physics, Advances in Chemical Physics, 97, eds. J. J. H. Miller, 1997, 181–362

[8] S. Li, Shishkin G. and Shishkina L., “Approximation of the solution and its derivative for the singularly perturbed Black-Scholes equation with nonsmooth initial data”, Comp. Math. Math. Phys., 47:3 (2007), 442–462 | DOI | MR | Zbl

[9] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennykh parabolicheskikh uravnenii konvektsii-diffuzii s kusochno-gladkim nachalnym usloviem”, Zhurn. vychisl. matematiki i mat. fiziki, 46:1 (2006), 52–76 | MR

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

[11] 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

[12] Marchuk G. I.,Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979 | MR

[13] Shishkin G. I., “Setochnaya approksimatsiya singulyarno vozmuschennoi kraevoi zadachi dlya kvazilineinogo ellipticheskogo uravneniya v sluchae polnogo vyrozhdeniya”, Zhurn. vychisl. matematiki i mat. fiziki, 31:12 (1991), 1808–1825 | MR | Zbl

[14] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967

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