Grid approximation of singularly perturbed parabolic reaction‐diffusion equations with piecewise smooth initial‐boundary conditions

References

• Bakhvalov , N.S. 1969 . On the optimization of methods for boundary‐value problems with boundary layers . Zh. Vychisl. Mat. Mat. Fiz. , 9 (4) : 841 – 859 . (in Russian)
   Google Scholar
• Farrell , P.A. , Hegarty , A.F. , Miller , J.J.H. , O'Riordan , E. and Shishkin , G.I. 2000 . Robust Computational Techniques for Boundary Layers , Boca Raton : Chapman and Hall/CRC Press .
   Google Scholar
• Hemker , P.W. and Shishkin , G.I. 1994 . Discrete approximation of singularly perturbed parabolic PDEs with a discontinuous initial condition . Computational Fluid Dynamics Journal , 2 (4) : 375 – 392 .
   Google Scholar
• Il'in , A.M. 1969 . Differencing scheme for a differential equation with a small parameter affecting the highest derivative . Math. Notes , 6 (2) : 596 – 602 .
   Google Scholar
• Kolmogorov , V.L. and Shishkin , G.I. 1997 . “ Numerical methods for singularly perturbed boundary value problems modeling diffusion processes ” . In Singular Perturbation Problems in Chemical Physics , Edited by: Miller , J.J.H. 181 – 362 . New York : Advances in Chemical Physics, John Willey & Sons .
   Google Scholar
• Ladyzhenskaya , O.A. , Solonnikov , V.A. and Ural'tseva , N.N. 1968 . Linear and Quasi‐linear Equations of Parabolic Type , Translations of Mathematical Monographs, 23 USA : Americal Mathematical Society .
   Google Scholar
• Li , S. , Shishkin , G. and Shishkina , L. 2007 . Approximation of the solution and its derivative for the singularly perturbed Black‐Scholes equation with nonsmooth initial data . Comp. Maths. Math. Phys. , 47 (3) accepted
   Google Scholar
• Miller , J.J.H. , O'Riordan , E. and Shishkin , G.I. 1996 . Fitted Numerical Methods for Singular Perturbation Problems , Singapore : World Scientific .
   Google Scholar
• Roos , H.‐G. , Stynes , M. and Tobiska , L. 1996 . Numerical Methods for Singularly Perturbed Differential Equations. Convection‐Diffusion and Flow Problems , Berlin : Springer‐Verlag .
   Google Scholar
• Samarskii , A.A. 2001 . The Theory of Difference Schemes , New York : Marcel Dekker .
   Google Scholar
• Shishkin , G.I. 1988 . A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous boundary condition . Comp. Maths. Math. Phys. , 28 (11) : 32 – 41 .
   Google Scholar
• Shishkin , G.I. 1988 . A difference scheme for a singularly perturbed equation of parabolic type with a discontinuous initial condition . Soviet Math. Dokl. , 37 (3) : 792 – 796 .
   Google Scholar
• Shishkin , G.I. 1991 . Grid approximation of a singularly perturbed boundary value problem for a quasilinear elliptic equation in the case of complete degeneration . Zh. Vychisl. Mat. Mat. Fiz. , 31 (12) : 1808 – 1825 . (in Russian)
   Google Scholar
• Shishkin , G.I. 1991 . Grid approximation of a singularly perturbed boundary value problem for quasilinear parabolic equations in the case of complete degeneracy in spatial variables . Sov. J. Numer Anal. Math. Modelling , 6 (3) : 243 – 261 .
   Google Scholar
• Shishkin , G.I. 1992 . Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations , Ural Section, Ekaterinburg : Russian Academy of Sciences . (in Russian)
   Google Scholar
• Shishkin , G.I. 1996 . Approximation of solutions and diffusion flows in the case of singularly perturbed boundary value problems with discontinuous initial conditions . Comp. Maths. Math. Phys. , 36 (9) : 1233 – 1250 .
   Google Scholar
• Shishkin , G.I. 1997 . Singularly perturbed boundary value problems with concentrated sources and discontinuous initial conditions . Comp. Maths. Math. Phys. , 37 (4) : 417 – 434 .
   Google Scholar
• Shishkin , G.I. 2005 . Grid approximation of parabolic convection‐diffusion equations with piecewise smooth initial conditions . Doklady Mathematics , 72 (3) : 850 – 853 .
   Google Scholar
• Shishkin , G.I. 2006 . Grid approximation of singularly perturbed parabolic convection‐diffusion equations subject to a piecewise smooth initial condition . Comp. Maths. Math. Phys. , 46 (1) : 49 – 72 .
   Google Scholar
• This research was supported by the Netherlands Research Organisation NWO under grant No. 047.016.008, the Russian Foundation for Basic Research under grants No. 04–01–00578, 04–01–89007‐NWO_a., the Boole Centre for Research in Informatics at the National University of Ireland, Cork and the Mathematics Applications Consortium for Science and Industry in Ireland (MACSI) under the Science Foundation Ireland (SFI) mathematics initiative.
   Google Scholar