Construction of difference schemes for singularly perturbed problems.
Investigation of a two-grid method.
The Richardson method
Main publications:
S. V. Tikhovskaya, “A Cascadic Multigrid Algorithm on the Shishkin Mesh for a Singularly Perturbed Elliptic Problem with regular layers”, XV International Scientific and Technical Conference “Applied Mechanics and Systems Dynamics” AMSD 2021 (Omsk, Russian Federation, 09.11 – 11.11.2021), Journal of Physics: Conference Series, 2182, 2022, 012034 , 9 pp.
S. V. Tikhovskaya, “A cascadic multigrid algorithm on the Shishkin mesh for a singularly perturbed elliptic problem”, V International Scientific and Technical Conference “Mechanical Science and Technology Update” MSTU-2021 (Omsk, Russian Federation, 16.03 – 17.03.2021), Journal of Physics: Conference Series, 1901, 2021, 012052 , 8 pp.
S. V. Tikhovskaya, M. F. Korbut, “Two-grid algorithm for a system of singularly perturbed reaction-diffusion equations on Shishkin mesh”, IV International Scientific and Technical Conference “Mechanical Science and Technology Update” (MSTU-2020) (Omsk, Russian Federation, 17.03 – 19.03.2020), Journal of Physics: Conference Series, 1546, 2020, 012104 , 8 pp.
S. V. Tikhovskaya, M. F. Korbut, “Two-grid algorithm for the solution of singularly perturbed two-parameter problem on Shishkin mesh”, 12th International Scientific and Technical Conference on Applied Mechanics and Systems Dynamics (AMSD 2018) (Omsk, Russian Federation, 13.11 – 15.11.2018), Journal of Physics: Conference Series, 1210, 2019, 012142 , 8 pp.
A. I. Zadorin, S. V. Tikhovskaya, “Formulas of numerical differentiation on a uniform mesh for functions with the exponential boundary layer”, International Journal of Numerical Analysis and Modeling, 16:4 (2019), 590–608www.math.ualberta.ca/ijnam/Volume-16-2019/No-4-19/2019-04-04.pdf
S. V. Tikhovskaya, “Solving a Singularly Perturbed Elliptic Problem by a Cascadic Multigrid Algorithm with Richardson Extrapolation”, Finite Difference Methods. Theory and Applications. FDM 2018. (Lozenetz, Bulgaria, 11.06 – 16.06.2018), Lecture Notes in Computer Science, 11386, 2019, 533–541
S. V. Tikhovskaya, “Analysis of the numerical differentiation formulas of functions with large gradients”, Ninth International Conference on Application of Mathematics in Technical and Natural Sciences (AMiTaNS-2017) (Albena, Bulgaria, 21.06 – 26.06.2017), AIP Conference Proceedings, 1895, 2017, 110010 , 9 pp.
S. V. Tikhovskaya, “Solving a singularly perturbed elliptic problem by a multigrid algorithm with Richardson extrapolation”, Sixth Conference on Numerical Analysis and Its Applications (NAA'2016) (Lozenetz, Bulgaria, 15.06 – 22.06.2016), Lecture Notes in Computer Science, 10187, 2017, 674–681
S. V. Tikhovskaya, A. I. Zadorin, “Analysis of polynomial interpolation of the function of two variables with large gradients in the parabolic boundary layers”, Eighth International Conference on Application of Mathematics in Technical and Natural Sciences (AMiTaNS-2016) (Albena, Bulgaria, 22.06 – 27.06.2016), AIP Conference Proceedings, 1773, 2016, 100008 , 9 pp.
S. V. Tikhovskaya , A. I. Zadorin, “A Two-grid Method with Richardson Extrapolation For a Semilinear Convection-diffusion Problem”, Seventh International Conference on Application of Mathematics in Technical and Natural Sciences (AMiTaNS-2015) (Albena, Bulgaria, 28.06 – 03.07.2015), AIP Conference Proceedings, 1684, 2015, 090007 , 8 pp.
S. V. Tikhovskaya, “Investigation of a two-grid method of improved accuracy for the elliptic reaction–diffusion equation with boundary layers”, Kazan. Gos. Univ. Uchen. Zap. Ser. Fiz.-Mat. Nauki, 157:1 (2015), 60–74
A. I. Zadorin, S. V. Tikhovskaya, N. A. Zadorin, “A two-grid method for elliptic problem with boundary layers”, Applied Numerical Mathematics, 93 (2015), 270–278
S. V. Tikhovskaya, Razrabotka raznostnykh skhem na sguschayuschikhsya setkakh dlya kraevykh zadach s pogranichnym sloem, dissertatsiya ... kandidata fiziko-matematicheskikh nauk, Omskii filial Instituta matematiki im. S.L. Soboleva, Omsk, 2013 , 105 pp. "www.sscc.ru/Diss_sov/Tikhovskaya.pdf
A. I. Zadorin, S. V. Tikhovskaya, “A two-grid method for a nonlinear singular perturbation boundary value problem on the Shishkin scheme”, Sib. Zh. Ind. Mat., 16:1 (2013), 42–55
A. I. Zadorin, S. V. Tikhovskaya, “Solving a second-order nonlinear singular perturbation ordinary differential equation by a Samarskii scheme”, Num. Anal. Appl., 6:1 (2013), 9–23 , WOS:000219643200002
S. V. Tikhovskaya, “A Two-Grid Method for an Elliptic Equation with Boundary Layers on a Shishkin Mesh”, Lobachevskii Journal of Mathematics, 35:4 (2014), 409–415 , WOS:000213043900019
A. I. Zadorin, S. V. Tikhovskaya, “Difference scheme on a uniform grid for the singularly perturbed Cauchy problem”, Journal of Mathematical Sciences, 195:6 (2013), 865–872
A. I. Zadorin, S. V. Tikhovskaya, “Analysis of a difference scheme for a singular perturbation Cauchy problem on refined grids”, Num. Anal. Appl., 4:1 (2011), 36–45 , WOS:000219603500004
