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Volume 19, Issue 1
Splitting Schemes for Some Second-Order Evolutionary Equations​

Petr N. Vabishchevich

Int. J. Numer. Anal. Mod., 19 (2022), pp. 19-32.

Published online: 2022-03

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  • Abstract

We consider the Cauchy problem for a second-order evolutionary equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of the operator $A$ by its conjugate operator $A^∗$. Time approximations are implemented so that the transition to a new level in time is associated with a separate solution of problems for operators $A$ and $A^∗$, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (well-posedness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.

  • Keywords

Second-order evolutionary equation, Cauchy problem, explicit schemes, splitting schemes, vibrations of a thin plate.

  • AMS Subject Headings

65J08, 65M06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-19-19, author = {Vabishchevich , Petr N.}, title = {Splitting Schemes for Some Second-Order Evolutionary Equations​}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2022}, volume = {19}, number = {1}, pages = {19--32}, abstract = {

We consider the Cauchy problem for a second-order evolutionary equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of the operator $A$ by its conjugate operator $A^∗$. Time approximations are implemented so that the transition to a new level in time is associated with a separate solution of problems for operators $A$ and $A^∗$, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (well-posedness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/20347.html} }
TY - JOUR T1 - Splitting Schemes for Some Second-Order Evolutionary Equations​ AU - Vabishchevich , Petr N. JO - International Journal of Numerical Analysis and Modeling VL - 1 SP - 19 EP - 32 PY - 2022 DA - 2022/03 SN - 19 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/20347.html KW - Second-order evolutionary equation, Cauchy problem, explicit schemes, splitting schemes, vibrations of a thin plate. AB -

We consider the Cauchy problem for a second-order evolutionary equation, in which the problem operator is the sum of two self-adjoint operators. The main feature of the problem is that one of the operators is represented in the form of the product of the operator $A$ by its conjugate operator $A^∗$. Time approximations are implemented so that the transition to a new level in time is associated with a separate solution of problems for operators $A$ and $A^∗$, not their products. The construction of unconditionally stable schemes is based on general results of the theory of stability (well-posedness) of operator-difference schemes in Hilbert spaces and is associated with the multiplicative perturbation of the problem operators, which lead to stable implicit schemes. As an example, the problem of the dynamics of a thin plate on an elastic foundation is considered.

Vabishchevich , Petr N.. (2022). Splitting Schemes for Some Second-Order Evolutionary Equations​. International Journal of Numerical Analysis and Modeling. 19 (1). 19-32. doi:
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