Abstract The problem of minimizing the root-mean-square error of the numerical-statistical projection estimation of the solution to an integral equation is solved. It is shown that the optimal estimator in this sense can be obtained by equalizing deterministic and stochastic components of the error in the case when the norm of the remainder of the utilized decomposition decreases inversely proportional to its length. As a test, the Milne problem of radiation transfer in a semi-infinite layer of matter is solved using Laguerre polynomials. To solve such a problem in the case of a finite layer, a special regularized projection algorithm is used.
{"title":"Construction and optimization of numerically-statistical projection algorithms for solving integral equations","authors":"A. S. Korda, G. A. Mikhailov, S. V. Rogasinsky","doi":"10.1515/rnam-2022-0018","DOIUrl":"https://doi.org/10.1515/rnam-2022-0018","url":null,"abstract":"Abstract The problem of minimizing the root-mean-square error of the numerical-statistical projection estimation of the solution to an integral equation is solved. It is shown that the optimal estimator in this sense can be obtained by equalizing deterministic and stochastic components of the error in the case when the norm of the remainder of the utilized decomposition decreases inversely proportional to its length. As a test, the Milne problem of radiation transfer in a semi-infinite layer of matter is solved using Laguerre polynomials. To solve such a problem in the case of a finite layer, a special regularized projection algorithm is used.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"213 - 219"},"PeriodicalIF":0.6,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42598300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract An iterative method with the number of iterations independent of the coefficient jumps is proposed for the boundary value problem for a diffusion equation with highly varying coefficient. The method applies one solution of the Poisson equation at each step of iteration. In the present paper we extend the class of domains the iterative method is justified for.
{"title":"Connection between the existence of a priori estimate for a flux and the convergence of iterative methods for diffusion equation with highly varying coefficients","authors":"G. Kobelkov, E. Schnack","doi":"10.1515/rnam-2022-0012","DOIUrl":"https://doi.org/10.1515/rnam-2022-0012","url":null,"abstract":"Abstract An iterative method with the number of iterations independent of the coefficient jumps is proposed for the boundary value problem for a diffusion equation with highly varying coefficient. The method applies one solution of the Poisson equation at each step of iteration. In the present paper we extend the class of domains the iterative method is justified for.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"143 - 147"},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44989861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A technique for constructing a sequence of difference schemes with the properties of a corrector and a predictor for integrating systems of the second-order ordinary differential equations is presented. The sequence of schemes begins with the explicit three-point Störmer method of the second order of approximation. Each subsequent scheme also implements the Störmer method corrected with additional terms calculated through the solution of the previous scheme. The stability of the resulting schemes and the increase in the order of convergence for the first of them are carefully substantiated. The results of calculations of the test problem are presented, confirming the increase in the order of accuracy of the constructed methods.
{"title":"Difference schemes for second-order ordinary differential equations with corrector and predictor properties","authors":"V. Shaidurov, A. Novikov","doi":"10.1515/rnam-2022-0015","DOIUrl":"https://doi.org/10.1515/rnam-2022-0015","url":null,"abstract":"Abstract A technique for constructing a sequence of difference schemes with the properties of a corrector and a predictor for integrating systems of the second-order ordinary differential equations is presented. The sequence of schemes begins with the explicit three-point Störmer method of the second order of approximation. Each subsequent scheme also implements the Störmer method corrected with additional terms calculated through the solution of the previous scheme. The stability of the resulting schemes and the increase in the order of convergence for the first of them are carefully substantiated. The results of calculations of the test problem are presented, confirming the increase in the order of accuracy of the constructed methods.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"175 - 187"},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45625216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A finite element method for a monolithic quasi-Lagrangian formulation of a fluid–porous structure interaction problem with a corrected balance of stresses on the fluid–structure interface is considered. Deformations of the elastic medium are not necessarily small and are modelled using Saint Venant–Kirchhoff (SVK) constitutive relation. The stability of the method is proved in a form of energy bound for the finite element solution.
{"title":"A finite element scheme for the numerical solution of the Navier–Stokes/Biot coupled problem","authors":"A. Lozovskiy, M. Olshanskii, Y. Vassilevski","doi":"10.1515/rnam-2022-0014","DOIUrl":"https://doi.org/10.1515/rnam-2022-0014","url":null,"abstract":"Abstract A finite element method for a monolithic quasi-Lagrangian formulation of a fluid–porous structure interaction problem with a corrected balance of stresses on the fluid–structure interface is considered. Deformations of the elastic medium are not necessarily small and are modelled using Saint Venant–Kirchhoff (SVK) constitutive relation. The stability of the method is proved in a form of energy bound for the finite element solution.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"159 - 174"},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44609689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
V. Agoshkov, V. Zalesny, V. Shutyaev, E. Parmuzin, N. Zakharova
Abstract The 4D variational data assimilation technique is presented for modelling the sea dynamics problems, developed at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS). The approach is based on the splitting method for the mathematical model of sea dynamics and the minimization of cost functionals related to the observation data by solving an optimality system that involves the adjoint equations and observation and background error covariances. Efficient algorithms for solving the variational data assimilation problems are presented based on iterative processes with a special choice of iterative parameters. The technique is illustrated for the Black Sea dynamics model with variational data assimilation to restore the sea surface heat fluxes.
{"title":"Variational data assimilation for a sea dynamics model","authors":"V. Agoshkov, V. Zalesny, V. Shutyaev, E. Parmuzin, N. Zakharova","doi":"10.1515/rnam-2022-0011","DOIUrl":"https://doi.org/10.1515/rnam-2022-0011","url":null,"abstract":"Abstract The 4D variational data assimilation technique is presented for modelling the sea dynamics problems, developed at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS). The approach is based on the splitting method for the mathematical model of sea dynamics and the minimization of cost functionals related to the observation data by solving an optimality system that involves the adjoint equations and observation and background error covariances. Efficient algorithms for solving the variational data assimilation problems are presented based on iterative processes with a special choice of iterative parameters. The technique is illustrated for the Black Sea dynamics model with variational data assimilation to restore the sea surface heat fluxes.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"131 - 142"},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49108249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The time-fractional phase transition problem, formulated in enthalpy form, is studied. This nonlinear problem with an unknown moving boundary includes, as an example, a mathematical model of one-phase Stefan problem with the latent heat accumulation memory. The posed problem is approximated by the backward Euler mesh scheme. The unique solvability of the mesh scheme is proved and a priori estimates for the solution are obtained. The properties of the mesh problem are studied, in particular, an estimate of movement rate for the mesh phase transition boundary is established. The proved estimate make it possible to localize the phase transition boundary and split the mesh scheme into the sum of a nonlinear problem of small algebraic dimension and a larger linear problem. This information can be used for further construction of efficient algorithms for implementing the mesh scheme. Several algorithms for implementing mesh scheme are briefly discussed.
{"title":"Mesh scheme for a phase transition problem with time-fractional derivative","authors":"A. Lapin","doi":"10.1515/rnam-2022-0013","DOIUrl":"https://doi.org/10.1515/rnam-2022-0013","url":null,"abstract":"Abstract The time-fractional phase transition problem, formulated in enthalpy form, is studied. This nonlinear problem with an unknown moving boundary includes, as an example, a mathematical model of one-phase Stefan problem with the latent heat accumulation memory. The posed problem is approximated by the backward Euler mesh scheme. The unique solvability of the mesh scheme is proved and a priori estimates for the solution are obtained. The properties of the mesh problem are studied, in particular, an estimate of movement rate for the mesh phase transition boundary is established. The proved estimate make it possible to localize the phase transition boundary and split the mesh scheme into the sum of a nonlinear problem of small algebraic dimension and a larger linear problem. This information can be used for further construction of efficient algorithms for implementing the mesh scheme. Several algorithms for implementing mesh scheme are briefly discussed.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"149 - 158"},"PeriodicalIF":0.6,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44817096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Idealized numerical experiments with the INM RAS climate model are used to study the potential predictability of the temperature in the upper 300-meter layer of the Arctic Ocean. It is shown that the heat content can be predictable for up to 4–6 years. Positive anomalies of the temperature and salinity are preceded for several years by a state in which the influx of Atlantic water into the Arctic Ocean exceeds the average value. Surface fields, including temperature, salinity, concentration and mass of ice, are less predictable than the heat content in the layer of 0–300 meters.
{"title":"On the multi-annual potential predictability of the Arctic Ocean climate state in the INM RAS climate model","authors":"E. Volodin, V. Vorobyeva","doi":"10.1515/rnam-2022-0010","DOIUrl":"https://doi.org/10.1515/rnam-2022-0010","url":null,"abstract":"Abstract Idealized numerical experiments with the INM RAS climate model are used to study the potential predictability of the temperature in the upper 300-meter layer of the Arctic Ocean. It is shown that the heat content can be predictable for up to 4–6 years. Positive anomalies of the temperature and salinity are preceded for several years by a state in which the influx of Atlantic water into the Arctic Ocean exceeds the average value. Surface fields, including temperature, salinity, concentration and mass of ice, are less predictable than the heat content in the layer of 0–300 meters.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"119 - 129"},"PeriodicalIF":0.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46289726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper is focused on the construction of a numerical stochastic model of the joint spatio-temporal fields of air temperature, wind speed vector with three-hour resolution, and semidiurnal precipitation amounts according to observation data at a group of weather stations located in the south of the Baikal natural territory. The model also takes into account the dependence of one-dimensional distributions on temporal and spatial coordinates. The heterogeneity of the field in spatial correlations and the periodical correlation in time are also taken into account. The results of calculations for verification of the model are presented. An example of using the developed model to study the properties of time series of the wind chill index is given.
{"title":"Development of a numerical stochastic model of joint spatio-temporal fields of weather parameters for the south part of the Baikal natural territory","authors":"M. S. Akenteva, N. Kargapolova, V. Ogorodnikov","doi":"10.1515/rnam-2022-0006","DOIUrl":"https://doi.org/10.1515/rnam-2022-0006","url":null,"abstract":"Abstract The paper is focused on the construction of a numerical stochastic model of the joint spatio-temporal fields of air temperature, wind speed vector with three-hour resolution, and semidiurnal precipitation amounts according to observation data at a group of weather stations located in the south of the Baikal natural territory. The model also takes into account the dependence of one-dimensional distributions on temporal and spatial coordinates. The heterogeneity of the field in spatial correlations and the periodical correlation in time are also taken into account. The results of calculations for verification of the model are presented. An example of using the developed model to study the properties of time series of the wind chill index is given.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"73 - 83"},"PeriodicalIF":0.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46037937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-01DOI: 10.1515/rnam-2022-frontmatter2
{"title":"Frontmatter","authors":"","doi":"10.1515/rnam-2022-frontmatter2","DOIUrl":"https://doi.org/10.1515/rnam-2022-frontmatter2","url":null,"abstract":"","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48557880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce discrete curvatures for planar curves based on the construction of sequences of pairs of mutually dual polylines. For piecewise-regular curves consisting of a finite number of fragments of regular generalized spirals with definite (positive or negative) curvatures our discrete curvatures approximate the exact averaged curvature from below and from above. In order to derive these estimates one should provide a distance function allowing to compute the closest point on the curve for an arbitrary point on the plane.With refinement of the polylines, the averaged curvature over refined curve segments converges to the pointwise values of the curvature and, thus, we obtain a good and stable local approximation of the curvature. For the important engineering case when the curve is approximated only by the inscribed (primal) polyline and the exact distance function is not available, we provide a comparative analysis for several techniques allowing to build dual polylines and discrete curvatures and evaluate their ability to create lower and upper estimates for the averaged curvature.
{"title":"Discrete curvatures for planar curves based on Archimedes’ duality principle","authors":"V. Garanzha, L. Kudryavtseva, Dmitry A. Makarov","doi":"10.1515/rnam-2022-0007","DOIUrl":"https://doi.org/10.1515/rnam-2022-0007","url":null,"abstract":"Abstract We introduce discrete curvatures for planar curves based on the construction of sequences of pairs of mutually dual polylines. For piecewise-regular curves consisting of a finite number of fragments of regular generalized spirals with definite (positive or negative) curvatures our discrete curvatures approximate the exact averaged curvature from below and from above. In order to derive these estimates one should provide a distance function allowing to compute the closest point on the curve for an arbitrary point on the plane.With refinement of the polylines, the averaged curvature over refined curve segments converges to the pointwise values of the curvature and, thus, we obtain a good and stable local approximation of the curvature. For the important engineering case when the curve is approximated only by the inscribed (primal) polyline and the exact distance function is not available, we provide a comparative analysis for several techniques allowing to build dual polylines and discrete curvatures and evaluate their ability to create lower and upper estimates for the averaged curvature.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"37 1","pages":"85 - 98"},"PeriodicalIF":0.6,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47959941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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