A test problem for the ‘particle-in-cell’ method is proposed which allows one to check individual errors on each stage of numerical implementation of the method. General testing usually controls only total (final) error. Using the test problem, we analyze errors of a difference method of MacCormack type and the CIC method being the most popular version of the ‘particle-in-cell’ method. It is shown that the CIC method having also the second formal order of accuracy is seriously inferior to the difference method under the same given error and same grid parameters. In particular, the results of the particle method lose their sense after approximately one calculation period, although the time interval of confidence in simulation data for the difference method is practically unlimited. The testing method proposed here is suitable for an arbitrary one-dimensional version of the ‘particle-in-cell’ method, and it allows one not only to compare numerical implementations of individual stages with each other, but also to verify available theoretical results.
{"title":"On a 1D-electrostatic test problem for the PIC method","authors":"Eugene V. Chizhonkov","doi":"10.1515/rnam-2024-0002","DOIUrl":"https://doi.org/10.1515/rnam-2024-0002","url":null,"abstract":"A test problem for the ‘particle-in-cell’ method is proposed which allows one to check individual errors on each stage of numerical implementation of the method. General testing usually controls only total (final) error. Using the test problem, we analyze errors of a difference method of MacCormack type and the CIC method being the most popular version of the ‘particle-in-cell’ method. It is shown that the CIC method having also the second formal order of accuracy is seriously inferior to the difference method under the same given error and same grid parameters. In particular, the results of the particle method lose their sense after approximately one calculation period, although the time interval of confidence in simulation data for the difference method is practically unlimited. The testing method proposed here is suitable for an arbitrary one-dimensional version of the ‘particle-in-cell’ method, and it allows one not only to compare numerical implementations of individual stages with each other, but also to verify available theoretical results.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"61 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756983","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}
Kseniya A. Alipova, Vasiliy G. Mizyak, Mikhail A. Tolstykh, Gordey S. Goyman
An algorithm for stochastic perturbation of the semi-Lagrangian trajectories is implemented in the ensemble weather prediction system based on the global atmosphere model SL-AV20 with a horizontal resolution of approximately 20 km, 51 vertical levels, and Local Ensemble Transform Kalman Filter (LETKF). The combined use of methods for stochastic perturbation of trajectories and the parameters and tendencies of subgrid-scale processes parameterizations allows to generate ensembles with a larger spread compared to ensembles without stochastic perturbations of trajectories. An improvement in probabilistic estimates of the ensemble forecasts for various variables is shown. The comparison of two versions of ensemble prediction system is presented.
{"title":"Stochastic perturbations in the semi-Lagrangian advection algorithm of the SL-AV global atmosphere model","authors":"Kseniya A. Alipova, Vasiliy G. Mizyak, Mikhail A. Tolstykh, Gordey S. Goyman","doi":"10.1515/rnam-2024-0001","DOIUrl":"https://doi.org/10.1515/rnam-2024-0001","url":null,"abstract":"An algorithm for stochastic perturbation of the semi-Lagrangian trajectories is implemented in the ensemble weather prediction system based on the global atmosphere model SL-AV20 with a horizontal resolution of approximately 20 km, 51 vertical levels, and Local Ensemble Transform Kalman Filter (LETKF). The combined use of methods for stochastic perturbation of trajectories and the parameters and tendencies of subgrid-scale processes parameterizations allows to generate ensembles with a larger spread compared to ensembles without stochastic perturbations of trajectories. An improvement in probabilistic estimates of the ensemble forecasts for various variables is shown. The comparison of two versions of ensemble prediction system is presented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"17 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756759","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}
Grigory V. Zasko, Andrey V. Boiko, Kirill V. Demyanko, Yuri M. Nechepurenko
The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances.
{"title":"Simulating the propagation of boundary-layer disturbances by solving boundary-value and initial-value problems","authors":"Grigory V. Zasko, Andrey V. Boiko, Kirill V. Demyanko, Yuri M. Nechepurenko","doi":"10.1515/rnam-2024-0005","DOIUrl":"https://doi.org/10.1515/rnam-2024-0005","url":null,"abstract":"The article deals with the downstream propagation of small–amplitude disturbances of viscous incompressible laminar boundary layers, using the linearized equations for disturbance amplitudes. Two different methods are proposed. The first one solves a two-dimensional boundary-value problem, using a buffer-domain technique to mimic the outflow boundary condition. The second one solves a streamwise initial-value problem, using a spectral parabolization at each integration step. Both methods show good performance in simulating the propagation of the Tollmien–Schlichting waves and the Görtler vortices and can be applied to compute the spatial optimal disturbances.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"9 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756791","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}
Aleksei F. Seleznev, Andrey S. Gavrilov, Dmitry N. Mukhin, Andrey S. Gritsun, Evgenii M. Volodin
Advanced numerical climate models are known to exhibit biases in simulating some features of El Niño–Southern Oscillation (ENSO), which is a key mode of interannual climate variability. In this study we analyze how two fundamental features of observed ENSO – asymmetry between hot and cold states and phase-locking to the annual cycle – are reflected in two different versions of the INMCM Earth system model (state-of-the-art Earth system model participating in the Coupled Model Intercomparison Project). We identify the above ENSO features using the conventional empirical orthogonal functions (EOF) analysis, which is applied to both observed and simulated upper ocean heat content (OHC) data in the tropical Pacific. We obtain that the observed tropical Pacific OHC variability is described well by two leading EOF-modes, which roughly reflect the fundamental recharge-discharge mechanism of ENSO. These modes exhibit strong seasonal cycles associated with ENSO phase locking while the revealed nonlinear dependencies between amplitudes of these cycles reflect ENSO asymmetry.We also assess and compare the predictability of observed and simulated ENSO based on linear inverse modelling. We find that the improved INMCM6 model has significant benefits in simulating described features of observed ENSO as compared with the previous INMCM5 model. The improvements of the INMCM6 model providing such benefits are discussed. We argue that proper cloud parameterization scheme is crucial for accurate simulation of ENSO dynamics with numerical climate models.
{"title":"ENSO phase locking, asymmetry and predictability in the INMCM Earth system model","authors":"Aleksei F. Seleznev, Andrey S. Gavrilov, Dmitry N. Mukhin, Andrey S. Gritsun, Evgenii M. Volodin","doi":"10.1515/rnam-2024-0004","DOIUrl":"https://doi.org/10.1515/rnam-2024-0004","url":null,"abstract":"Advanced numerical climate models are known to exhibit biases in simulating some features of El Niño–Southern Oscillation (ENSO), which is a key mode of interannual climate variability. In this study we analyze how two fundamental features of observed ENSO – asymmetry between hot and cold states and phase-locking to the annual cycle – are reflected in two different versions of the INMCM Earth system model (state-of-the-art Earth system model participating in the Coupled Model Intercomparison Project). We identify the above ENSO features using the conventional empirical orthogonal functions (EOF) analysis, which is applied to both observed and simulated upper ocean heat content (OHC) data in the tropical Pacific. We obtain that the observed tropical Pacific OHC variability is described well by two leading EOF-modes, which roughly reflect the fundamental recharge-discharge mechanism of ENSO. These modes exhibit strong seasonal cycles associated with ENSO phase locking while the revealed nonlinear dependencies between amplitudes of these cycles reflect ENSO asymmetry.We also assess and compare the predictability of observed and simulated ENSO based on linear inverse modelling. We find that the improved INMCM6 model has significant benefits in simulating described features of observed ENSO as compared with the previous INMCM5 model. The improvements of the INMCM6 model providing such benefits are discussed. We argue that proper cloud parameterization scheme is crucial for accurate simulation of ENSO dynamics with numerical climate models.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":"18 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139756795","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}
We show theoretically and numerically that the lowest non-trivial eigenvector function for a specific eigenproblem has almost constant values in high conductivity channels, which are different in separate channels. Therefore, based on these distinct values, all separate connected clusters of open pores can be identified in digital cores.
{"title":"Extracting connectivity paths in digital core images using solution of partial minimum eigenvalue problem","authors":"Serguei Yu. Maliassov, Yuri V. Vassilevski","doi":"10.1515/rnam-2023-0028","DOIUrl":"https://doi.org/10.1515/rnam-2023-0028","url":null,"abstract":"We show theoretically and numerically that the lowest non-trivial eigenvector function for a specific eigenproblem has almost constant values in high conductivity channels, which are different in separate channels. Therefore, based on these distinct values, all separate connected clusters of open pores can be identified in digital cores.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 5","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492818","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}
Ilya D. Tretyak, Gordey S. Goyman, Vladimir V. Shashkin
We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.
{"title":"Multiresolution approximation for shallow water equations using summation-by-parts finite differences","authors":"Ilya D. Tretyak, Gordey S. Goyman, Vladimir V. Shashkin","doi":"10.1515/rnam-2023-0030","DOIUrl":"https://doi.org/10.1515/rnam-2023-0030","url":null,"abstract":"We present spatial approximation for shallow water equations on a mesh of multiple rectangular blocks with different resolution in Cartesian geometry. The approximation is based on finite-difference operators that fulfill Summation By Parts (SBP) property – a discrete analogue of integration by parts. The solution continuity conditions between mesh blocks are imposed in a weak form using Simultaneous Approximation Terms (SAT) method.We show that the resulting discrete divergence and gradient operators are anti-conjugate. The important consequences are the discrete analogues for mass and energy conservation laws along with the proof of stability for linearized equations. The numerical shallow water equations model based on the presented spatial approximation is tested using problems with meteorological context. Test results prove high-order accuracy of SBP-SAT discretization. The interfaces between mesh blocks of different resolution produce no significant noise. The local mesh refinement is shown to have positive effect on the solution both locally inside the refined region and globally in the dynamically coupled areas.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492819","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}
Dmitry V. Kulyamin, Sergey V. Kostrykin, Pavel A. Ostanin, Valentin P. Dymnikov
The paper provides a detailed description of the numerical implementation of the transport scheme in the Earth’s ionosphere three-dimensional dynamical model of the Institute of Numerical Mathematics (INMIM). The presented version of INM-IM model takes into account the global dynamical processes of ion transport and ambipolar diffusion in the altitude range between 100 and 500 km (corresponding to F region). Based upon the splitting method the model solves the equations of ambipolar diffusion on the first split step and incorporates specifically designed advective transport scheme on the second step. The accuracy of transport scheme implementation in the model has been investigated through analytical solutions. The stability of the numerical algorithm has been demonstrated even for cases when transport velocities approaching extreme values.
{"title":"Numerical model of Earth ionosphere F region based on three-dimensional transport and ambipolar diffusion equations","authors":"Dmitry V. Kulyamin, Sergey V. Kostrykin, Pavel A. Ostanin, Valentin P. Dymnikov","doi":"10.1515/rnam-2023-0027","DOIUrl":"https://doi.org/10.1515/rnam-2023-0027","url":null,"abstract":"The paper provides a detailed description of the numerical implementation of the transport scheme in the Earth’s ionosphere three-dimensional dynamical model of the Institute of Numerical Mathematics (INMIM). The presented version of INM-IM model takes into account the global dynamical processes of ion transport and ambipolar diffusion in the altitude range between 100 and 500 km (corresponding to F region). Based upon the splitting method the model solves the equations of ambipolar diffusion on the first split step and incorporates specifically designed advective transport scheme on the second step. The accuracy of transport scheme implementation in the model has been investigated through analytical solutions. The stability of the numerical algorithm has been demonstrated even for cases when transport velocities approaching extreme values.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 9","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492814","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}
Dmitry V. Blagodatskikh, Nikolay G. Iakovlev, Evgenii M. Volodin, Andrey S. Gritsun
The present paper considers numerical properties of two different approaches to discretization of the isoneutral diffusion. The necessity of an alternative treatment of the isoneutral diffusion in a terrain-following climate ocean model as opposed to the more convenient rotated tensor formalism is studied. A new method of the approximation of the isoneutral diffusion based on a non-local computational stencil is formulated. The validity of the non-local discretization of the isoneutral diffusion operator with regard to a terrain-following vertical coordinate in the INMCM ocean model is demonstrated.
{"title":"Non-local discretization of the isoneutral diffusion operator in a terrain-following climate ocean model","authors":"Dmitry V. Blagodatskikh, Nikolay G. Iakovlev, Evgenii M. Volodin, Andrey S. Gritsun","doi":"10.1515/rnam-2023-0026","DOIUrl":"https://doi.org/10.1515/rnam-2023-0026","url":null,"abstract":"The present paper considers numerical properties of two different approaches to discretization of the isoneutral diffusion. The necessity of an alternative treatment of the isoneutral diffusion in a terrain-following climate ocean model as opposed to the more convenient rotated tensor formalism is studied. A new method of the approximation of the isoneutral diffusion based on a non-local computational stencil is formulated. The validity of the non-local discretization of the isoneutral diffusion operator with regard to a terrain-following vertical coordinate in the INMCM ocean model is demonstrated.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 8","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492815","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}
Shale rock, being a common caprock for carbon dioxide reservoirs, is subject to extensive research. One of the topics is breakthrough phenomena during injection of supercritical carbon dioxide in shale, the nature of which is still to be fully understood. In the present paper, a two-phase flow model, which may possibly be used to explain the breakthrough behaviour is examined. Capillary effects and geomechanics are included in the model, and influential model parameters are determined.
{"title":"Explaining breakthrough behaviour in shale rock: influence of capillary effects and geomechanics","authors":"Denis Anuprienko, Valentina Svitelman","doi":"10.1515/rnam-2023-0025","DOIUrl":"https://doi.org/10.1515/rnam-2023-0025","url":null,"abstract":"Shale rock, being a common caprock for carbon dioxide reservoirs, is subject to extensive research. One of the topics is breakthrough phenomena during injection of supercritical carbon dioxide in shale, the nature of which is still to be fully understood. In the present paper, a two-phase flow model, which may possibly be used to explain the breakthrough behaviour is examined. Capillary effects and geomechanics are included in the model, and influential model parameters are determined.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 6","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492817","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}
Victor P. Shutyaev, Eugene I. Parmuzin, Igor Yu. Gejadze
The problem of variational assimilation of observational data for the sea thermodynamics model is considered with the aim to reconstruct heat fluxes on the sea surface. The local sensitivity of functionals of the solution to the observational data and input data for heat fluxes is studied in the considered problem and the results of numerical experiments are presented for the Black Sea dynamics model.
{"title":"The study of the local sensitivity of functionals of the optimal solution to observational data and the heat flux input data in a variational assimilation problem for the sea thermodynamics model","authors":"Victor P. Shutyaev, Eugene I. Parmuzin, Igor Yu. Gejadze","doi":"10.1515/rnam-2023-0029","DOIUrl":"https://doi.org/10.1515/rnam-2023-0029","url":null,"abstract":"The problem of variational assimilation of observational data for the sea thermodynamics model is considered with the aim to reconstruct heat fluxes on the sea surface. The local sensitivity of functionals of the solution to the observational data and input data for heat fluxes is studied in the considered problem and the results of numerical experiments are presented for the Black Sea dynamics model.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":" 7","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138492816","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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