Abstract The paper is focused on stochastic modelling of the process of transition of infected cells and virions of HIV-1 infection between two lymph nodes. The model is based on the following assumptions: (1) the duration of transition of infected cells and virions between two lymph nodes is set using a time-dependent function, (2) infected cells produce virions in the process of transition between two lymph nodes, (3) infected cells and virions may die when moving between two lymph nodes. The methods of the theory of branching random processes are used to study analytically the model variables. An algorithm for statistical modelling of the number of infected cells and virions in the second lymph node is presented. The results of computational experiments studying the distribution law of the number of virions produced by one infected cell depending on the duration of movement between two lymph nodes are presented.
{"title":"Numerical modelling of the transition of infected cells and virions between two lymph nodes in a stochastic model of HIV-1 infection","authors":"N. Pertsev, V. Topchii, K. Loginov","doi":"10.1515/rnam-2021-0024","DOIUrl":"https://doi.org/10.1515/rnam-2021-0024","url":null,"abstract":"Abstract The paper is focused on stochastic modelling of the process of transition of infected cells and virions of HIV-1 infection between two lymph nodes. The model is based on the following assumptions: (1) the duration of transition of infected cells and virions between two lymph nodes is set using a time-dependent function, (2) infected cells produce virions in the process of transition between two lymph nodes, (3) infected cells and virions may die when moving between two lymph nodes. The methods of the theory of branching random processes are used to study analytically the model variables. An algorithm for statistical modelling of the number of infected cells and virions in the second lymph node is presented. The results of computational experiments studying the distribution law of the number of virions produced by one infected cell depending on the duration of movement between two lymph nodes are presented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45439560","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}
S. Simakov, T. Gamilov, F. Liang, D. Gognieva, Mariam K. Gappoeva, P. Kopylov
Abstract In the present paper we construct the model of coronary flow which utilizes the patient’s CT data on both large CA and perfusion. We reconstruct large coronary vessels based on the CT data and extrude possibly invisible vessels from the branches of the left coronary artery so that every terminal point supply every of standardized zone of myocardium. We apply a previously developed and validated 1D model of haemodynamics. Utilizing the patient-specific CTP data for modifying outflow terminal resistance at rest and under stress conditions is the major novelty of the present work. From our results it follows that acceptable values of fractional flow reserve (FFR) after the stenotic treatment may produce false negative output. Therefore healthy FFR values may be observed after ineffective revascularization. We also conclude that microvascular perfusion impairment plays significant role in correct computational estimation of haemodynamic indices before stenotic treatment. The advantage of the presented approach is the availability of transmural perfusion ratio assessment in everyday practice.
{"title":"Numerical evaluation of the effectiveness of coronary revascularization","authors":"S. Simakov, T. Gamilov, F. Liang, D. Gognieva, Mariam K. Gappoeva, P. Kopylov","doi":"10.1515/rnam-2021-0025","DOIUrl":"https://doi.org/10.1515/rnam-2021-0025","url":null,"abstract":"Abstract In the present paper we construct the model of coronary flow which utilizes the patient’s CT data on both large CA and perfusion. We reconstruct large coronary vessels based on the CT data and extrude possibly invisible vessels from the branches of the left coronary artery so that every terminal point supply every of standardized zone of myocardium. We apply a previously developed and validated 1D model of haemodynamics. Utilizing the patient-specific CTP data for modifying outflow terminal resistance at rest and under stress conditions is the major novelty of the present work. From our results it follows that acceptable values of fractional flow reserve (FFR) after the stenotic treatment may produce false negative output. Therefore healthy FFR values may be observed after ineffective revascularization. We also conclude that microvascular perfusion impairment plays significant role in correct computational estimation of haemodynamic indices before stenotic treatment. The advantage of the presented approach is the availability of transmural perfusion ratio assessment in everyday practice.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47683635","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 In the present paper, we consider a compartmental epidemic model which simulates the co-circulation of three influenza strains, A(H1N1)pdm09, A(H3N2), and B, in a population with the history of exposure to these virus strains. A strain-specific incidence data for the model input was generated using long-term weekly ARI incidence and virologic testing data. The algorithm for model calibration was developed as a combination of simulated annealing and BFGS optimization methods. Two simulations were carried out, assuming the absence and the presence of protected individuals in the population, with 2017– 2018 and 2018–2019 epidemic seasons in Moscow as a case study. It was shown that strain-specific immune levels defined by virologic studies might be used in the model to obtain plausible incidence curves. However, different output parameter values, such as fractions of individuals exposed to particular virus strain in the previous epidemic season, can correspond to similar incidence trajectories, which complicates the assessment of herd immunity levels based on the model calibration. The results of the study will be used in the research of the interplay between the immunity formation dynamics and the circulation of influenza strains in Russian cities.
{"title":"Herd immunity levels and multi-strain influenza epidemics in Russia: a modelling study","authors":"V. Leonenko","doi":"10.1515/rnam-2021-0023","DOIUrl":"https://doi.org/10.1515/rnam-2021-0023","url":null,"abstract":"Abstract In the present paper, we consider a compartmental epidemic model which simulates the co-circulation of three influenza strains, A(H1N1)pdm09, A(H3N2), and B, in a population with the history of exposure to these virus strains. A strain-specific incidence data for the model input was generated using long-term weekly ARI incidence and virologic testing data. The algorithm for model calibration was developed as a combination of simulated annealing and BFGS optimization methods. Two simulations were carried out, assuming the absence and the presence of protected individuals in the population, with 2017– 2018 and 2018–2019 epidemic seasons in Moscow as a case study. It was shown that strain-specific immune levels defined by virologic studies might be used in the model to obtain plausible incidence curves. However, different output parameter values, such as fractions of individuals exposed to particular virus strain in the previous epidemic season, can correspond to similar incidence trajectories, which complicates the assessment of herd immunity levels based on the model calibration. The results of the study will be used in the research of the interplay between the immunity formation dynamics and the circulation of influenza strains in Russian cities.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41840094","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}
T. Dobroserdova, Y. Vassilevski, S. Simakov, T. Gamilov, A. Svobodov, L. Yurpolskaya
Abstract Palliation of congenital single ventricle heart defects suggests multi-stage surgical interventions that divert blood flow from the inferior and superior vena cava directly to the right and left pulmonary arteries, skipping the right ventricle. Such system with cavopulmonary anastomoses and single left ventricle is called Fontan circulation, and the region of reconnection is called the total cavopulmonary connection (TCPC). Computational blood flow models allow clinicians to predict the results of the Fontan operation, to choose an optimal configuration of TCPC and thus to reduce negative postoperative consequences. We propose a two-scale (1D3D) haemodynamic model of systemic circulation for a patient who has underwent Fontan surgical operation. We use CT and 4D flow MRI data to personalize the model. The model is tuned to patient’s data and is able to represent measured time-averaged flow rates at the inlets and outlets of TCPC, as well as pressure in TCPC for the patient in horizontal position.We demonstrate that changing to quiescent standing position leads to other patterns of blood flow in regional (TCPC) and global haemodynamics. This confirms clinical data on exercise intolerance of Fontan patients.
{"title":"Two-scale haemodynamic modelling for patients with Fontan circulation","authors":"T. Dobroserdova, Y. Vassilevski, S. Simakov, T. Gamilov, A. Svobodov, L. Yurpolskaya","doi":"10.1515/rnam-2021-0022","DOIUrl":"https://doi.org/10.1515/rnam-2021-0022","url":null,"abstract":"Abstract Palliation of congenital single ventricle heart defects suggests multi-stage surgical interventions that divert blood flow from the inferior and superior vena cava directly to the right and left pulmonary arteries, skipping the right ventricle. Such system with cavopulmonary anastomoses and single left ventricle is called Fontan circulation, and the region of reconnection is called the total cavopulmonary connection (TCPC). Computational blood flow models allow clinicians to predict the results of the Fontan operation, to choose an optimal configuration of TCPC and thus to reduce negative postoperative consequences. We propose a two-scale (1D3D) haemodynamic model of systemic circulation for a patient who has underwent Fontan surgical operation. We use CT and 4D flow MRI data to personalize the model. The model is tuned to patient’s data and is able to represent measured time-averaged flow rates at the inlets and outlets of TCPC, as well as pressure in TCPC for the patient in horizontal position.We demonstrate that changing to quiescent standing position leads to other patterns of blood flow in regional (TCPC) and global haemodynamics. This confirms clinical data on exercise intolerance of Fontan patients.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49603452","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 Pathophysiology of ischemic heart disease is a complex phenomenon determined by the interaction of multiple processes including the inflammatory, immunological, infectious, mechanical, biochemical and epigenetic ones. A predictive clinically relevant modelling of the entire trajectory of the human organism, from the initial alterations in lipid metabolism through to atherosclerotic plaque formation and finally to the pathologic state of the ischemic heart disease, is an open insufficiently explored problem. In the present review, we consider the existing mathematical frameworks which are used to describe, analyze and predict the dynamics of various processes related to cardiovascular diseases at the molecular, cellular, tissue, and holistic human organism level. The mechanistic, statistical and machine learning models are discussed in detail with special focus on the underlying assumptions and their clinical relevance. All together, they provide a solid computational platform for further expansion and tailoring for practical applications.
{"title":"Frontiers in mathematical modelling of the lipid metabolism under normal conditions and its alterations in heart diseases","authors":"G. Bocharov, D. Grebennikov, R. Savinkov","doi":"10.1515/rnam-2021-0021","DOIUrl":"https://doi.org/10.1515/rnam-2021-0021","url":null,"abstract":"Abstract Pathophysiology of ischemic heart disease is a complex phenomenon determined by the interaction of multiple processes including the inflammatory, immunological, infectious, mechanical, biochemical and epigenetic ones. A predictive clinically relevant modelling of the entire trajectory of the human organism, from the initial alterations in lipid metabolism through to atherosclerotic plaque formation and finally to the pathologic state of the ischemic heart disease, is an open insufficiently explored problem. In the present review, we consider the existing mathematical frameworks which are used to describe, analyze and predict the dynamics of various processes related to cardiovascular diseases at the molecular, cellular, tissue, and holistic human organism level. The mechanistic, statistical and machine learning models are discussed in detail with special focus on the underlying assumptions and their clinical relevance. All together, they provide a solid computational platform for further expansion and tailoring for practical applications.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45682959","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 present paper is devoted to a model describing a two-phase isothermal mixture, in which one of the phases obeys solid-like (namely, elastic) rheology. A fully Eulerian description is considered. To describe the stress–strain behaviour of the solid phase the elastic energy term is added to the Helmholtz free energy. The term depends on Almansi strain tensor. In its turn, the strain tensor is defined as the solution of the corresponding evolutionary equation. Considered model belongs to the phase field family. Formally it describes two-component mixture and uses mass densities of the components as order parameters. A distinctive feature of the considered model is its preliminary regularization according to the quasi-hydrodynamic framework. The dissipativity in total energy is proved when periodic boundary conditions are imposed. A spatial dissipative semi-discrete (continuous in time and discrete in space) scheme based on staggered grids is suggested. The theoretical results remain valid in the absence of the regularization. The results of a numerical study in a 2D setting are presented.
{"title":"A regularized isothermal phase-field model of two-phase solid–fluid mixture and its spatial dissipative discretization equations","authors":"V. Balashov","doi":"10.1515/rnam-2021-0017","DOIUrl":"https://doi.org/10.1515/rnam-2021-0017","url":null,"abstract":"Abstract The present paper is devoted to a model describing a two-phase isothermal mixture, in which one of the phases obeys solid-like (namely, elastic) rheology. A fully Eulerian description is considered. To describe the stress–strain behaviour of the solid phase the elastic energy term is added to the Helmholtz free energy. The term depends on Almansi strain tensor. In its turn, the strain tensor is defined as the solution of the corresponding evolutionary equation. Considered model belongs to the phase field family. Formally it describes two-component mixture and uses mass densities of the components as order parameters. A distinctive feature of the considered model is its preliminary regularization according to the quasi-hydrodynamic framework. The dissipativity in total energy is proved when periodic boundary conditions are imposed. A spatial dissipative semi-discrete (continuous in time and discrete in space) scheme based on staggered grids is suggested. The theoretical results remain valid in the absence of the regularization. The results of a numerical study in a 2D setting are presented.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43536046","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 stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.
{"title":"Stability analysis of implicit semi-Lagrangian methods for numerical solution of non-hydrostatic atmospheric dynamics equations","authors":"V. Shashkin","doi":"10.1515/rnam-2021-0020","DOIUrl":"https://doi.org/10.1515/rnam-2021-0020","url":null,"abstract":"Abstract The stability of implicit semi-Lagrangian schemes for time-integration of the non-hydrostatic atmosphere dynamics equations is analyzed in the present paper. The main reason for the instability of the considered class of schemes is the semi-Lagrangian advection of stratified thermodynamic variables coupled to the fixed point iteration method used to solve the implicit in time upstream trajectory computation problem. We identify two types of unstable modes and obtain stability conditions in terms of the scheme parameters. Stabilization of sound modes requires the use of a pressure reference profile and time off-centering. Gravity waves are stable only for an even number of fixed point method iterations. The maximum time step is determined by inverse buoyancy frequency in the case when the reference profile of the potential temperature is not used. Generally, applying time off-centering and reference profile to pressure variable is necessary for stability. Using reference profile for potential temperature and an even number of the iterations allows one to significantly increase the maximum time-step value.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45859562","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 Realizations of the numerical solution of the scalar transport equation on the sphere, written in divergent form, are presented. Various temporal discretizations are considered: the one-step Taylor–Galerkin method (TG2), the two-step Taylor–Galerkin method of the second (TTG2), third (TTG3), and fourth (TTG4) orders. The standard Finite-Element Galerkin method with linear basis functions on a triangle is applied as spatial discretization. The flux correction technique (FCT) is implemented. Test runs are carried out with different initial profiles: a function from C∞ (Gaussian profile) and a discontinuous function (slotted cylinder). The profiles are advected by reversible, nondivergent velocity fields, therefore the initial distribution coincides with the final one. The case of a divergent velocity field is also considered to test the conservation and positivity properties of the schemes. It is demonstrated that TG2, TTG3, and TTG4 schemes with FCT applied give the best result for small Courant numbers, and TTG2, TTG4 are preferable in case of large Courant number. However, TTG2+FCT scheme has the worst stability. The use of FCT increases the integral errors, but ensures that the solution is positive with high accuracy. The implemented schemes are included in the dynamic core of a new sea ice model developed using the INMOST package. The acceleration of the parallel program and solution convergence with spatial resolution are demonstrated.
{"title":"The suite of Taylor–Galerkin class schemes for ice transport on sphere implemented by the INMOST package","authors":"Sergey S. Petrov, N. Iakovlev","doi":"10.1515/rnam-2021-0019","DOIUrl":"https://doi.org/10.1515/rnam-2021-0019","url":null,"abstract":"Abstract Realizations of the numerical solution of the scalar transport equation on the sphere, written in divergent form, are presented. Various temporal discretizations are considered: the one-step Taylor–Galerkin method (TG2), the two-step Taylor–Galerkin method of the second (TTG2), third (TTG3), and fourth (TTG4) orders. The standard Finite-Element Galerkin method with linear basis functions on a triangle is applied as spatial discretization. The flux correction technique (FCT) is implemented. Test runs are carried out with different initial profiles: a function from C∞ (Gaussian profile) and a discontinuous function (slotted cylinder). The profiles are advected by reversible, nondivergent velocity fields, therefore the initial distribution coincides with the final one. The case of a divergent velocity field is also considered to test the conservation and positivity properties of the schemes. It is demonstrated that TG2, TTG3, and TTG4 schemes with FCT applied give the best result for small Courant numbers, and TTG2, TTG4 are preferable in case of large Courant number. However, TTG2+FCT scheme has the worst stability. The use of FCT increases the integral errors, but ensures that the solution is positive with high accuracy. The implemented schemes are included in the dynamic core of a new sea ice model developed using the INMOST package. The acceleration of the parallel program and solution convergence with spatial resolution are demonstrated.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49105113","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 Correlative randomized algorithms are constructed by simple randomization of the algorithm of maximum cross-section (equalization, delta tracking) with the use of a one-dimensional distribution and the correlation function or only correlation length of a random medium. The value of the used correlation length can be adjusted using simple test studies. The calculations carried out confirmed the practical effectiveness of the new algorithms.
{"title":"New correlative randomized algorithms for statistical modelling of radiation transfer in stochastic medium","authors":"G. Mikhailov, I. N. Medvedev","doi":"10.1515/rnam-2021-0018","DOIUrl":"https://doi.org/10.1515/rnam-2021-0018","url":null,"abstract":"Abstract Correlative randomized algorithms are constructed by simple randomization of the algorithm of maximum cross-section (equalization, delta tracking) with the use of a one-dimensional distribution and the correlation function or only correlation length of a random medium. The value of the used correlation length can be adjusted using simple test studies. The calculations carried out confirmed the practical effectiveness of the new algorithms.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49259976","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}
I. Timokhin, S. Matveev, E. Tyrtyshnikov, A. Smirnov
Abstract In this paper we consider the problem of modelling a system of aggregating particles, that are being transported with stationary velocities dependent on masses of the particles in one-dimensional case. A numerical method based on the ideas of POD (Proper Orthogonal Decomposition) is constructed, and its capacity to speed up the solution up to 40 times is demonstrated.
{"title":"Model reduction for Smoluchowski equations with particle transfer","authors":"I. Timokhin, S. Matveev, E. Tyrtyshnikov, A. Smirnov","doi":"10.1515/rnam-2021-0015","DOIUrl":"https://doi.org/10.1515/rnam-2021-0015","url":null,"abstract":"Abstract In this paper we consider the problem of modelling a system of aggregating particles, that are being transported with stationary velocities dependent on masses of the particles in one-dimensional case. A numerical method based on the ideas of POD (Proper Orthogonal Decomposition) is constructed, and its capacity to speed up the solution up to 40 times is demonstrated.","PeriodicalId":49585,"journal":{"name":"Russian Journal of Numerical Analysis and Mathematical Modelling","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rnam-2021-0015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49368466","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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