We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $epsll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+infty$ as $eps$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-delta/eps})$, $delta>0$, while the length of the next spectral segments is $O(eps)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{Om}$ in the geometry $Om$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^{Om}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{Om}$, that is no non trivial bounded solution at the threshold frequency for $A^{Om}$. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.
{"title":"Spectrum of the Dirichlet Laplacian in a thin cubic lattice","authors":"Lucas Chesnel, Sergei A Nazarov","doi":"10.1051/m2an/2023082","DOIUrl":"https://doi.org/10.1051/m2an/2023082","url":null,"abstract":"We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $epsll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+infty$ as $eps$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-delta/eps})$, $delta>0$, while the length of the next spectral segments is $O(eps)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{Om}$ in the geometry $Om$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^{Om}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{Om}$, that is no non trivial bounded solution at the threshold frequency for $A^{Om}$. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"380 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134946892","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}
Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains.
{"title":"Stable approximation of Helmholtz solutions in the disk by evanescent plane waves","authors":"Emile Parolin, Daan Huybrechs, Andrea Moiola","doi":"10.1051/m2an/2023081","DOIUrl":"https://doi.org/10.1051/m2an/2023081","url":null,"abstract":"Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548675","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 extend the 2-parameter Weibull to the generalized gamma distribution by adding a new partial parameter. The new shape parameter can be used to easily generate generalized gamma distributions with different shapes of the density function, hazard rate, and mean residual lifetimes that are useful in simulating various business processes such as manufacturing processes, and reliability systems. We derived some theoretical results and created visual presentations to demonstrate the behaviors of this new shape parameter as well. A new Monte Carlo simulation based on the new parameter was proposed to assess the discrepancy between the generalized gamma and its subfamilies. The power analysis of the proposed test was evaluated via simulation studies. We also present some numerical examples.
{"title":"A new formulation of generalized gamma: Some results and applications","authors":"Cheng Peng","doi":"10.1051/ps/2023018","DOIUrl":"https://doi.org/10.1051/ps/2023018","url":null,"abstract":"We extend the 2-parameter Weibull to the generalized gamma distribution by adding a new partial parameter. The new shape parameter can be used to easily generate generalized gamma distributions with different shapes of the density function, hazard rate, and mean residual lifetimes that are useful in simulating various business processes such as manufacturing processes, and reliability systems. We derived some theoretical results and created visual presentations to demonstrate the behaviors of this new shape parameter as well. A new Monte Carlo simulation based on the new parameter was proposed to assess the discrepancy between the generalized gamma and its subfamilies. The power analysis of the proposed test was evaluated via simulation studies. We also present some numerical examples.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689763","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}
Sudarshan Kumar Kenettinkara, Nikhil Manoj, Veerappa Gowda G. D.
In this article, we present the convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws. We combine a MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme. The resulting scheme is shown to converge to a weak solution by establishing the maximum principle, bounded variation estimates and L1 Lipschitz continuity in time. Further, using a space-step dependent slope limiter, we prove its convergence to the entropy solution. We also propose a MUSCL-Hancock type second-order scheme which requires only one intermediate stage unlike the Runge-Kutta schemes and is easier to implement. The performance of the proposed second-order schemes in comparison to a first-order scheme is demonstrated through several numerical experiments.
{"title":"Convergence of a second-order scheme for non-local conservation laws","authors":"Sudarshan Kumar Kenettinkara, Nikhil Manoj, Veerappa Gowda G. D.","doi":"10.1051/m2an/2023080","DOIUrl":"https://doi.org/10.1051/m2an/2023080","url":null,"abstract":"In this article, we present the convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws. We combine a MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme. The resulting scheme is shown to converge to a weak solution by establishing the maximum principle, bounded variation estimates and L1 Lipschitz continuity in time. Further, using a space-step dependent slope limiter, we prove its convergence to the entropy solution. We also propose a MUSCL-Hancock type second-order scheme which requires only one intermediate stage unlike the Runge-Kutta schemes and is easier to implement. The performance of the proposed second-order schemes in comparison to a first-order scheme is demonstrated through several numerical experiments.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135791508","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}
This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on [[EQUATION]] that do not satisfy the classical Harnack inequalities, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.
{"title":"Exponential quasi-ergodicity for processes with discontinuous trajectories","authors":"Aurélien Velleret","doi":"10.1051/ps/2023016","DOIUrl":"https://doi.org/10.1051/ps/2023016","url":null,"abstract":"This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on [[EQUATION]] that do not satisfy the classical Harnack inequalities, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135131268","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}
This paper presents the lowest-order nonconforming immersed virtual element method for solving elliptic interface problems on unfitted polygonal meshes. The local discrete space on each interface mesh element consists of the solutions of local interface problems with Neumann boundary conditions, and the elliptic projection is modified so that its range is the space of broken linear polynomials satisfying the interface conditions. We derive optimal error estimates in the broken H1-norm and L2-norm, under the piecewise H2regulartiy assumption. The mesh assumptions in our scheme allow small cut elements and do not require additional mesh procedures. Several numerical experiments are provided to confirm the theoretical results.
{"title":"A nonconforming immersed virtual element method for elliptic interface problems","authors":"Hyeokjoo Park, Do Young Kwak","doi":"10.1051/m2an/2023078","DOIUrl":"https://doi.org/10.1051/m2an/2023078","url":null,"abstract":"This paper presents the lowest-order nonconforming immersed virtual element method for solving elliptic interface problems on unfitted polygonal meshes. The local discrete space on each interface mesh element consists of the solutions of local interface problems with Neumann boundary conditions, and the elliptic projection is modified so that its range is the space of broken linear polynomials satisfying the interface conditions. We derive optimal error estimates in the broken H1-norm and L2-norm, under the piecewise H2regulartiy assumption. The mesh assumptions in our scheme allow small cut elements and do not require additional mesh procedures. Several numerical experiments are provided to confirm the theoretical results.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353906","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}
In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior $L^infty$ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior $L^infty$ error estimates for $L^2$ initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements.
{"title":"Fully discrete pointwise smoothing error estimates for measure valued initial data","authors":"Boris Vexler, Dmitriy Leykekhman, Jakob Wagner","doi":"10.1051/m2an/2023076","DOIUrl":"https://doi.org/10.1051/m2an/2023076","url":null,"abstract":"In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior $L^infty$ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior $L^infty$ error estimates for $L^2$ initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135098725","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}
In this paper we analyze a lowest order virtual element method for the classic load reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration.
{"title":"VEM discretization allowing small edges for the reaction-convection-diffusion equation: source and spectral problems","authors":"Felipe Lepe, Gonzalo Rivera","doi":"10.1051/m2an/2023069","DOIUrl":"https://doi.org/10.1051/m2an/2023069","url":null,"abstract":"In this paper we analyze a lowest order virtual element method for the classic load reaction-convection-diffusion problem and the convection-diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135254643","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}
In this paper, we study the pattern formation of the Cucker–Smale type kinetic models. Two distributed Cucker–Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results.
{"title":"Pattern formation of the Cucker-Smale type kinetic models based on gradient flow","authors":"Xinyu Wang, Xiaoping Xue","doi":"10.1051/m2an/2023079","DOIUrl":"https://doi.org/10.1051/m2an/2023079","url":null,"abstract":"In this paper, we study the pattern formation of the Cucker–Smale type kinetic models. Two distributed Cucker–Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135588744","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}
Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda
Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.
{"title":"Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors","authors":"Raimund Bürger, Julio Careaga, Stefan Diehl, Romel Pineda","doi":"10.1051/m2an/2023068","DOIUrl":"https://doi.org/10.1051/m2an/2023068","url":null,"abstract":"Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion-reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved under a Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than for the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236636","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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