Abstract We present an implicit–explicit finite volume scheme for isentropic two phase flow in all Mach number regimes. The underlying model belongs to the class of symmetric hyperbolic thermodynamically compatible models. The key element of the scheme consists of a linearisation of pressure and enthalpy terms at a reference state. The resulting stiff linear parts are integrated implicitly, whereas the non-linear higher order and transport terms are treated explicitly. Due to the flux splitting, the scheme is stable under a CFL condition which is determined by the resolution of the slow material waves and allows large time steps even in the presence of fast acoustic waves. Further the singular Mach number limits of the model are studied and the asymptotic preserving property of the scheme is proven. In numerical simulations the consistency with single phase flow, accuracy and the approximation of material waves in different Mach number regimes are assessed.
{"title":"An all Mach number finite volume method for isentropic two-phase flow","authors":"M. Lukáčová-Medvid’ová, G. Puppo, Andrea Thomann","doi":"10.1515/jnma-2022-0015","DOIUrl":"https://doi.org/10.1515/jnma-2022-0015","url":null,"abstract":"Abstract We present an implicit–explicit finite volume scheme for isentropic two phase flow in all Mach number regimes. The underlying model belongs to the class of symmetric hyperbolic thermodynamically compatible models. The key element of the scheme consists of a linearisation of pressure and enthalpy terms at a reference state. The resulting stiff linear parts are integrated implicitly, whereas the non-linear higher order and transport terms are treated explicitly. Due to the flux splitting, the scheme is stable under a CFL condition which is determined by the resolution of the slow material waves and allows large time steps even in the presence of fast acoustic waves. Further the singular Mach number limits of the model are studied and the asymptotic preserving property of the scheme is proven. In numerical simulations the consistency with single phase flow, accuracy and the approximation of material waves in different Mach number regimes are assessed.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81161006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.
{"title":"A C0-conforming DG finite element method for biharmonic equations on triangle/tetrahedron","authors":"X. Ye, Shangyou Zhang","doi":"10.1515/jnma-2021-0012","DOIUrl":"https://doi.org/10.1515/jnma-2021-0012","url":null,"abstract":"Abstract A C0-conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the biharmonic equation. The first strong gradient of C0 finite element functions is a vector of discontinuous piecewise polynomials. The second gradient is the weak gradient of discontinuous piecewise polynomials. This method, by its name, uses nonconforming (non C1) approximations and keeps simple formulation of conforming finite element methods without any stabilizers. Optimal order error estimates in both a discrete H2-norm and the L2-norm are established for the corresponding finite element solutions. Numerical results are presented to confirm the theory of convergence.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76207936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1515/jnma-2021-frontmatter4
Article Frontmatter was published on December 1, 2021 in the journal Journal of Numerical Mathematics (volume 29, issue 4).
文章Frontmatter于2021年12月1日发表在《journal of Numerical Mathematics》第29卷第4期。
{"title":"Frontmatter","authors":"","doi":"10.1515/jnma-2021-frontmatter4","DOIUrl":"https://doi.org/10.1515/jnma-2021-frontmatter4","url":null,"abstract":"Article Frontmatter was published on December 1, 2021 in the journal Journal of Numerical Mathematics (volume 29, issue 4).","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138530248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.
{"title":"A posteriori error analysis of an enriched Galerkin method of order one for the Stokes problem","authors":"V. Girault, María González, F. Hecht","doi":"10.1515/jnma-2020-0100","DOIUrl":"https://doi.org/10.1515/jnma-2020-0100","url":null,"abstract":"Abstract We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86537613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.
{"title":"Adaptive space–time finite element methods for parabolic optimal control problems","authors":"U. Langer, Andreas Schafelner","doi":"10.1515/jnma-2021-0059","DOIUrl":"https://doi.org/10.1515/jnma-2021-0059","url":null,"abstract":"Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73149704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abhinav K. Jha, Ondvrej P'artl, N. Ahmed, D. Kuzmin
Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.
{"title":"An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations","authors":"Abhinav K. Jha, Ondvrej P'artl, N. Ahmed, D. Kuzmin","doi":"10.1515/jnma-2021-0123","DOIUrl":"https://doi.org/10.1515/jnma-2021-0123","url":null,"abstract":"Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87512565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. Burman, R. Durst, Miguel A. Fern'andez, Johnny Guzm'an
Abstract We present a loosely coupled, non-iterative time-splitting scheme based on Robin–Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a parabolic/parabolic coupled system and a parabolic/hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as O(ΔtT+log(1Δt)), $mathcal{O}big({Delta t} sqrt{T +log(frac{1}{{Delta t}})}big),$where Δt is the time step.
{"title":"Loosely coupled, non-iterative time-splitting scheme based on Robin–Robin coupling: Unified analysis for parabolic/parabolic and parabolic/hyperbolic problems","authors":"E. Burman, R. Durst, Miguel A. Fern'andez, Johnny Guzm'an","doi":"10.1515/jnma-2021-0119","DOIUrl":"https://doi.org/10.1515/jnma-2021-0119","url":null,"abstract":"Abstract We present a loosely coupled, non-iterative time-splitting scheme based on Robin–Robin coupling conditions. We apply a novel unified analysis for this scheme applied to both a parabolic/parabolic coupled system and a parabolic/hyperbolic coupled system. We show for both systems that the scheme is stable, and the error converges as O(ΔtT+log(1Δt)), $mathcal{O}big({Delta t} sqrt{T +log(frac{1}{{Delta t}})}big),$where Δt is the time step.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88449232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Martin Hutzenthaler, Arnulf Jentzen, T. Kruse, T. Nguyen
Abstract Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal 1/ε of the prescribed approximation accuracy ε ∈ (0, ∞) and the dimension d ∈ N = {1, 2, 3, . . .} of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.
摘要倒向随机微分方程(BSDEs)是目前随机分析和计算随机学中研究最多的方程之一。应用中的bsde通常是非线性和高维的。在几乎所有的情况下，这种非线性高维BSDEs都不能显式求解，设计和分析数值逼近方法来近似求解非线性高维BSDEs一直是并且仍然是一个非常活跃的研究课题。虽然科学文献中有大量的研究文章分析了非线性BSDEs的数值逼近方法，直到今天还没有数值近似方法在科学文献中已被证明能克服的诅咒维度的数值近似非线性元,计算操作的数量的数值逼近方法近似地计算研究的一个样本路径解决多项式增长最多的倒数1 /ε规定的近似精度ε∈(0,∞)和维d∈N = {1, 2,3、…}的BSDE。本文提出了一种新的高维BSDEs Monte carlo型数值逼近方法，并证明了这种Monte carlo型数值逼近方法确实克服了BSDEs解路径近似计算中的维数诅咒，这是克服这一障碍的关键贡献。
{"title":"Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations","authors":"Martin Hutzenthaler, Arnulf Jentzen, T. Kruse, T. Nguyen","doi":"10.1515/jnma-2021-0111","DOIUrl":"https://doi.org/10.1515/jnma-2021-0111","url":null,"abstract":"Abstract Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approximation methods for nonlinear BSDEs, until today there has been no numerical approximation method in the scientific literature which has been proven to overcome the curse of dimensionality in the numerical approximation of nonlinear BSDEs in the sense that the number of computational operations of the numerical approximation method to approximatively compute one sample path of the BSDE solution grows at most polynomially in both the reciprocal 1/ε of the prescribed approximation accuracy ε ∈ (0, ∞) and the dimension d ∈ N = {1, 2, 3, . . .} of the BSDE. It is the key contribution of this article to overcome this obstacle by introducing a new Monte Carlo-type numerical approximation method for high-dimensional BSDEs and by proving that this Monte Carlo-type numerical approximation method does indeed overcome the curse of dimensionality in the approximative computation of solution paths of BSDEs.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80828978","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux (F∗) and a non-oscillatory flux (Fs). This novel approach paves a way to construct non-oscillatory entropy stable flux (F̂) as a simple combination of (F∗ and Fs) which inherently optimize the numerical diffusion in the entropy stable flux (F̂) such that it reduces to the underlying non-oscillatory flux (Fs) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair (F∗, Fs), (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that fully discrete schemes using these entropy stable fluxes do not exhibit nonphysical spurious oscillations in approximating the discontinuities compared to the non-oscillatory schemes using underlying fluxes (Fs) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.
{"title":"Entropy stable non-oscillatory fluxes: An optimized wedding of entropy conservative flux with non-oscillatory flux","authors":"R. Dubey","doi":"10.1515/jnma-2022-0075","DOIUrl":"https://doi.org/10.1515/jnma-2022-0075","url":null,"abstract":"Abstract This work frames the problem of constructing non-oscillatory entropy stable fluxes as a least square optimization problem. A flux sign stability condition is defined for a pair of entropy conservative flux (F∗) and a non-oscillatory flux (Fs). This novel approach paves a way to construct non-oscillatory entropy stable flux (F̂) as a simple combination of (F∗ and Fs) which inherently optimize the numerical diffusion in the entropy stable flux (F̂) such that it reduces to the underlying non-oscillatory flux (Fs) in the flux sign stable region. This robust approach is (i) agnostic to the choice of flux pair (F∗, Fs), (ii) does not require the computation of costly dissipation operator and high order reconstruction of scaled entropy variable to construct the diffusion term. Various non-oscillatory entropy stable fluxes are constructed and exhaustive computational results for standard test problems are given which show that fully discrete schemes using these entropy stable fluxes do not exhibit nonphysical spurious oscillations in approximating the discontinuities compared to the non-oscillatory schemes using underlying fluxes (Fs) only. Moreover, these entropy stable schemes maintain the formal order of accuracy of the lower order flux in the pair.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72466961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A diffuse interface model for tumour growth in the presence of a nutrient consumed by the tumour is considered. The system of equations consists of a Cahn–Hilliard equation with source terms for the tumour cells and a reaction–diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.
{"title":"Numerical analysis for a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport","authors":"H. Garcke, D. Trautwein","doi":"10.1515/jnma-2021-0094","DOIUrl":"https://doi.org/10.1515/jnma-2021-0094","url":null,"abstract":"Abstract A diffuse interface model for tumour growth in the presence of a nutrient consumed by the tumour is considered. The system of equations consists of a Cahn–Hilliard equation with source terms for the tumour cells and a reaction–diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.","PeriodicalId":50109,"journal":{"name":"Journal of Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":3.0,"publicationDate":"2021-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90888163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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