Pub Date : 2023-12-29DOI: 10.1007/s10543-023-01001-w
Fredrik Hellman, Axel Målqvist, Malin Mosquera
In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.
{"title":"Well-posedness and finite element approximation of mixed dimensional partial differential equations","authors":"Fredrik Hellman, Axel Målqvist, Malin Mosquera","doi":"10.1007/s10543-023-01001-w","DOIUrl":"https://doi.org/10.1007/s10543-023-01001-w","url":null,"abstract":"<p>In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"178 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139070858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-27DOI: 10.1007/s10543-023-01000-x
Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao
The well-known Ait-Sahalia-type interest model, arising in mathematical finance, has some typical features: polynomial drift that blows up at the origin, highly nonlinear diffusion, and positive solution. The known explicit numerical methods including truncated/tamed Euler–Maruyama (EM) applied to it do not preserve its positivity. The main interest of this work is to investigate the numerical conservation of positivity of the solution of generalised Ait-Sahalia-type model. By modifying the truncated EM method to generate positive sequences of numerical approximations, we obtain the rate of convergence of the numerical algorithm not only at time T but also over the time interval [0, T]. Numerical experiments confirm the theoretical results.
{"title":"Positivity-preserving truncated Euler–Maruyama method for generalised Ait-Sahalia-type interest model","authors":"Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao","doi":"10.1007/s10543-023-01000-x","DOIUrl":"https://doi.org/10.1007/s10543-023-01000-x","url":null,"abstract":"<p>The well-known Ait-Sahalia-type interest model, arising in mathematical finance, has some typical features: polynomial drift that blows up at the origin, highly nonlinear diffusion, and positive solution. The known explicit numerical methods including truncated/tamed Euler–Maruyama (EM) applied to it do not preserve its positivity. The main interest of this work is to investigate the numerical conservation of positivity of the solution of generalised Ait-Sahalia-type model. By modifying the truncated EM method to generate positive sequences of numerical approximations, we obtain the rate of convergence of the numerical algorithm not only at time <i>T</i> but also over the time interval [0, <i>T</i>]. Numerical experiments confirm the theoretical results.</p>","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"16 3‐4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1007/s10543-023-00999-3
Malak Diab, Andreas Frommer, Karsten Kahl
Abstract For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.
{"title":"A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port-Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian","authors":"Malak Diab, Andreas Frommer, Karsten Kahl","doi":"10.1007/s10543-023-00999-3","DOIUrl":"https://doi.org/10.1007/s10543-023-00999-3","url":null,"abstract":"Abstract For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"47 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1007/s10543-023-00998-4
Joackim Bernier, S Blanes, Fernando Casas, A Escorihuela-Tomàs
Abstract We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.
{"title":"Symmetric-conjugate splitting methods for linear unitary problems","authors":"Joackim Bernier, S Blanes, Fernando Casas, A Escorihuela-Tomàs","doi":"10.1007/s10543-023-00998-4","DOIUrl":"https://doi.org/10.1007/s10543-023-00998-4","url":null,"abstract":"Abstract We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation.","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"78 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135087342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s10543-023-00994-8
Laurent O. Jay, Olga Sokratova
{"title":"A note on approximate Jacobians of implicit Runge–Kutta methods and convergence of modified Newton iterations","authors":"Laurent O. Jay, Olga Sokratova","doi":"10.1007/s10543-023-00994-8","DOIUrl":"https://doi.org/10.1007/s10543-023-00994-8","url":null,"abstract":"","PeriodicalId":55351,"journal":{"name":"BIT Numerical Mathematics","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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