In this work a class of finite volume schemes is proposed to numerically solve equations involving propagating fronts. They fall into the class of Hamilton-Jacobi equations. Finite volume schemes based on staggered grids, and initially developed to compute fluid flows, are adapted to the G-equation, using the Hamilton-Jacobi theoretical framework. The designed scheme has a maximum principle property and is consistent an monotonous on Cartesian grids. A convergence property is then obtained for the scheme on Cartesian grids and numerical experiments evidence the convergence of the scheme on more general meshes.
{"title":"A class of robust numerical schemes to compute front propagation","authors":"N. Therme","doi":"10.5802/SMAI-JCM.39","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.39","url":null,"abstract":"In this work a class of finite volume schemes is proposed to numerically solve equations involving propagating fronts. They fall into the class of Hamilton-Jacobi equations. Finite volume schemes based on staggered grids, and initially developed to compute fluid flows, are adapted to the G-equation, using the Hamilton-Jacobi theoretical framework. The designed scheme has a maximum principle property and is consistent an monotonous on Cartesian grids. A convergence property is then obtained for the scheme on Cartesian grids and numerical experiments evidence the convergence of the scheme on more general meshes.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133487383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An extension and numerical approximation of the shear shallow water equations model recently proposed in [21] is considered in this work. The model equations are able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical shallow water equations in describing such phenomena. The model equations, orig- inally developed for horizontal flow or flows occurring over small constant slopes, are straight- forwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations are derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different conditions.
{"title":"Numerical simulations of hydraulic jumps with the Shear Shallow Water model","authors":"A. Delis, H. Guillard, Y. Tai","doi":"10.5802/SMAI-JCM.37","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.37","url":null,"abstract":"An extension and numerical approximation of the shear shallow water equations\u0000model recently proposed in [21] is considered in this work. The model equations are able to\u0000describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of\u0000the classical shallow water equations in describing such phenomena. The model equations, orig-\u0000inally developed for horizontal flow or flows occurring over small constant slopes, are straight-\u0000forwardly extended here for modeling flows over non-constant slopes and numerically solved\u0000by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to\u0000the extended model equations are derived and several numerical tests are performed to validate\u0000the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under\u0000different conditions.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131827107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Reflective tomography recovers the surfaces of a scene to be imaged, from optical images: a tomographic algorithm computes a full volumic reconstruction and then the surfaces are extracted from this reconstruction. For better performance, we would like to avoid computing accurately the full reconstruction, and we want to focus computations on the sought surfaces. For that purpose we propose an iterative multiresolution process. The initialization computes a coarse reconstruction, and the iterations refines it. To identify the voxels to be refined, we take advantage of the asymptotic behaviour of the reconstruction, with respect to its cut-off frequency: it discriminates the surfaces to be extracted. By the way the proposed algorithm is greedy: each iteration maximizes the accumulated intensity of the selected voxels, with prescribed volume. The combination of the complexity analysis and the numerical results shows that this novel approach succeeds in reconstructing surfaces and is relatively efficient compared with the standard method. These works pave the way towards accelerated algorithms in reflective tomography. They can be extended to a general class of problems concerning the determination of asymptotically discriminated sets, what is related to the computation of singular support of distributions. 2010 Mathematics Subject Classification. 78A97, 94A12, 65B99, 65Y20.
{"title":"Multiresolution greedy algorithm dedicated to reflective tomography","authors":"Jean-Baptiste Bellet","doi":"10.5802/SMAI-JCM.35","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.35","url":null,"abstract":"Reflective tomography recovers the surfaces of a scene to be imaged, from optical images: a tomographic algorithm computes a full volumic reconstruction and then the surfaces are extracted from this reconstruction. For better performance, we would like to avoid computing accurately the full reconstruction, and we want to focus computations on the sought surfaces. For that purpose we propose an iterative multiresolution process. The initialization computes a coarse reconstruction, and the iterations refines it. To identify the voxels to be refined, we take advantage of the asymptotic behaviour of the reconstruction, with respect to its cut-off frequency: it discriminates the surfaces to be extracted. By the way the proposed algorithm is greedy: each iteration maximizes the accumulated intensity of the selected voxels, with prescribed volume. The combination of the complexity analysis and the numerical results shows that this novel approach succeeds in reconstructing surfaces and is relatively efficient compared with the standard method. These works pave the way towards accelerated algorithms in reflective tomography. They can be extended to a general class of problems concerning the determination of asymptotically discriminated sets, what is related to the computation of singular support of distributions. 2010 Mathematics Subject Classification. 78A97, 94A12, 65B99, 65Y20.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126870176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An improved understanding of the divergence-free constraint for the incompressible Navier--Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of {em pressure-robustness} allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order $k$ are comparably accurate than non-pressure-robust methods of formal order $2k$ on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.
{"title":"On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond","authors":"N. Gauger, A. Linke, Philipp W. Schroeder","doi":"10.5802/SMAI-JCM.44","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.44","url":null,"abstract":"An improved understanding of the divergence-free constraint for the incompressible Navier--Stokes equations leads to the observation that a semi-norm and corresponding equivalence classes of forces are fundamental for their nonlinear dynamics. The recent concept of {em pressure-robustness} allows to distinguish between space discretisations that discretise these equivalence classes appropriately or not. This contribution compares the accuracy of pressure-robust and non-pressure-robust space discretisations for transient high Reynolds number flows, starting from the observation that in generalised Beltrami flows the nonlinear convection term is balanced by a strong pressure gradient. Then, pressure-robust methods are shown to outperform comparable non-pressure-robust space discretisations. Indeed, pressure-robust methods of formal order $k$ are comparably accurate than non-pressure-robust methods of formal order $2k$ on coarse meshes. Investigating the material derivative of incompressible Euler flows, it is conjectured that strong pressure gradients are typical for non-trivial high Reynolds number flows. Connections to vortex-dominated flows are established. Thus, pressure-robustness appears to be a prerequisite for accurate incompressible flow solvers at high Reynolds numbers. The arguments are supported by numerical analysis and numerical experiments.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132803311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a "computational basis" of shape functions for element families. The tensor product family, $Q^-_rLambda^k$, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, $S^-_rLambda^k$ and $S_rLambda^k$ respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order $rgeq 1$ and for any differential form order $0leq kleq n$, for problems in dimension $n=2$ or $3$. The bases are defined via shared subspace structures, allowing easy comparison of elements across families. We use and include code in SageMath to find, list, and verify these computational basis functions.
{"title":"Computational Serendipity and Tensor Product Finite Element Differential Forms","authors":"A. Gillette, Tyler Kloefkorn, Victoria Sanders","doi":"10.5802/SMAI-JCM.41","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.41","url":null,"abstract":"Many conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a \"computational basis\" of shape functions for element families. The tensor product family, $Q^-_rLambda^k$, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, $S^-_rLambda^k$ and $S_rLambda^k$ respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order $rgeq 1$ and for any differential form order $0leq kleq n$, for problems in dimension $n=2$ or $3$. The bases are defined via shared subspace structures, allowing easy comparison of elements across families. We use and include code in SageMath to find, list, and verify these computational basis functions.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133865739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gypsilab is a Matlab toolbox which aims at simplifying the development of numerical methods that apply to the resolution of problems in multiphysics, in particular, those involving FEM or BEM simulations. The specifities of the toolbox, in particular its ease of use, are shown together with the methodology we have followed for its development. Example codes that are short though representative enough are given both for FEM and BEM applications. A performance comparison with FreeFem++ is also provided.
{"title":"FEM and BEM simulations with the Gypsilab framework","authors":"F. Alouges, M. Aussal","doi":"10.5802/SMAI-JCM.36","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.36","url":null,"abstract":"Gypsilab is a Matlab toolbox which aims at simplifying the development of numerical methods that apply to the resolution of problems in multiphysics, in particular, those involving FEM or BEM simulations. The specifities of the toolbox, in particular its ease of use, are shown together with the methodology we have followed for its development. Example codes that are short though representative enough are given both for FEM and BEM applications. A performance comparison with FreeFem++ is also provided.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123207527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots of specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.
{"title":"Maximum-principle-satisfying second-order Intrusive Polynomial Moment scheme","authors":"J. Kusch, G. Alldredge, M. Frank","doi":"10.5802/SMAI-JCM.42","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.42","url":null,"abstract":"Using standard intrusive techniques when solving hyperbolic conservation laws with uncertainties can lead to oscillatory solutions as well as nonhyperbolic moment systems. The Intrusive Polynomial Moment (IPM) method ensures hyperbolicity of the moment system while restricting oscillatory over- and undershoots of specified bounds. In this contribution, we derive a second-order discretization of the IPM moment system which fulfills the maximum principle. This task is carried out by investigating violations of the specified bounds due to the errors from the numerical optimization required by the scheme. This analysis gives weaker conditions on the entropy that is used, allowing the choice of an entropy which enables choosing the exact minimal and maximal value of the initial condition as bounds. Solutions calculated with the derived scheme are nonoscillatory while fulfilling the maximum principle. The second-order accuracy of our scheme leads to significantly reduced numerical costs.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128232734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.
{"title":"Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations","authors":"Ward Melis, Thomas Rey, G. Samaey","doi":"10.5802/smai-jcm.43","DOIUrl":"https://doi.org/10.5802/smai-jcm.43","url":null,"abstract":"We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128966698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that an analog of the Scott-Vogelius finite elements are inf-sup stable on certain nondegenerate meshes for piecewise cubic velocity fields. We also characterize the divergence of the velocity space on such meshes. In addition, we show how such a characterization relates to the dimension of C^1 piecewise quartics on the same mesh.
{"title":"Cubic Lagrange elements satisfying exact incompressibility","authors":"J. Guzmán, R. Scott","doi":"10.5802/SMAI-JCM.38","DOIUrl":"https://doi.org/10.5802/SMAI-JCM.38","url":null,"abstract":"We prove that an analog of the Scott-Vogelius finite elements are inf-sup stable on certain nondegenerate meshes for piecewise cubic velocity fields. We also characterize the divergence of the velocity space on such meshes. In addition, we show how such a characterization relates to the dimension of C^1 piecewise quartics on the same mesh.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125403709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eigenvalue analysis based methods are well suited for the reconstruction of finitely supported measures from their moments up to a certain degree. We give a precise description when Prony's method succeeds in terms of an interpolation condition. In particular, this allows for the unique reconstruction of a measure from its trigonometric moments whenever its support is separated and also for the reconstruction of a measure on the unit sphere from its moments with respect to spherical harmonics. Both results hold in arbitrary dimensions and also yield a certificate for popular semidefinite relaxations of these reconstruction problems.
{"title":"Prony’s method on the sphere","authors":"Stefan Kunis, H. Möller, Ulrich von der Ohe","doi":"10.5802/smai-jcm.53","DOIUrl":"https://doi.org/10.5802/smai-jcm.53","url":null,"abstract":"Eigenvalue analysis based methods are well suited for the reconstruction of finitely supported measures from their moments up to a certain degree. We give a precise description when Prony's method succeeds in terms of an interpolation condition. In particular, this allows for the unique reconstruction of a measure from its trigonometric moments whenever its support is separated and also for the reconstruction of a measure on the unit sphere from its moments with respect to spherical harmonics. Both results hold in arbitrary dimensions and also yield a certificate for popular semidefinite relaxations of these reconstruction problems.","PeriodicalId":376888,"journal":{"name":"The SMAI journal of computational mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131020279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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