Mathematik und Statistikhttps://kops.uni-konstanz.de:443/handle/123456789/82020-04-02T17:21:21Z2020-04-02T17:21:21ZViscous Profiles for Shock Waves in Isothermal MagnetohydrodynamicsKlaiber, Andreaspop213418123456789/491542020-03-27T10:26:28Z2012-09Viscous Profiles for Shock Waves in Isothermal Magnetohydrodynamics
Klaiber, Andreas
We prove existence of viscous profiles for slow and fast shock waves in isothermal magnetohydrodynamics with a general pressure law both using the Conley index and connection matrices.
2012-09Klaiber, Andreas510We prove existence of viscous profiles for slow and fast shock waves in isothermal magnetohydrodynamics with a general pressure law both using the Conley index and connection matrices.SpringerJOURNAL_ARTICLEeng10.1007/s10884-012-9266-11040-72941572-9222663683243Journal of Dynamics and Differential Equations2020-03-27T11:26:28+01:00123456789/39Journal of Dynamics and Differential Equations ; 24 (2012), 3. - S. 663-683. - Springer. - ISSN 1040-7294. - eISSN 1572-9222unknown2020-03-27T10:26:28ZFirst-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrowsSadiq, I Mohammed Rizwan123456789/491492020-03-27T09:46:17Z2012-03First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows
Sadiq, I Mohammed Rizwan
An average modeling methodology under the lubrication approach is used to formulate a set of three coupled nonlinear partial differential equations based on the Nusselt scales. This system, known as the energy-integral method in literature, simplifies the Navier-Stokes equation at the first order and analyzes the dynamics of a thin sheet of fluid flowing over a topography with sinusoidally varying longitudinal furrows. Limiting cases of the linear stability results are mathematically discussed and the complete linear system is numerically handled by means of finite differences to approximate the eigenfunctions and their derivatives in a periodic domain. In a geometry which resembles a vertical shift of a topography, with the amplitude being equal to the shift length, it is found that such a geometry stabilizes the flow compared to its counterpart with no shift, such that the wave characteristics get affected. To confirm the stability results, a numerical investigation is performed.
2012-03Sadiq, I Mohammed Rizwan510An average modeling methodology under the lubrication approach is used to formulate a set of three coupled nonlinear partial differential equations based on the Nusselt scales. This system, known as the energy-integral method in literature, simplifies the Navier-Stokes equation at the first order and analyzes the dynamics of a thin sheet of fluid flowing over a topography with sinusoidally varying longitudinal furrows. Limiting cases of the linear stability results are mathematically discussed and the complete linear system is numerically handled by means of finite differences to approximate the eigenfunctions and their derivatives in a periodic domain. In a geometry which resembles a vertical shift of a topography, with the amplitude being equal to the shift length, it is found that such a geometry stabilizes the flow compared to its counterpart with no shift, such that the wave characteristics get affected. To confirm the stability results, a numerical investigation is performed.JOURNAL_ARTICLEeng10.1103/PhysRevE.85.0363091539-37551550-2376853 Pt 2Physical review. E, Statistical, nonlinear, and soft matter physics2020-03-27T10:46:16+01:00123456789/39Physical review. E, Statistical, nonlinear, and soft matter physics ; 85 (2012), 3 Pt 2. - 036309. - ISSN 1539-3755. - eISSN 1550-2376true2020-03-27T09:46:16ZPositivity of continuous piecewise polynomialsPlaumann, Danielpop93081123456789/489592020-03-06T08:10:04Z2012Positivity of continuous piecewise polynomials
Plaumann, Daniel
Real algebraic geometry provides certificates for the positivity of polynomials on semialgebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalities. We show how Putinar's theorem for strictly positive polynomials on compact sets can be applied in the case of strictly positive piecewise polynomials on a simplicial complex. In the one‐dimensional case, we improve this result to cover all non‐negative piecewise polynomials and give explicit degree bounds.
2012Plaumann, Daniel510Real algebraic geometry provides certificates for the positivity of polynomials on semialgebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalities. We show how Putinar's theorem for strictly positive polynomials on compact sets can be applied in the case of strictly positive piecewise polynomials on a simplicial complex. In the one‐dimensional case, we improve this result to cover all non‐negative piecewise polynomials and give explicit degree bounds.WileyJOURNAL_ARTICLEeng10.1112/blms/bds0070024-60931469-2120749757444Bulletin of the London Mathematical Society2020-03-06T09:09:27+01:00123456789/39Bulletin of the London Mathematical Society ; 44 (2012), 4. - S. 749-757. - Wiley. - ISSN 0024-6093. - eISSN 1469-2120true2020-03-06T08:09:27ZPOD-based mixed-integer optimal control of evolution systemsJäkle, Christianpop256999Volkwein, Stefanpop214121123456789/489572020-03-07T02:02:27Z2020POD-based mixed-integer optimal control of evolution systems
Jäkle, Christian; Volkwein, Stefan
In this chapter the authors consider the numerical treatment of a mixed- integer optimal control problem governed by linear convection-diffusion equations and binary control variables. Using relaxation techniques (introduced by [31] for ordinary differential equations) the original mixed-integer optimal control problem is transferred into a relaxed optimal control problem with no integrality constraints. After an optimal solution to the relaxed problem has been computed, binary admis- sible controls are constructed by a sum-up rounding technique. This allows us to construct – in an iterative process – binary admissible controls such that the cor- responding optimal state and the optimal cost value approximate the original ones with arbitrary accuracy. However, using finite element (FE) methods to discretize the state and adjoint equations yield often to extensive systems which make the frequently calculations time-consuming. Therefore, a model-order reduction based on the proper orthogonal decomposition (POD) method is applied. Compared to the FE case, the POD approach yields to a significant acceleration of the CPU times while the error stays sufficiently small.
2020Jäkle, ChristianVolkwein, Stefan510In this chapter the authors consider the numerical treatment of a mixed- integer optimal control problem governed by linear convection-diffusion equations and binary control variables. Using relaxation techniques (introduced by [31] for ordinary differential equations) the original mixed-integer optimal control problem is transferred into a relaxed optimal control problem with no integrality constraints. After an optimal solution to the relaxed problem has been computed, binary admis- sible controls are constructed by a sum-up rounding technique. This allows us to construct – in an iterative process – binary admissible controls such that the cor- responding optimal state and the optimal cost value approximate the original ones with arbitrary accuracy. However, using finite element (FE) methods to discretize the state and adjoint equations yield often to extensive systems which make the frequently calculations time-consuming. Therefore, a model-order reduction based on the proper orthogonal decomposition (POD) method is applied. Compared to the FE case, the POD approach yields to a significant acceleration of the CPU times while the error stays sufficiently small.INPROCEEDINGSurn:nbn:de:bsz:352-2-1o0d721ili9ux8eng2020-03-06T08:52:09+01:00123456789/39Kolloquium zum 60. Geburtstag von Herrn Prof. Dr. Michael Dellnitz2020-03-06T07:52:09Z