Habilitationenhttps://kobra.uni-kassel.de:443/handle/123456789/20130124424502023-12-05T09:39:45Z2023-12-05T09:39:45ZComputing Ground States for Fermi-Bose Mixtures through Efficient Numerical MethodsÁvila, Andrés I.https://kobra.uni-kassel.de:443/handle/123456789/148842023-07-10T11:41:24Z2023-05-01T00:00:00ZIn this work, we will first review the Quantum Mechanics theory to derive the main equations. Next, we will analyze these equations by Functional Analysis methods to find conditions for existence, uniqueness, multiplicity, and other properties as positivity. Next, we will review and develop some numerical methods for solving the nonlinear Schrödinger equation, its time version, generalizations with rotational terms, and systems of NLSE (NLSS). We notice that the main problem to run numerical methods is the memory use, which can be a bottleneck for simulations involving very large linear systems. Finally, we will address this problem of Computing Efficiency and learn some techniques and tools to understand code behavior and memory use to improve our methods and study the effect of using numerical libraries.
2023-05-01T00:00:00ZÁvila, Andrés I.In this work, we will first review the Quantum Mechanics theory to derive the main equations. Next, we will analyze these equations by Functional Analysis methods to find conditions for existence, uniqueness, multiplicity, and other properties as positivity. Next, we will review and develop some numerical methods for solving the nonlinear Schrödinger equation, its time version, generalizations with rotational terms, and systems of NLSE (NLSS). We notice that the main problem to run numerical methods is the memory use, which can be a bottleneck for simulations involving very large linear systems. Finally, we will address this problem of Computing Efficiency and learn some techniques and tools to understand code behavior and memory use to improve our methods and study the effect of using numerical libraries.Numerical Methods for Fluid Flow: High Order SBP Schemes, IMEX Advection-Diffusion Splitting and Positivity Preservation for Production-Destruction PDEsOrtleb, Sigrunhttps://kobra.uni-kassel.de:443/handle/123456789/143232023-01-03T17:33:33Z2021-01-01T00:00:00ZThe current demands regarding the numerical simulation of fluid flow often require highly accurate computations to obtain a detailed resolution of the occurring physical phenomena. The basic concept for the construction of a fluid solver is to transfer the physical model into a numerical scheme which complies with the underlying physical principles such as conservation and balances of certain quantities. In addition, the numerical methods are required to be stable and efficient. Again, stability is often determined by physically motivated quantities such as energy or entropy and it is generally easier to be achieved for low order schemes. Furthermore, the need for efficiency and possible implementation in parallel hardware environment has led to the development of sophisticated schemes in space with compact stencils such as discontinuous Galerkin (DG) methods and flux reconstruction schemes which extend classical space discretization methods.
In this work, the newly found generalized SBP properties of DG schemes on Legendre-Gauss nodes pave the way to their application to skew-symmetric forms. Since their quadrature rule possesses a higher degree of exactness, DG schemes on Legendre-Gauss nodes are usually more accurate than those on Legendre-Gauss-Lobatto nodes and might be preferable for long-time simulations, which is precisely the situation in which the preservation of secondary quantities should be most beneficial.
2021-01-01T00:00:00ZOrtleb, SigrunThe current demands regarding the numerical simulation of fluid flow often require highly accurate computations to obtain a detailed resolution of the occurring physical phenomena. The basic concept for the construction of a fluid solver is to transfer the physical model into a numerical scheme which complies with the underlying physical principles such as conservation and balances of certain quantities. In addition, the numerical methods are required to be stable and efficient. Again, stability is often determined by physically motivated quantities such as energy or entropy and it is generally easier to be achieved for low order schemes. Furthermore, the need for efficiency and possible implementation in parallel hardware environment has led to the development of sophisticated schemes in space with compact stencils such as discontinuous Galerkin (DG) methods and flux reconstruction schemes which extend classical space discretization methods.
In this work, the newly found generalized SBP properties of DG schemes on Legendre-Gauss nodes pave the way to their application to skew-symmetric forms. Since their quadrature rule possesses a higher degree of exactness, DG schemes on Legendre-Gauss nodes are usually more accurate than those on Legendre-Gauss-Lobatto nodes and might be preferable for long-time simulations, which is precisely the situation in which the preservation of secondary quantities should be most beneficial.Numerical Methods for the Unsteady Compressible Navier-Stokes EquationsBirken, Philipphttps://kobra.uni-kassel.de:443/handle/123456789/20130130424852021-06-23T14:24:24Z2013-01-30T00:00:00ZWe consider numerical methods for the compressible time dependent Navier-Stokes equations, discussing the spatial discretization by Finite Volume and Discontinuous Galerkin methods, the time integration by time adaptive implicit Runge-Kutta and Rosenbrock methods and the solution of the appearing nonlinear and linear equations systems by preconditioned Jacobian-Free Newton-Krylov, as well as Multigrid methods. As applications, thermal Fluid structure interaction and other unsteady flow problems are considered. The text is aimed at both mathematicians and engineers.
2013-01-30T00:00:00ZBirken, PhilippWe consider numerical methods for the compressible time dependent Navier-Stokes equations, discussing the spatial discretization by Finite Volume and Discontinuous Galerkin methods, the time integration by time adaptive implicit Runge-Kutta and Rosenbrock methods and the solution of the appearing nonlinear and linear equations systems by preconditioned Jacobian-Free Newton-Krylov, as well as Multigrid methods. As applications, thermal Fluid structure interaction and other unsteady flow problems are considered. The text is aimed at both mathematicians and engineers.