3 research outputs found
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Multiple time scales and pressure forcing in discontinuous Galerkin approximations to layered ocean models
Abstract This paper addresses some issues involving the application of discontinuous Galerkin (DG) methods to ocean circulation models having a generalized vertical coordinate. These issues include the following. (1) Determine the pressure forcing at cell edges, where the dependent variables can be discontinuous. In principle, this could be accomplished by solving a Riemann problem for the full system, but some ideas related to barotropic-baroclinic time splitting can be used to reduce the Riemann problem to a much simpler system of lower dimension. Such splittings were originally developed in order to address the multiple time scales that are present in the system. (2) Adapt the general idea of barotropic-baroclinic splitting to a DG implementation. A significant step is enforcing consistency between the numerical solution of the layer equations and the numerical solution of the vertically-integrated barotropic equations. The method used here has the effect of introducing a type of time filtering into the forcing for the layer equations, which are solved with a long time step. (3) Test these ideas in a model problem involving geostrophic adjustment in a multilayer fluid. In certain situations, the DG formulation can give significantly better results than those obtained with a standard finite difference formulation
Das unstetige Galerkinverfahren fĂŒr Strömungen mit freier OberflĂ€che und im Grundwasserbereich in geophysikalischen Anwendungen
Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory.
Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing.
This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.FĂŒr viele geophysikalische Anwendungen spielen Strömungen mit freier OberflĂ€che und im Grundwasserbereich oder sogar die Kopplung dieser beiden eine zentrale Rolle. Oftmals charakteristisch fĂŒr diese Anwendungsszenarien sind groĂe Rechengebiete und lange Simulationszeiten. Folglich ist das Berechnen akkurater Lösungen mit betrĂ€chtlichem Rechenaufwand verbunden und der Einsatz effizienter Lösungsverfahren sowie von Techniken des Hochleistungsrechnens obligatorisch, um Ergebnisse innerhalb eines annehmbaren Zeitrahmens zu erhalten.
Unstetige Galerkinverfahren stellen eine Gruppe numerischer Verfahren zum Lösen von Differentialgleichungen dar, und kombinieren Eigenschaften von Methoden der Finiten Volumen- und Finiten Elementeverfahren. Sie ermöglichen hohe Approximationsordnungen, bieten einen hohen Grad an FlexibilitĂ€t und sind fĂŒr paralleles Rechnen gut geeignet.
Diese Dissertation besteht aus acht Artikeln und einer erweiterten Zusammenfassung, in diesen die Anwendung unstetiger Galerkinverfahren auf mathematische Modelle inklusive solcher fĂŒr Strömungen mit freier OberflĂ€che und im Grundwasserbereich beschrieben wird. Die behandelten Themen umfassen Diskretisierungs- und Implementierungsaspekte, die Parallelisierung der Methode sowie eine diskrete StabilitĂ€tsanalyse des gekoppelten Modells