14 research outputs found
Self-Evaluation Applied Mathematics 2003-2008 University of Twente
This report contains the self-study for the research assessment of the Department of Applied Mathematics (AM) of the Faculty of Electrical Engineering, Mathematics and Computer Science (EEMCS) at the University of Twente (UT). The report provides the information for the Research Assessment Committee for Applied Mathematics, dealing with mathematical sciences at the three universities of technology in the Netherlands. It describes the state of affairs pertaining to the period 1 January 2003 to 31 December 2008
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High-order (hybridized) discontinuous Galerkin method for geophysical flows
As computational research has grown, simulation has become a standard tool in many fields of academic and industrial areas. For example, computational fluid dynamics (CFD) tools in aerospace and research facilities are widely used to evaluate the aerodynamic performance of aircraft or wings. Weather forecasts are highly dependent on numerical weather prediction (NWP) model. However, it is still difficult to simulate the complex physical phenomena of a wide range of length and time scales with modern computational resources. In this study, we develop a robust, efficient and high-order accurate numerical methods and techniques to tackle the challenges. First, we use high-order spatial discretization using (hybridized) discontinuous Galerkin (DG) methods. The DG method combines the advantages of finite volume and finite element methods. As such, it is well-suited to problems with large gradients including shocks and with complex geometries, and large-scale simulations. However, DG typically has many degrees-of-freedoms. To mitigate the expense, we use hybridized DG (HDG) method that introduces new “trace unknowns” on the mesh skeleton (mortar interfaces) to eliminate the local “volume unknowns” with static condensation procedure and reduces globally coupled system when implicit time-stepping is required. Also, since the information between the elements is exchanged through the mesh skeleton, the mortar interfaces can be used as a glue to couple multi-phase regions, e.g., solid and fluid regions, or non-matching grids, e.g., a rotating mesh and a stationary mesh. That is the HDG method provides an efficient and flexible coupling environment compared to standard DG methods. Second, we develop an HDG-DG IMEX scheme for an efficient time integrating scheme. The idea is to divide the governing equations into stiff and nonstiff parts, implicitly treat the former with HDG methods, and explicitly treat the latter with DG methods. The HDG-DG IMEX scheme facilitates high-order temporal and spatial solutions, avoiding too small a time step. Numerical results show that the HDG-DG IMEX scheme is comparable to an explicit Runge-Kutta DG scheme in terms of accuracy while allowing for much larger timestep sizes. We also numerically observe that IMEX HDG-DG scheme can be used as a tool to suppress the high-frequency modes such as acoustic waves or fast gravity waves in atmospheric or ocean models. In short, IMEX HDG-DG methods are attractive for applications in which a fast and stable solution is important while permitting inaccurate processing of the fast modes. Third, we also develop an EXPONENTIAL DG scheme for an efficient time integrators. Similar to the IMEX method, the governing equations are separated into linear and nonlinear parts, then the two parts are spatially discretized with DG methods. Next, we analytically integrate the linear term and approximate the nonlinear term with respect to time. This method accurately handles the fast wave modes in the linear operator. To efficiently evaluate a matrix exponential, we employ the cutting-edge adaptive Krylov subspace method. Finally, we develop a sliding-mesh interface by combining nonconforming treatment and the arbitrary Lagrangian-Eulerian (ALE) scheme for simulating rotating flows, which are important to estimate the characteristics of a rotating wind turbine or understanding vortical structures shown in atmospheric or astronomical phenomena. To integrate the rotating motion of the domain, we use the ALE formulation to map the governing equation to the stationary reference domain and introduce mortar interfaces between the stationary mesh and the rotating mesh. The mortar structure on the sliding interface changes dynamically as the mesh rotatesEngineering Mechanic
On leapfrog-Chebyshev schemes for second-order differential equations
In this thesis the efficient time integration of semilinear second-order ordinary differential equations is investigated. Based on the leapfrog (Störmer, Verlet) scheme a new class of explicit two-step schemes is constructed by utilizing Chebyshev polynomials. For deriving rigorous error bounds of these leapfrog-Chebyshev (LFC) schemes a more general class of two-step schemes is introduced. Precise conditions are stated for this general class guaranteeing stability as well as second-order convergence in time. In addition, the influence of the starting value is analyzed in detail. Furthermore, by combining the leapfrog scheme with this general class of schemes a class of multirate two-step methods is constructed. Sufficient conditions for the stability of these schemes are derived as well as error bounds showing the second-order convergence in time. For both the LFC schemes and the multirate schemes if equipped with the LFC schemes it is shown that in specific situations they outperform the leapfrog scheme. Numerical examples are provided to illustrate the theoretical results
High-Order Finite-Volume Schemes for Magnetohydrodynamics
New high-order finite-volume numerical schemes for the magnetohydrodynamics equations are proposed in two and three dimensions. Two different sets of magnetohydrodynamics equations are considered. The first set is the ideal magnetohydrodynamics system, which assumes that the fluid can be treated as a perfect conductor. The second set is resistive MHD, which involves non-zero resistivity. A high-order central essentially nonoscillatory (CENO) approach is employed, which combines unlimited k-exact polynomial reconstruction with a monotonicity preserving scheme. The CENO schemes, which were originally developed for compressible fluid flow, are applied to the MHD equations, along with two possible control mechanisms for divergence error of the magnetic field. The hyperbolic fluxes are calculated by solving a Riemann problem at each cell interface, and elliptic fluxes are computed through k-exact gradient interpolation where point-wise values of the gradients are required. Smooth test problems and test cases with discontinuities (weak or strong) are considered, and convergence studies are presented for both the ideal and resistive MHD systems. Several potential space physics applications are explored. For these simulations, cubed-sphere grids are used to model the interaction of the solar wind with planetary bodies or their satellites. The basic cubed-sphere grid discretizes a simulation domain between two concentric spheres using six root blocks (corresponding to the six faces of a cube). Conditions describing the atmosphere of the inner body can be applied at the boundary of the inner sphere. For some problems we also need to solve equations within the inner sphere, for which we develop a seven-block cubed-sphere grid where the empty space inside the interior sphere is discretized as a seventh root block. We consider lunar flow problems for which we employ the seven-block cubed-sphere mesh. Ideal MHD is solved between the inner and outer spheres of the grid, and the magnetic diffusion equations are solved within the inner sphere, which represents the lunar interior. Two cases are considered: one is without intrinsic magnetic field, where only a wake is expected without any bow shock forming ahead of the Moon, and the second is with a small dipole moment to model a lunar crustal magnetic anomaly, in which case a small-scale magnetosphere is expected ahead of the region with the magnetic anomaly