801 research outputs found
High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes
We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE)
finite volume scheme on unstructured triangular meshes that is high order
accurate in space and time and that also allows for time-accurate local time
stepping (LTS). The new scheme uses the following basic ingredients: a high
order WENO reconstruction in space on unstructured meshes, an element-local
high-order accurate space-time Galerkin predictor that performs the time
evolution of the reconstructed polynomials within each element, the computation
of numerical ALE fluxes at the moving element interfaces through approximate
Riemann solvers, and a one-step finite volume scheme for the time update which
is directly based on the integral form of the conservation equations in
space-time. The inclusion of the LTS algorithm requires a number of crucial
extensions, such as a proper scheduling criterion for the time update of each
element and for each node; a virtual projection of the elements contained in
the reconstruction stencils of the element that has to perform the WENO
reconstruction; and the proper computation of the fluxes through the space-time
boundary surfaces that will inevitably contain hanging nodes in time due to the
LTS algorithm. We have validated our new unstructured Lagrangian LTS approach
over a wide sample of test cases solving the Euler equations of compressible
gasdynamics in two space dimensions, including shock tube problems, cylindrical
explosion problems, as well as specific tests typically adopted in Lagrangian
calculations, such as the Kidder and the Saltzman problem. When compared to the
traditional global time stepping (GTS) method, the newly proposed LTS algorithm
allows to reduce the number of element updates in a given simulation by a
factor that may depend on the complexity of the dynamics, but which can be as
large as 4.7.Comment: 31 pages, 13 figure
Variational Multiscale Modeling and Memory Effects in Turbulent Flow Simulations
Effective computational models of multiscale problems have to account for the impact of unresolved physics on the resolved scales. This dissertation advances our fundamental understanding of multiscale models and develops a mathematically rigorous closure modeling framework by combining the Mori-Zwanzig (MZ) formalism of Statistical Mechanics with the variational multiscale (VMS) method. This approach leverages scale-separation projectors as well as phase-space projectors to provide a systematic modeling approach that is applicable to complex non-linear partial differential equations. %The MZ-VMS framework is investigated in the context of turbulent flows. Spectral as well as continuous and discontinuous finite element methods are considered.
The MZ-VMS framework leads to a closure term that is non-local in time and appears as a convolution or memory integral. The resulting non-Markovian system is used as a starting point for model development. Several new insights are uncovered: It is shown that unresolved scales lead to memory effects that are driven by an orthogonal projection of the coarse-scale residual and, in the case of finite elements, inter-element jumps. Connections between MZ-based methods, artificial viscosity, and VMS models are explored.
The MZ-VMS framework is investigated in the context of turbulent flows. Large eddy simulations of Burgers' equation, turbulent flows, and magnetohydrodynamic turbulence using spectral and discontinuous Galerkin methods are explored. In the spectral method case, we show that MZ-VMS models lead to substantial improvements in the prediction of coarse-grained quantities of interest. Applications to discontinuous Galerkin methods show that modern flux schemes can inherently capture memory effects, and that it is possible to guarantee non-linear stability and conservation via the MZ-VMS approach. We conclude by demonstrating how ideas from MZ-VMS can be adapted for shock-capturing and filtering methods.PHDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145847/1/parish_1.pd
Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes
In this article we present a new class of high order accurate
Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for
solving nonlinear hyperbolic systems of conservation laws on moving two
dimensional unstructured triangular meshes. A WENO reconstruction algorithm is
used to achieve high order accuracy in space and a high order one-step time
discretization is achieved by using the local space-time Galerkin predictor.
For that purpose, a new element--local weak formulation of the governing PDE is
adopted on moving space--time elements. The space-time basis and test functions
are obtained considering Lagrange interpolation polynomials passing through a
predefined set of nodes. Moreover, a polynomial mapping defined by the same
local space-time basis functions as the weak solution of the PDE is used to map
the moving physical space-time element onto a space-time reference element. To
maintain algorithmic simplicity, the final ALE one-step finite volume scheme
uses moving triangular meshes with straight edges. This is possible in the ALE
framework, which allows a local mesh velocity that is different from the local
fluid velocity. We present numerical convergence rates for the schemes
presented in this paper up to sixth order of accuracy in space and time and
show some classical numerical test problems for the two-dimensional Euler
equations of compressible gas dynamics.Comment: Accepted by "Communications in Computational Physics
A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
The Koopman operator is a linear but infinite dimensional operator that
governs the evolution of scalar observables defined on the state space of an
autonomous dynamical system, and is a powerful tool for the analysis and
decomposition of nonlinear dynamical systems. In this manuscript, we present a
data driven method for approximating the leading eigenvalues, eigenfunctions,
and modes of the Koopman operator. The method requires a data set of snapshot
pairs and a dictionary of scalar observables, but does not require explicit
governing equations or interaction with a "black box" integrator. We will show
that this approach is, in effect, an extension of Dynamic Mode Decomposition
(DMD), which has been used to approximate the Koopman eigenvalues and modes.
Furthermore, if the data provided to the method are generated by a Markov
process instead of a deterministic dynamical system, the algorithm approximates
the eigenfunctions of the Kolmogorov backward equation, which could be
considered as the "stochastic Koopman operator" [1]. Finally, four illustrative
examples are presented: two that highlight the quantitative performance of the
method when presented with either deterministic or stochastic data, and two
that show potential applications of the Koopman eigenfunctions
Recommended from our members
Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws
The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling.
Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms
- …