140 research outputs found
Moment Approximations and Model Cascades for Shallow Flow
Shallow flow models are used for a large number of applications including
weather forecasting, open channel hydraulics and simulation-based natural
hazard assessment. In these applications the shallowness of the process
motivates depth-averaging. While the shallow flow formulation is advantageous
in terms of computational efficiency, it also comes at the price of losing
vertical information such as the flow's velocity profile. This gives rise to a
model error, which limits the shallow flow model's predictive power and is
often not explicitly quantifiable.
We propose the use of vertical moments to overcome this problem. The shallow
moment approximation preserves information on the vertical flow structure while
still making use of the simplifying framework of depth-averaging. In this
article, we derive a generic shallow flow moment system of arbitrary order
starting from a set of balance laws, which has been reduced by scaling
arguments. The derivation is based on a fully vertically resolved reference
model with the vertical coordinate mapped onto the unit interval. We specify
the shallow flow moment hierarchy for kinematic and Newtonian flow conditions
and present 1D numerical results for shallow moment systems up to third order.
Finally, we assess their performance with respect to both the standard shallow
flow equations as well as with respect to the vertically resolved reference
model. Our results show that depending on the parameter regime, e.g. friction
and slip, shallow moment approximations significantly reduce the model error in
shallow flow regimes and have a lot of potential to increase the predictive
power of shallow flow models, while keeping them computationally cost
efficient
Hybrid Riemann Solvers for Large Systems of Conservation Laws
In this paper we present a new family of approximate Riemann solvers for the
numerical approximation of solutions of hyperbolic conservation laws. They are
approximate, also referred to as incomplete, in the sense that the solvers
avoid computing the characteristic decomposition of the flux Jacobian. Instead,
they require only an estimate of the globally fastest wave speeds in both
directions. Thus, this family of solvers is particularly efficient for large
systems of conservation laws, i.e. with many different propagation speeds, and
when no explicit expression for the eigensystem is available. Even though only
fastest wave speeds are needed as input values, the new family of Riemann
solvers reproduces all waves with less dissipation than HLL, which has the same
prerequisites, requiring only one additional flux evaluation.Comment: 9 page
Convergence Analysis of the Grad's Hermite Approximation to the Boltzmann Equation
In (Commun Pure Appl Math 2(4):331-407, 1949), Grad proposed a Hermite series
expansion for approximating solutions to kinetic equations that have an
unbounded velocity space. However, for initial boundary value problems, poorly
imposed boundary conditions lead to instabilities in Grad's Hermite expansion,
which could result in non-converging solutions. For linear kinetic equations, a
method for posing stable boundary conditions was recently proposed for
(formally) arbitrary order Hermite approximations. In the present work, we
study -convergence of these stable Hermite approximations, and prove
explicit convergence rates under suitable regularity assumptions on the exact
solution. We confirm the presented convergence rates through numerical
experiments involving the linearised-BGK equation of rarefied gas dynamics
Relations between WENO3 and Third-order Limiting in Finite Volume Methods
Weighted essentially non-oscillatory (WENO) and finite volume (FV) methods
employ different philosophies in their way to perform limiting. We show that a
generalized view on limiter functions, which considers a two-dimensional,
rather than a one-dimensional dependence on the slopes in neighboring cells,
allows to write WENO3 and -order FV schemes in the same fashion.
Within this framework, it becomes apparent that the classical approach of FV
limiters to only consider ratios of the slopes in neighboring cells, is overly
restrictive. The hope of this new perspective is to establish new connections
between WENO3 and FV limiter functions, which may give rise to improvements for
the limiting behavior in both approaches.Comment: 22 page
On Third-Order Limiter Functions for Finite Volume Methods
In this article, we propose a finite volume limiter function for a
reconstruction on the three-point stencil. Compared to classical limiter
functions in the MUSCL framework, which yield -order accuracy,
the new limiter is -order accurate for smooth solutions. In an
earlier work, such a -order limiter function was proposed and
showed successful results [2]. However, it came with unspecified parameters. We
close this gap by giving information on these parameters.Comment: 8 pages, conference proceeding
Third-order Limiting for Hyperbolic Conservation Laws applied to Adaptive Mesh Refinement and Non-Uniform 2D Grids
In this paper we extend the recently developed third-order limiter function
[J. Sci. Comput., (2016), 68(2), pp.~624--652] to make it
applicable for more elaborate test cases in the context of finite volume
schemes. This work covers the generalization to non-uniform grids in one and
two space dimensions, as well as two-dimensional Cartesian grids with adaptive
mesh refinement (AMR). The extension to 2D is obtained by the common approach
of dimensional splitting. In order to apply this technique without loss of
third-order accuracy, the order-fix developed by Buchm\"uller and Helzel [J.
Sci. Comput., (2014), 61(2), pp.~343--368] is incorporated into the scheme.
Several numerical examples on different grid configurations show that the
limiter function maintains the optimal third-order
accuracy on smooth profiles and avoids oscillations in case of discontinuous
solutions
On curl-preserving finite volume discretizations for shallow water equations
The preservation of intrinsic or inherent constraints, like divergence-conditions, has gained increasing interest in numerical simulations of various physical evolution equations. In Torrilhon and Fey, SIAM J. Numer. Anal. (42/4) 2004, a general framework is presented how to incorporate the preservation of a discrete constraint into upwind finite volume methods. This paper applies this framework to the wave equation system and the system of shallow water equations. For the wave equation a curl-preservation for the momentum variable is present and easily identified. The preservation in case of the shallow water system is more involved due to the presence of convection. It leads to the vorticity evolution as generalized curl-constraint. The mechanisms of vorticity generation are discussed. For the numerical discretization special curl-preserving flux distributions are discussed and their incorporation into a finite volume scheme described. This leads to numerical discretizations which are exactly curl-preserving for a specific class of discrete curl-operators. The numerical experiments for the wave equation show a significant improvement of the new method against classical schemes. The extension of the curl-free numerical discretization to the shallow water case is possible after isolating the pressure flux. Simulation examples demonstrate the influence of the modification. The vortex structure is more clearly resolve
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