1,790 research outputs found
2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame
In this work, an arbitrary order augmented WENO-ADER scheme for the resolution of the 2D Shallow Water Equations (SWE) with geometric source term is presented and its application to other shallow water models involving non-geometric sources is explored. This scheme is based in the 1D Augmented Roe Linearized-ADER (ARL-ADER) scheme, presented by the authors in a previous work and motivated by a suitable compromise between accuracy and computational cost. It can be regarded as an arbitrary order version of the Augmented Roe solver, which accounts for the contribution of continuous and discontinuous geometric source terms at cell interfaces in the resolution of the Derivative Riemann Problem (DRP). The main novelty of this work is the extension of the ARL-ADER scheme to 2 dimensions, which involves the design of a particular procedure for the integration of the source term with arbitrary order that ensures an exact balance between flux fluctuations and sources. This procedure makes the scheme preserve equilibrium solutions with machine precision and capture the transient waves accurately. The scheme is applied to the SWE with bed variation and is extended to handle non-geometric source terms such as the Coriolis source term. When considering the SWE with bed variation and Coriolis, the most relevant equilibrium states are the still water at rest and the geostrophic equilibrium. The traditional well-balanced property is extended to satisfy the geostrophic equilibrium. This is achieved by means of a geometric reinterpretation of the Coriolis source term. By doing this, the formulation of the source terms is unified leading to a single geometric source regarded as an apparent topography. The numerical scheme is tested for a broad variety of situations, including some cases where the first order scheme ruins the solution
Atmospheric Circulation of Exoplanets
We survey the basic principles of atmospheric dynamics relevant to explaining
existing and future observations of exoplanets, both gas giant and terrestrial.
Given the paucity of data on exoplanet atmospheres, our approach is to
emphasize fundamental principles and insights gained from Solar-System studies
that are likely to be generalizable to exoplanets. We begin by presenting the
hierarchy of basic equations used in atmospheric dynamics, including the
Navier-Stokes, primitive, shallow-water, and two-dimensional nondivergent
models. We then survey key concepts in atmospheric dynamics, including the
importance of planetary rotation, the concept of balance, and scaling arguments
to show how turbulent interactions generally produce large-scale east-west
banding on rotating planets. We next turn to issues specific to giant planets,
including their expected interior and atmospheric thermal structures, the
implications for their wind patterns, and mechanisms to pump their east-west
jets. Hot Jupiter atmospheric dynamics are given particular attention, as these
close-in planets have been the subject of most of the concrete developments in
the study of exoplanetary atmospheres. We then turn to the basic elements of
circulation on terrestrial planets as inferred from Solar-System studies,
including Hadley cells, jet streams, processes that govern the large-scale
horizontal temperature contrasts, and climate, and we discuss how these
insights may apply to terrestrial exoplanets. Although exoplanets surely
possess a greater diversity of circulation regimes than seen on the planets in
our Solar System, our guiding philosophy is that the multi-decade study of
Solar-System planets reviewed here provides a foundation upon which our
understanding of more exotic exoplanetary meteorology must build.Comment: In EXOPLANETS, edited by S. Seager, to be published in the Spring of
2010 in the Space Science Series of the University of Arizona Press (Tucson,
AZ) (refereed; accepted for publication
To Split or Not to Split, That Is the Question in Some Shallow Water Equations
In this paper we analyze the use of time splitting techniques for solving
shallow water equation. We discuss some properties that these schemes should
satisfy so that interactions between the source term and the shock waves are
controlled. This paper shows that these schemes must be well balanced in the
meaning expressed by Greenberg and Leroux [5]. More specifically, we analyze in
what cases it is enough to verify an Approximate C-property and in which cases
it is required to verify an Exact C-property (see [1], [2]). We also include
some numerical tests in order to justify our reasoning
Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction
We present a novel approach for solving the shallow water equations using a
discontinuous Galerkin spectral element method. The method we propose has three
main features. First, it enjoys a discrete well-balanced property, in a spirit
similar to the one of e.g. [20]. As in the reference, our scheme does not
require any a-priori knowledge of the steady equilibrium, moreover it does not
involve the explicit solution of any local auxiliary problem to approximate
such equilibrium. The scheme is also arbitrarily high order, and verifies a
continuous in time cell entropy equality. The latter becomes an inequality as
soon as additional dissipation is added to the method. The method is
constructed starting from a global flux approach in which an additional flux
term is constructed as the primitive of the source. We show that, in the
context of nodal spectral finite elements, this can be translated into a simple
modification of the integral of the source term. We prove that, when using
Gauss-Lobatto nodal finite elements this modified integration is equivalent at
steady state to a high order Gauss collocation method applied to an ODE for the
flux. This method is superconvergent at the collocation points, thus providing
a discrete well-balanced property very similar in spirit to the one proposed in
[20], albeit not needing the explicit computation of a local approximation of
the steady state. To control the entropy production, we introduce artificial
viscosity corrections at the cell level and incorporate them into the scheme.
We provide theoretical and numerical characterizations of the accuracy and
equilibrium preservation of these corrections. Through extensive numerical
benchmarking, we validate our theoretical predictions, with considerable
improvements in accuracy for steady states, as well as enhanced robustness for
more complex scenario
A Well-Balanced Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations
We develop a well-balanced central-upwind scheme for rotating shallow water
model with horizontal temperature and/or density gradients---the thermal
rotating shallow water (TRSW). The scheme is designed using the flux
globalization approach: first, the source terms are incorporated into the
fluxes, which results in a hyperbolic system with global fluxes; second, we
apply the Riemann-problem-solver-free central-upwind scheme to the rewritten
system. We ensure that the resulting method is well-balanced by switching off
the numerical diffusion when the computed solution is near (at)
thermo-geostrophic equilibria.
The designed scheme is successfully tested on a series of numerical examples.
Motivated by future applications to large-scale motions in the ocean and
atmosphere, the model is considered on the tangent plane to a rotating planet
both in mid-latitudes and at the Equator. The numerical scheme is shown to be
capable of quite accurately maintaining the equilibrium states in the presence
of nontrivial topography and rotation. Prior to numerical simulations, an
analysis of the TRSW model based on the use of Lagrangian variables is
presented, allowing one to obtain criteria of existence and uniqueness of the
equilibrium state, of the wave-breaking and shock formation, and of instability
development out of given initial conditions. The established criteria are
confirmed in the conducted numerical experiments
Reducing Numerical Artifacts by Sacrificing Well-Balance for Rotating Shallow-Water Flow
We consider the problem of rotational shallow-water flow for which non-trivial rotating steady-state solutions are of great importance. In particular, we investigate a high-resolution central-upwind scheme that is well-balanced for a subset of these stationary solutions and show that the well-balanced design is the source of numerical artifacts when applied to more general problems. We propose an alternative flux evaluation that sacrifices the well-balanced property and demonstrate that this gives qualitatively better results for relevant test cases and real-world oceanographic simulations.acceptedVersio
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