797 research outputs found
Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics
For surface gravity waves propagating in shallow water, we propose a variant
of the fully nonlinear Serre-Green-Naghdi equations involving a free parameter
that can be chosen to improve the dispersion properties. The novelty here
consists in the fact that the new model conserves the energy, contrary to other
modified Serre's equations found in the literature. Numerical comparisons with
the Euler equations show that the new model is substantially more accurate than
the classical Serre equations, specially for long time simulations and for
large amplitudes.Comment: 24 pages, 4 figures, 41 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
A Constrained Transport Scheme for MHD on Unstructured Static and Moving Meshes
Magnetic fields play an important role in many astrophysical systems and a
detailed understanding of their impact on the gas dynamics requires robust
numerical simulations. Here we present a new method to evolve the ideal
magnetohydrodynamic (MHD) equations on unstructured static and moving meshes
that preserves the magnetic field divergence-free constraint to machine
precision. The method overcomes the major problems of using a cleaning scheme
on the magnetic fields instead, which is non-conservative, not fully Galilean
invariant, does not eliminate divergence errors completely, and may produce
incorrect jumps across shocks. Our new method is a generalization of the
constrained transport (CT) algorithm used to enforce the condition on fixed Cartesian grids. Preserving at the discretized level is necessary to maintain the
orthogonality between the Lorentz force and . The possibility of
performing CT on a moving mesh provides several advantages over static mesh
methods due to the quasi-Lagrangian nature of the former (i.e., the mesh
generating points move with the flow), such as making the simulation
automatically adaptive and significantly reducing advection errors. Our method
preserves magnetic fields and fluid quantities in pure advection exactly.Comment: 13 pages, 9 figures, accepted to MNRAS. Animations available at
http://www.cfa.harvard.edu/~pmocz/research.htm
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
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