We present a fully adaptive multiresolution scheme for spatially
one-dimensional quasilinear strongly degenerate parabolic equations
with zero-flux and periodic boundary conditions. The numerical scheme
is based on a finite volume discretization using the Engquist-Osher
numerical flux and explicit time stepping. An adaptive multiresolution
scheme based on cell averages is then used to speed up the CPU time and
the memory requirements of the underlying finite volume scheme, whose
first-order...
These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian geometries. The discretization schemes are based either on finite volume or finite difference schemes. The concept of multiresolution analyses, including Harten’s approach for point and cell averages, is described in some detail. Then the sparse point representation method is discussed. Different strategies for adaptive time-stepping, like local scale dependent time stepping and time step control, are...
Anisotropy and intermittency of quasi-static magnetohydrodynamic (MHD) turbulence in an
imposed magnetic field are examined, using three-dimensional orthonormal wavelet analysis.
Wavelets are an efficient tool to examine directional scale-dependent statistics, since
they are based on well-localized functions in space, scale and direction. The analysis is
applied to two turbulent MHD flows computed by direct numerical simulation with
512 grid points...
A new numerical scheme called particle-in-wavelets is proposed for the Vlasov-Poisson
equations, and tested in the simplest case of one spatial dimension. The plasma
distribution function is discretized using tracer particles, and the charge distribution
is reconstructed using wavelet-based density estimation. The latter consists in projecting
the Delta distributions corresponding to the particles onto a finite dimensional linear
space spanned by...
The Lagrangian statistics in rotating Saint-Venant turbulence are studied by means of
direct numerical simulation using a pseudo-spectral discretization fully resolving, both
in time and space, all the inertio-gravity waves present in the system. To understand the
influence of waves, three initial conditions are considered, one which is dominated by
waves, one which is dominated by vortices, and one which is intermediate between these two
extreme...
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