2,293 research outputs found
Maximum-principle preserving space-time isogeometric analysis
In this work we propose a nonlinear stabilization technique for
convection-diffusion-reaction and pure transport problems discretized with
space-time isogeometric analysis. The stabilization is based on a
graph-theoretic artificial diffusion operator and a novel shock detector for
isogeometric analysis. Stabilization in time and space directions are performed
similarly, which allow us to use high-order discretizations in time without any
CFL-like condition. The method is proven to yield solutions that satisfy the
discrete maximum principle (DMP) unconditionally for arbitrary order. In
addition, the stabilization is linearity preserving in a space-time sense.
Moreover, the scheme is proven to be Lipschitz continuous ensuring that the
nonlinear problem is well-posed. Solving large problems using a space-time
discretization can become highly costly. Therefore, we also propose a
partitioned space-time scheme that allows us to select the length of every time
slab, and solve sequentially for every subdomain. As a result, the
computational cost is reduced while the stability and convergence properties of
the scheme remain unaltered. In addition, we propose a twice differentiable
version of the stabilization scheme, which enjoys the same stability properties
while the nonlinear convergence is significantly improved. Finally, the
proposed schemes are assessed with numerical experiments. In particular, we
considered steady and transient pure convection and convection-diffusion
problems in one and two dimensions
Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations
We introduce a \textit{non-modal} analysis technique that characterizes the
diffusion properties of spectral element methods for linear
convection-diffusion systems. While strictly speaking only valid for linear
problems, the analysis is devised so that it can give critical insights on two
questions: (i) Why do spectral element methods suffer from stability issues in
under-resolved computations of nonlinear problems? And, (ii) why do they
successfully predict under-resolved turbulent flows even without a
subgrid-scale model? The answer to these two questions can in turn provide
crucial guidelines to construct more robust and accurate schemes for complex
under-resolved flows, commonly found in industrial applications. For
illustration purposes, this analysis technique is applied to the hybridized
discontinuous Galerkin methods as representatives of spectral element methods.
The effect of the polynomial order, the upwinding parameter and the P\'eclet
number on the so-called \textit{short-term diffusion} of the scheme are
investigated. From a purely non-modal analysis point of view, polynomial orders
between and with standard upwinding are well suited for under-resolved
turbulence simulations. For lower polynomial orders, diffusion is introduced in
scales that are much larger than the grid resolution. For higher polynomial
orders, as well as for strong under/over-upwinding, robustness issues can be
expected. The non-modal analysis results are then tested against under-resolved
turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While
devised in the linear setting, our non-modal analysis succeeds to predict the
behavior of the scheme in the nonlinear problems considered
Spatially partitioned embedded Runge-Kutta Methods
We study spatially partitioned embedded Runge–Kutta (SPERK) schemes for partial differential equations (PDEs), in which each of the component schemes is applied over a different part of the spatial domain. Such methods may be convenient for problems in which the smoothness of the solution or the magnitudes of the PDE coefficients vary strongly in space. We focus on embedded partitioned methods as they offer greater efficiency and avoid the order reduction that may occur in non-embedded schemes. We demonstrate that the lack of conservation in partitioned schemes can lead to non-physical effects and propose conservative additive schemes based on partitioning the fluxes rather than the ordinary differential equations. A variety of SPERK schemes are presented, including an embedded pair suitable for the time evolution of fifth-order weighted non-oscillatory (WENO) spatial discretizations. Numerical experiments are provided to support the theory
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