1,087 research outputs found
A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes
We are interested in the approximation of a steady hyperbolic problem. In
some cases, the solution can satisfy an additional conservation relation, at
least when it is smooth. This is the case of an entropy. In this paper, we
show, starting from the discretisation of the original PDE, how to construct a
scheme that is consistent with the original PDE and the additional conservation
relation. Since one interesting example is given by the systems endowed by an
entropy, we provide one explicit solution, and show that the accuracy of the
new scheme is at most degraded by one order. In the case of a discontinuous
Galerkin scheme and a Residual distribution scheme, we show how not to degrade
the accuracy. This improves the recent results obtained in [1, 2, 3, 4] in the
sense that no particular constraints are set on quadrature formula and that a
priori maximum accuracy can still be achieved. We study the behavior of the
method on a non linear scalar problem. However, the method is not restricted to
scalar problems
Runge-Kutta residual distribution schemes
We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given
Numerical simulation of conservation laws with moving grid nodes: Application to tsunami wave modelling
In the present article we describe a few simple and efficient finite volume
type schemes on moving grids in one spatial dimension combined with appropriate
predictor-corrector method to achieve higher resolution. The underlying finite
volume scheme is conservative and it is accurate up to the second order in
space. The main novelty consists in the motion of the grid. This new dynamic
aspect can be used to resolve better the areas with large solution gradients or
any other special features. No interpolation procedure is employed, thus
unnecessary solution smearing is avoided, and therefore, our method enjoys
excellent conservation properties. The resulting grid is completely
redistributed according the choice of the so-called monitor function. Several
more or less universal choices of the monitor function are provided. Finally,
the performance of the proposed algorithm is illustrated on several examples
stemming from the simple linear advection to the simulation of complex shallow
water waves. The exact well-balanced property is proven. We believe that the
techniques described in our paper can be beneficially used to model tsunami
wave propagation and run-up.Comment: 46 pages, 7 figures, 7 tables, 94 references. Accepted to
Geosciences. Other author's papers can be downloaded at
http://www.denys-dutykh.com
Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)
This workshop brought together
leading experts, as well as the most
promising young researchers, working on nonlinear
hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in
modeling, analysis, and numerics. Particular topics included ill-/well-posedness,
randomness and multiscale modeling, flows in a moving domain, free boundary problems,
games and control
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