Th. Sonar

On meshless collocation approximations of conservation laws : Preliminary investigations on positive schemes and dissipation models

We consider meshless collocation, methods for the numerical solution of transport processes described by hyperbolic conservation laws. The future goal is the construction of a robust and reliable meshfree discretization method for the equations of gas dynamics in complex geometrics. In this paper we start with the simplest scalar model problems and analyze basic problems occurring in the grid-free approach. A topological condition on clouds of points is derived and several possible versions of a generalized Lax-Friedrichs scheme are discussed with respect to their numerical dissipation. A moving least-squares approach is followed to construct a positive discretization for solutions with shocks which is thoroughly analyzed and applied to test problems.

From continuous recovery to discrete filtering in numerical approximations of conservation laws

Modern numerical approximations of conservation laws rely on numerical dissipation as a means of stabilization. The older, alternative approach is the use of central differencing with a dose of artificial dissipation. In this paper we review the successful class of weighted essentially non-oscillatory finite volume schemes which comprise sophisticated methods of the first kind. New developments in image processing have made new devices possible which can serve as highly nonlinear artificial dissipation terms. We view artificial dissipation as discrete filter operation and introduce several new algorithms inspired by image processing.

Adaptive Spectral Filtering and Digital Total Variation Postprocessing for the DG Method on Triangular Grids: Application to the Euler Equations

With respect to the possible presence of discontinuities in the solutions of nonlinear wave propagation problems high order methods have to be provided with a dose of supplementary numerical dissipation, otherwise the approximate solution may severely suffer from the presence of Gibbs oscillations. To prevent these oscillations from rendering the scheme unstable we apply the spectral filtering framework to the DG method on triangular grids. The corresponding spectral filter has been derived in [18] from a spectral viscosity formulation and is applied adaptively in order to restrict artificial viscosity to shock locations. Furthermore, the image processing technique of DTV filtering is shown to be a useful postprocessor. Numerical experiments are carried out for the two-dimensional Euler equations where we show results for the Shu-Osher shock–density wave interaction problem as well as the interaction of a moving vortex with a stationary shock.

Adaptive Spectral and DTV Filtering on Triangular Grids for the DG Method Applied to Compressible Fluid Flow

We apply the spectral filtering framework to stabilize the DG method on triangular grids in case of strong shocks. The corresponding modal filter is applied adaptively in order to restrict artificial viscosity to shock locations. Furthermore, the digital total variation filter originally developed for image denoising is used to postprocess the slightly oscillatory approximation at output times. Numerical experiments are carried out for the two‐dimensional Euler equations.

Numerical Methods for Time‐Dependent PDEs

The minisymposium aims at presenting recent research in the area of numerical methods for time‐dependent partial differential equations. The topics of time integration, space discretizations, and convergence issues like that of convergence to a steady state are considered as well as new algorithms for detecting and treating singularities like shock waves.