Residual equilibrium schemes for time dependent partial differential equations

Many applications involve partial differential equations which admits nontrivial steady state solutions. The design of schemes which are able to describe correctly these equilibrium states may be challenging for numerical methods, in particular for high order ones. In this paper, inspired by micro-macro decomposition methods for kinetic equations, we present a class of schemes which are capable to preserve the steady state solution and achieve high order accuracy for a class of time dependent partial differential equations including nonlinear diffusion equations and kinetic equations. Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure

Parallelization of a relaxation scheme modelling the bedload transport of sediments in shallow water flow

In this work we are interested in numerical simulations for bedload erosion processes. We present a relaxation solver that we apply to moving dunes test cases in one and two dimensions. In particular we retrieve the so-called anti-dune process that is well described in the experiments. In order to be able to run 2D test cases with reasonable CPU time, we also describe and apply a parallelization procedure by using domain decomposition based on the classical MPI library.Comment: 19 page

Modelling Shallow Water Flows by a High Resolution Riemann Solver

Shock-capturing methods originally developed for compressible gas dynamics has been applied to many other non-linear hyperbolic systems of conservation equations, like reactive flows, two-phase flow, porous media flow and finally shallow water flows, with applications to rivers, estuaries, dam break and in particular tsunami propagation resulting from earthquake and landslide. The basic ingredient of the proposed model for solving the shallow water equations is a finite volume Flux Vector Splitting Riemann Solver with a second order resolution scheme and implicit treatment of the source terms. The model includes entropy fix and step bed treatments, and finally shore line tracking, giving the ability to capture local discontinuities - like shock waves - and reducing numerical diffusion and unphysical viscosity effects which dominates in all finite difference methods. The numerical model has been validated in respect to different numerical test cases and comparisons with the exact solution of the Riemann problems are presented.JRC.G.7-Traceability and vulnerability assessmen

Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws

The development of reliable numerical methods for the simulation of real life problems requires both a fundamental knowledge in the field of numerical analysis and a proper experience in practical applications as well as their mathematical modeling. Thus, the purpose of the workshop was to bring together experts not only from the field of applied mathematics but also from civil and mechanical engineering working in the area of modern high order methods for the solution of partial differential equations or even approximation theory necessary to improve the accuracy as well as robustness of numerical algorithms

Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove that the scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be applied to models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of numerical examples

Stratified shallow flow modelling

Environmental hydraulics covers a very wide range of applications including free surface flows in rivers. estuaries and lakes. To find engineering solutions to environmental hydraulics problems. 3D numerical modelling is nowadays widely used. However. the computation of sharp spatial gradients (such as found in stratified estuaries and lakes. around plumes near outfalls along rivers and coasts or in exchange areas of high shear). and the modelling of these processes along steep bathymetric slopes (such as found at the edge of dredged channels or of the continental shelf) remains a challenge. In addition. crude assumptions (such as the hydrostatic assumption) are often made to the primary differential equations in order to simplify the problem and enable long term prediction of environmental hydraulic changes. In this thesis. a robust adaptive mesh displacement (AMD) method is implemented and validated against the lock exchange case in particular. The AMD method aims at vertically focusing nodes within each water column to capture sharp gradients. while reducing the number of nodes or requiring prior knowledge of the flow structure. Second. a direct computation of dynamic pressure is introduced based on the equation of vertical momentum and validated against the analytical potential flow theory solution of a source-sink pair. Dynamic pressure is necessary to model destratification recirculation devices. or flow over dredge channel. or solitary waves. for instance. This direct computation method makes the hydrostatic assumption redundant. Third. a new advection scheme is implemented. whose main advantage is simplicity averaging over Riemann problems without solving them. while excessive numerical viscosity is compensated for by using high-resolution MUSCL type reconstruction. Recommendations are made in this thesis to extend the advection scheme developed herein for tracer advection to the non-linear shallow water equations. to the diffusion terms and to turbulence closure laws within the same finite element framework

ICASE/LaRC Workshop on Adaptive Grid Methods

Solution-adaptive grid techniques are essential to the attainment of practical, user friendly, computational fluid dynamics (CFD) applications. In this three-day workshop, experts gathered together to describe state-of-the-art methods in solution-adaptive grid refinement, analysis, and implementation; to assess the current practice; and to discuss future needs and directions for research. This was accomplished through a series of invited and contributed papers. The workshop focused on a set of two-dimensional test cases designed by the organizers to aid in assessing the current state of development of adaptive grid technology. In addition, a panel of experts from universities, industry, and government research laboratories discussed their views of needs and future directions in this field