The convergence rate of approximate solutions for nonlinear scalar conservation laws

The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates

Forces on Bins - The Effect of Random Friction

In this note we re-examine the classic Janssen theory for stresses in bins, including a randomness in the friction coefficient. The Janssen analysis relies on assumptions not met in practice; for this reason, we numerically solve the PDEs expressing balance of momentum in a bin, again including randomness in friction.Comment: 11 pages, LaTeX, with 9 figures encoded, gzippe

A Parallel Mesh-Adaptive Framework for Hyperbolic Conservation Laws

We report on the development of a computational framework for the parallel, mesh-adaptive solution of systems of hyperbolic conservation laws like the time-dependent Euler equations in compressible gas dynamics or Magneto-Hydrodynamics (MHD) and similar models in plasma physics. Local mesh refinement is realized by the recursive bisection of grid blocks along each spatial dimension, implemented numerical schemes include standard finite-differences as well as shock-capturing central schemes, both in connection with Runge-Kutta type integrators. Parallel execution is achieved through a configurable hybrid of POSIX-multi-threading and MPI-distribution with dynamic load balancing. One- two- and three-dimensional test computations for the Euler equations have been carried out and show good parallel scaling behavior. The Racoon framework is currently used to study the formation of singularities in plasmas and fluids.Comment: late submissio

Large Eddy Simulation of acoustic pulse propagation and turbulent flow interaction in expansion mufflers

A novel hybrid pressure-based compressible solver is developed and validated for low Mach number acoustic flow simulation. The solver is applied to the propagation of an acoustic pulse in a simple expansion muffler, a configuration frequently employed in HVAC and automotive exhaust systems. A set of benchmark results for experimental analysis of the simple expansion muffler both with and without flow are obtained to compare attenuation in forced pulsation for various mean-flow velocities. The experimental results are then used for validation of the proposed pressure-based compressible solver. Compressible, Unsteady Reynolds Averaged Navier-Stokes (URANS) simulation of a muffler with a mean through flow is conducted and results are presented to demonstrate inherent limitations associated with this approach. Consequently, a mixed synthetic inflow boundary condition is developed and validated for compressible Large Eddy Simulation (LES) of channel flow. The mixed synthetic boundary is then employed for LES of a simple expansion muffler to analyse the flow-acoustic and acoustic-pulse interactions inside the expansion muffler. The improvement in the prediction of vortex shedding inside the chamber is highlighted in comparison to the URANS method. Further, the effect of forced pulsation on flow-acoustic is observed in regard to the shift in Strouhal number inside the simple expansion muffler

Applications of the DFLU flux to systems of conservation laws

The DFLU numerical flux was introduced in order to solve hyperbolic scalar conservation laws with a flux function discontinuous in space. We show how this flux can be used to solve systems of conservation laws. The obtained numerical flux is very close to a Godunov flux. As an example we consider a system modeling polymer flooding in oil reservoir engineering

On the Divergence-Free Condition in Godunov-Type Schemes for Ideal Magnetohydrodynamics: the Upwind Constrained Transport Method

We present a general framework to design Godunov-type schemes for multidimensional ideal magnetohydrodynamic (MHD) systems, having the divergence-free relation and the related properties of the magnetic field B as built-in conditions. Our approach mostly relies on the 'Constrained Transport' (CT) discretization technique for the magnetic field components, originally developed for the linear induction equation, which assures div(B)=0 and its preservation in time to within machine accuracy in a finite-volume setting. We show that the CT formalism, when fully exploited, can be used as a general guideline to design the reconstruction procedures of the B vector field, to adapt standard upwind procedures for the momentum and energy equations, avoiding the onset of numerical monopoles of O(1) size, and to formulate approximate Riemann solvers for the induction equation. This general framework will be named here 'Upwind Constrained Transport' (UCT). To demonstrate the versatility of our method, we apply it to a variety of schemes, which are finally validated numerically and compared: a novel implementation for the MHD case of the second order Roe-type positive scheme by Liu and Lax (J. Comp. Fluid Dynam. 5, 133, 1996), and both the second and third order versions of a central-type MHD scheme presented by Londrillo and Del Zanna (Astrophys. J. 530, 508, 2000), where the basic UCT strategies have been first outlined

Finite volume methods for unidirectional dispersive wave models

We extend the framework of the finite volume method to dispersive unidirectional water wave propagation in one space dimension. In particular we consider a KdV-BBM type equation. Explicit and IMEX Runge-Kutta type methods are used for time discretizations. The fully discrete schemes are validated by direct comparisons to analytic solutions. Invariants conservation properties are also studied. Main applications include important nonlinear phenomena such as dispersive shock wave formation, solitary waves and their various interactions.Comment: 25 pages, 12 figures, 51 references. Other authors papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws

We introduce a new methodology for adding localized, space-time smooth, artificial viscosity to nonlinear systems of conservation laws which propagate shock waves, rarefactions, and contact discontinuities, which we call the $C$-method. We shall focus our attention on the compressible Euler equations in one space dimension. The novel feature of our approach involves the coupling of a linear scalar reaction-diffusion equation to our system of conservation laws, whose solution $C(x,t)$ is the coefficient to an additional (and artificial) term added to the flux, which determines the location, localization, and strength of the artificial viscosity. Near shock discontinuities, $C(x,t)$ is large and localized, and transitions smoothly in space-time to zero away from discontinuities. Our approach is a provably convergent, spacetime-regularized variant of the original idea of Richtmeyer and Von Neumann, and is provided at the level of the PDE, thus allowing a host of numerical discretization schemes to be employed. We demonstrate the effectiveness of the $C$-method with three different numerical implementations and apply these to a collection of classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a classical continuous finite-element implementation using second-order discretization in both space and time, FEM-C. Second, we use a simplified WENO scheme within our $C$-method framework, WENO-C. Third, we use WENO with the Lax-Friedrichs flux together with the $C$-equation, and call this WENO-LF-C. All three schemes yield higher-order discretization strategies, which provide sharp shock resolution with minimal overshoot and noise, and compare well with higher-order WENO schemes that employ approximate Riemann solvers, outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure

Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations

We assess the suitability of a recent high-resolution central scheme developed by Kurganov & Tadmor (2000) for the solution of the relativistic hydrodynamics equations. The novelty of this approach relies on the absence of Riemann solvers in the solution procedure. The computations we present are performed in one and two spatial dimensions in Minkowski spacetime. Standard numerical experiments such as shock tubes and the relativistic flat-faced step test are performed. As an astrophysical application the article includes two-dimensional simulations of the propagation of relativistic jets using both Cartesian and cylindrical coordinates. The simulations reported clearly show the capabilities of the numerical scheme to yield satisfactory results, with an accuracy comparable to that obtained by the so-called high-resolution shock-capturing schemes based upon Riemann solvers (Godunov-type schemes), even well inside the ultrarelativistic regime. Such central scheme can be straightforwardly applied to hyperbolic systems of conservation laws for which the characteristic structure is not explicitly known, or in cases where the exact solution of the Riemann problem is prohibitively expensive to compute numerically. Finally, we present comparisons with results obtained using various Godunov-type schemes as well as with those obtained using other high-resolution central schemes which have recently been reported in the literature.Comment: 14 pages, 12 figures, to appear in A&

Numerical simulations of shocks encountering clumpy regions

We present numerical simulations of the adiabatic interaction of a shock with a clumpy region containing many individual clouds. Our work incorporates a sub-grid turbulence model which for the first time makes this investigation feasible. We vary the Mach number of the shock, the density contrast of the clouds, and the ratio of total cloud mass to inter-cloud mass within the clumpy region. Cloud material becomes incorporated into the flow. This "mass-loading" reduces the Mach number of the shock, and leads to the formation of a dense shell. In cases in which the mass-loading is sufficient the flow slows enough that the shock degenerates into a wave. The interaction evolves through up to four stages: initially the shock decelerates; then its speed is nearly constant; next the shock accelerates as it leaves the clumpy region; finally it moves at a constant speed close to its initial speed. Turbulence is generated in the post-shock flow as the shock sweeps through the clumpy region. Clouds exposed to turbulence can be destroyed more rapidly than a similar cloud in an "isolated" environment. The lifetime of a downstream cloud decreases with increasing cloud-to-intercloud mass ratio. We briefly discuss the significance of these results for starburst superwinds and galaxy evolution.Comment: 17 pages, 19 figures, accepted for publication in MNRA