Local error analysis for approximate solutions of hyperbolic conservation laws

We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted into L loc ∞ estimates, following the Lip′ convergence theory developed by Tadmor et al. Comparisons between the local truncation error and the L loc ∞ -error show remarkably similar behavior. Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341].Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41709/1/10444_2005_Article_7099.pd

Accelerated convergence to steady state by gradual far-field damping

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76495/1/AIAA-11054-168.pd

Multicomponent Flow Calculations by a Consistent Primitive Algorithm

The dynamics of inviscid multicomponent fluids may be modelled by the Euler equations, augmented by one (or more) additional species equation(s). Attempts to compute solutions for extended Euler models in conservation form, show strong oscillations and other computational inaccuracies near material interfaces. These are due to erroneous pressure fluctuations generated by the conservative wave model. This problem does not occur in single component computations and arises only in the presence of several species. A nonconservative (primitive) Euler formulation is proposed, which results in complete elimination of the oscillations. The numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation. Numerical experiments show clean monotonic solution profiles, with acceptably small conservation error for shocks of weak to moderate strengths.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31590/1/0000519.pd

On the Dynamics of a Shock-Bubble Interaction

We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity. Such interactions have been studied experimentally in an attempt to elucidate the mechanisms whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas and Sturtevant: a M S = 1:22 planar shock wave, moving through air, impinges on a cylindrical bubble which contains either helium or Refrigerant 22. These flows are modelled using the two-dimensional, compressible Euler equations for a two component fluid (air-helium or air-Refrigerant 22). Although simulations of shock wave phenomena are now fairly commonplace, they are mostly restricted to single component flows. Unfortunately, multi-component extensions of ..

A comment on the computation of non-conservative products.

Adaptive Schemes for Deterministic and Stochastic Flow Problem

A relaxation scheme for the two-layer shallow water system.

International audienc

A Central Scheme for Shallow Water Flows along Channels with Irregular Geometry

We present a new semi-discrete central scheme for one-dimensional shallow water flows along channels with non-uniform rectangular cross sections and bottom topography. The scheme preserves the positivity of the water height, and it is preserves steady-states of rest (i.e. it is well-balanced). Along with a detailed description of the scheme, numerous numerical examples are presented for unsteady and steady flows. Comparison with exact solutions illustrate the accuracy and robustness of the numerical algorithm. AMS subject classification: Primary 65M99; Secondary 35L65 Key words: Hyperbolic systems of conservation and balance laws, semi-discrete schemes, Saint-Venant system of Shallow Water equations, non-oscillatory reconstructions, channels with irregular geometry. 1 The Shallow-water Model We consider the shallow water equations along channels with non-uniform rectangular cross sections and bottom topography. The model describes flows that are nearly horizontal and can be obtained by averaging the Euler equations over the channel cross section [3], resulting in the balance law ∂A ∂

On the Dynamics of a Shock-Bubble Interaction

We present a detailed numerical study of the interaction of a weak shock wave with an isolated cylindrical gas inhomogeneity. Such interactions have been studied experimentally in an attempt to elucidate the mechanisms whereby shock waves propagating through random media enhance mixing. Our study concentrates on the early phases of the interaction process which are dominated by repeated refractions and reflections of acoustic fronts at the bubble interface. Specifically, we have reproduced two of the experiments performed by Haas and Sturtevant: a Ms- 1.22 planar shock wave, moving through air, impinges on a cylindrical bubble which contains either helium or Refrigerant 22. These flows are modelled using the two-dimensional, compressible Euler equations for a two component fluid (air-helium or air-Refrigerant 22). Although simulations of shock wave phenomena are now fairly commonplace, they are mostly restricted to single component flows. Unfortunately, multi-component extensions of successful single component schemes often suffer from spurious oscillations which are generated at material interfaces. Here we avoid such problems by employing a novel, nonconservative shock-capturing scheme. In addition, we have utilized a sophis-ticated adaptive mesh refinement algorithm which enables extremely high resolution simulations t

Interface tracking method for compressible multifluids

This paper is concerned with numerical methods for compressible multicomponent fluids. The fluid components are assumed immiscible, and are separated by material interfaces, each endowed with its own equation of state (EOS). Cell averages of computational cells that are occupied by several fluid components require a “mixed-cell” EOS, which may not always be physically meaningful, and often leads to spurious oscillations. We present a new interface tracking algorithm, which avoids using mixed-cell information by solving the Riemann problem between its single-fluid neighboring cells. The resulting algorithm is oscillation-free for isolated material interfaces, conservative, and tends to produce almost perfect jumps across material fronts. The computational framework is general and may be used in conjunction with one's favorite finite-volume method. The robustness of the method is illustrated on shock-interface interaction in one space dimension, oscillating bubbles with radial symmetry and shock-bubble interaction in two space dimensions