Richard Liska

Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations

The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

Composite Schemes for Conservation Laws

Global composition of several time steps of the two-step Lax--Wendroff scheme followed by a Lax--Friedrichs step seems to enhance the best features of both, although it is only first order accurate. We show this by means of some examples of one-dimensional shallow water flow over an obstacle. In two dimensions we present a new version of Lax--Friedrichs and an associated second order predictor-corrector method. Composition of these schemes is shown to be effective and efficient for some two-dimensional Riemann problems and for Nohs infinite strength cylindrical shock problem. We also show comparable results for composition of the predictor-corrector scheme with a modified second order accurate weighted essentially nonoscillatory (WENO) scheme. That composition is second order but is more efficient and has better symmetry properties than WENO alone. For scalar advection in two dimensions the optimal stability of the new predictor-corrector scheme is shown using computer algebra. We also show that the generalization of this scheme to three dimensions is unstable, but by using sampling we are able to show that the composites are suboptimally stable.

Testing Stability by Quantifier Elimination

For initial and initial-boundary value problems described by differential equations, stability requires the solutions to behave well for large times. For linear constant-coefficient problems, Fourier and Laplace transforms are used to convert stability problems to questions about roots of polynomials. Many of these questions can be viewed, in a natural way, as quantifier-elimination problems. The Tarski?Seidenberg theorem shows that quantifier-elimination problems are solvable in a finite number of steps. However, the complexity of this algorithm makes it impractical for even the simplest problems. The newer Quantifier Elimination by Partial Algebraic Decomposition (QEPCAD) algorithm is far more practical, allowing the solution of some non-trivial problems. In this paper, we show how to write all common stability problems as quantifier-elimination problems, and develop a set of computer-algebra tools that allows us to find analytic solutions to simple stability problems in a few seconds, and to solve some interesting problems in from a few minutes to a few hours.

Two-dimensional shallow water equations by composite schemes

Composite schemes are formed by global composition of several Lax–Wendroff steps followed by a diffusive Lax–Friedrichs or WENO step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. These schemes are applied to the shallow water equations in two dimensions. The Lax–Friedrichs composite is also formulated for a trapezoidal mesh, which is necessary in several example problems. The suitability of the composite schemes for the shallow water equations is demonstrated on several examples, including the circular dam break problem, the shock focusing problem and supercritical channel flow problems. Copyright

Optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian methods

A new optimization-based synchronized flux-corrected conservative interpolation (remapping) of mass and momentum for arbitrary Lagrangian-Eulerian hydro methods is described. Fluxes of conserved variables - mass and momentum - are limited in a synchronous way to preserve local bounds of primitive variables - density and velocity.

Highly efficient accelerator of dense matter using laser-induced cavity pressure acceleration

Acceleration of dense matter to high velocities is of high importance for high energy density physics, inertial confinement fusion, or space research. The acceleration schemes employed so far are capable of accelerating dense microprojectiles to velocities approaching 1000 km/s; however, the energetic efficiency of acceleration is low. Here, we propose and demonstrate a highly efficient scheme of acceleration of dense matter in which a projectile placed in a cavity is irradiated by a laser beam introduced into the cavity through a hole and then accelerated in a guiding channel by the pressure of a hot plasma produced in the cavity by the laser beam or by the photon pressure of the ultra-intense laser radiation trapped in the cavity. We show that the acceleration efficiency in this scheme can be much higher than that achieved so far and that sub-relativisitic projectile velocities are feasible in the radiation pressure regime.

Recent results from experimental studies on laser?plasma coupling in a shock ignition relevant regime

Shock ignition (SI) is an appealing approach in the inertial confinement scenario for the ignition and burn of a pre-compressed fusion pellet. In this scheme, a strong converging shock is launched by laser irradiation at an intensity Iλ 2 >10 15 Wc m −2 µm 2 at the end of the compression phase. In this intensity regime, laser–plasma interactions are characterized by the onset of a variety of instabilities, including stimulated Raman scattering, Brillouin scattering and the two plasmon decay, accompanied by the generation of a population of fast electrons. The effect of the fast electrons on the efficiency of the shock wave production is investigated in a series of dedicated experiments at the Prague Asterix Laser Facility (PALS). We study the laser–plasma coupling in a SI relevant regime in a planar geometry by creating an extended preformed plasma with a laser beam at ∼7 × 10 13 Wc m −2 (250 ps, 1315 nm). A strong shock is launched by irradiation with a second laser beam at intensities in the range 10 15 –10 16 Wc m −2 (250 ps, 438 nm) at various delays with respect to the first beam. The pre-plasma is characterized using x-ray spectroscopy, ion diagnostics and interferometry. Spectroscopy and calorimetry of the backscattered radiation is performed in the spectral range 250–850 nm, including (3/2)ω, ω and ω/2 emission. The fast electron production is characterized through spectroscopy and imaging of the Kα emission. Information on the shock pressure is obtained using shock breakout chronometry and measurements of the craters produced by the shock in a massive target. Preliminary results show that the backscattered energy is in the range 3–15%, mainly due to backscattered light at the laser wavelength (438 nm), which increases with increasing the delay between the two laser beams. The values of the peak shock pressures inferred from the shock breakout times are lower than expected from 2D numerical simulations. The same simulations reveal that the 2D effects play a major role in these experiments, with the laser spot size comparable with the distance between critical and ablation layers.

Applying quantifier elimination to stability analysis of difference schemes

Stability analysis is an important tool for constructing time-stepping finite difference schemes for partial differential equations. This paper describes how von Neumann stability analysis can be reduced to a quantifier elimination problem over the reals. We report our experience in analyzing some difference schemes by using a quantifier elimination package based on the partial cylindricl algebraic decomposition algorithm

Symmetry- and essentially-bound-preserving flux-corrected remapping of momentum in staggered ALE hydrodynamics

We present a new flux-corrected approach for remapping of velocity in the framework of staggered arbitrary Lagrangian-Eulerian methods. The main focus of the paper is the definition and preservation of coordinate invariant local bounds for velocity vector and development of momentum remapping method such that the radial symmetry of the radially symmetric flows is preserved when remapping from one equiangular polar mesh to another. The properties of this new method are demonstrated on a set of selected numerical cyclic remapping tests and a full hydrodynamic example.

Selected laser plasma simulations by ALE method

A novel 2D ALE code in Cartesian and cylindrical geometries on logically rectangular quadrilateral meshes was developed for laser plasma applications. We present here simulations of laser interactions with three different types (disc flyer, double foil and foam) of targets used in experiments at the PALS laser. The application of the ALE method proved to be essential as the pure Lagrangian method for these problems fails due to mesh degeneration.