Shock Waves; Rarefaction Waves; Equations of State A Solution Guide
Short course for LANL summer students in XCP & T.
42 ENGINEERING↗
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Equation of state of boron carbide B 4 C
We present the results of recent experiments conducted on the Sandia Z machine and a new tabular equation of state for B 4 C. The equation of state was calibrated to a combination of density functional calculations reported here and fits to preexisting data. It was constructed partly to recover the effects of a shock-driven, polymorphic phase transition of unknown character beginning at particle velocities of just under 3 km/s (shock pressures of 95 GPa). Some of the Z experiments included sound speeds determined by the overtaking rarefaction method, from which we calculate the Grüneisen parameter and compare with previous experiments conducted at the OMEGA laser [Fratanduono et al ., Phys. Rev. B 94 , 184107 (2016)], our own first principles calculations, and another recent tabular equation of state [Zhang et al ., Phys. Rev. E 102 , 053203 (2020)]. We also compare our results with previous static compression, thermophysical, and melt studies, finding mixed consistency. We predict the onset and completion of shock melting at 225 and 265 GPa, respectively, and predict a melt curve that is largely flat to pressures of several hundred GPa.
36 MATERIALS SCIENCE↗
Euler equations and the Sod shock tube problem
The Euler equations are a subset of the magnetohydrodynamic (MHD) equations in the infinitely collisional, unmagnetized limit. MHD modeling is central to many areas of plasma physics, ranging from low-temperature glow discharges to inertial confinement fusion. An important aspect of the Euler equations is their ability to describe states with discontinuities, such as shock waves. A standard benchmark test for numerical implementation of the Euler equations is the Sod shock tube. In this test, the system is initialized at rest with a pressure and density discontinuity, which results in a shock wave traveling into the low-pressure region and a rarefaction wave traveling into the high-pressure region. Starting with the presentation of the Euler equations, a numerical algorithm is presented here to solve these equations in one dimension. This is followed by an overview of the Sod shock tube problem that includes the precise initial setup and the analytic solution. Finally, the analytic solution is compared with results from numerical simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Parallel-in-Time Solution of Hyperbolic PDE Systems via Characteristic-Variable Block Preconditioning
We consider the parallel-in-time solution of both linear and nonlinear hyperbolic partial differential equation (PDE) systems in one spatial dimension. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that intervariable coupling between characteristic variables is weak, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small, while that between the original variables is not. For an ℓ-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of ℓ scalar linear(ized)-advection-like problems, each associated with a different characteristic wave-speed in the underlying linear(ized) PDE. Furthermore, we approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions. For many test problems, the solver converges in just a handful of iterations and with mesh-independent convergence rates.
97 MATHEMATICS AND COMPUTING↗
Parallel-in-Time Solution of Scalar Nonlinear Conservation Laws
Here, we consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially nonoscillatory (WENO) reconstructions, and in time with high-order explicit Runge–Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of nondifferentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton’s method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration; however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for nonconservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley–Leverett equations. For many test problems, the solver converges in just a handful of iterations with a convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.
97 MATHEMATICS AND COMPUTING↗