Algorithm of radiation hydrodynamics with nonorthogonal mesh for 3D implosion problem

Highlights

• A mesh generation based on the cube spherical projection is proposed in ICF.

• A new system of the non-equilibrium radiation hydrodynamics equations is presented.

• The explicit-implicit numerical scheme is presented in spherical coordinates.

Abstract

It is crucial to understand the degradation of implosion performance caused by three-dimensional physics and the implementation of 3-D spherical mesh is challenging. In this paper a mesh generation method based on the cube spherical projection is proposed for 3D implosion problem in Inertial Confinement Fusion (ICF). This kind of mesh has the advantage of good quality, convenient adding radiation source and adaptive moving in the radial direction. Together, a mathematical model of radiation hydrodynamics with ion-electron non-equilibrium is present. The equations of radiation hydrodynamics are solved with the operator splitting method, such that the hydrodynamics equations are solved explicitly making use of the capability of well understood explicit schemes while the radiation diffusion and electron/ion conduction parts are solved implicitly by an eleven-point scheme on the non-orthogonal mesh. We present results from several benchmark problems in order to validate our numerical scheme. And finally, we investigate the development of fourth-order modes and compute the disturbance growth process of hot spot boundary, which shown that it conforms to the physical law.

Introduction

Inertial Confinement Fusion (ICF) involves very complicated physical processes, in particular for the indirect drive approach including the hohlraum physics and capsule physics. Where, the capsule physics includes radiation ablation, drive symmetry and hydrodynamic instability. These processes with different characteristic length scales and time scales are coupled and compete with each other. The symmetry concern arises from the fact that the spherical shells, must remain as concentric as possible, otherwise compression will be limited, yield will be reduced, and the target may even break up before ignition. Therefore, ICF is a complicated multi-material, multi-physics processes [1], [2], [3], [26]. In order to study the physical process of ICF carefully, especially the dynamic process of implosion, it is necessary to provide meaningful experimental data, reduce the number of experiments, save funds and so on. So, numerical simulation has become more and more important in the study of ICF. Highly reliable numerical methods are required to analyze experimental results and to predict the behavior of physics [4], [5].

Radiation hydrodynamical equations have been successfully applied in ICF or astrophysical. In general, radiation hydrodynamics concerns on the propagation of thermal radiation through a fluid and the effect of this radiation on the hydrodynamics describing the fluid motion. When the radiation field is in the thermodynamic non-equilibrium, by some assumptions and simplification techniques, the process of radiation transfer can be described approximately by the three-temperature (3-T) energy equations, where the unknowns are the temperature of the electron, ion, and radiation. So, Radiation hydrodynamical equations may be written as a hyperbolic system of conservation laws plus energy diffusion equations (see, e.g. [5], [6], [7], [8], [9], [10], [11], [12], [13], [18]).

Because of the dramatically different time scales of energy transport and fluid advection, radiation hydrodynamics equations calculations are usually treated using an implicit-explicit (IMEX) scheme. In such algorithms, the fluid advection component is treated explicitly, whereas the energy transport and exchange components are treated implicitly. That is to say, operator splitting is adopted. In 1998, Wenlong Dai and Paul R. Woodward proposed a second order accurate finite difference scheme for multidimensional radiation hydrodynamical equations in a diffusion limit, a Godunov scheme including linear and nonlinear Riemann solvers is proposed for the set of conservation laws, the radiative heat conduction is treated implicitly. In this paper, the most notable are the second order accuracy in both space and time, quick damping of numerical errors when the size of time steps is large, but its energy equation is described by one temperature [18]. A second order self-consistent IMEX methods time integration technique for solving radiation hydrodynamics problems is presented in 2010 [9], the idea of this method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations, but this paper only considers a high energy density radiation diffusion hydrodynamics model. There are also other discrete method for solving Radiation hydrodynamical equations [6], [7], [12], [16].

Many radiation hydrodynamics algorithms are directly integrated into the code for physical applications, such as code RAGE, HYDRA, LASNEX, PLUTO, LARED etc [4], [8], [19], [20], [21], where some are based on Euler grids which are fixed, and others on Lagrangian grids which move with the fluid. The RAGE code is implemented in ICF with Euler grids, the hydrodynamics is a basic Godunov solver, which is made significant improvements to increase the advection algorithms robustness, and it converges nonlinearities in a novel manner to allow larger timesteps and more robust behavior [21].

The main contribution of the present paper is to present a non-equilibrium radiation hydrodynamics equations for simulating the implosion problem in ICF. Firstly, we develop a mesh generation based on the cube spherical projection which is very convenient to compute the problem of radiation source in the radius direction. Then, on this non-orthogonal mesh, we propose an effective numerical scheme to solve the equations. Our scheme is based on operator splitting method, and the hydrodynamics equations are solved explicitly making use of the capability of some well understood explicit schemes, while the radiation diffusion and electron/ion conduction parts are solved implicitly by an eleven-point scheme on the non-orthogonal mesh.

The paper is organized as follows: In Section 2, we introduce the radiation hydrodynamics equations; In Section 3, we give a mesh generation method; In Section 4, numerical algorithms are given, including the advection term, pressure term and diffusion term, etc; In Section 5, several numerical tests are presented which demonstrate the accuracy of the present scheme; Finally we conclude in section 6.