A Godunov-type tensor artificial viscosity for staggered Lagrangian hydrodynamics

Highlights

• A Godunov type artificial viscosity is derived in tensor space.

• The developed viscosity is proved to preserve the cylindrical symmetry.

• The developed viscosity performs well in a series of typical cases.

Abstract

This paper describes a tensor artificial viscosity for staggered Lagrangian hydrodynamics. Specifically, two viscous tensors are constructed using a discrete velocity gradient method. Under the condition of a piecewise constant distribution of velocity in each subcell, the Generalized Riemann Invariant relation, in the spirit of the Godunov methods, is applied to determine the coefficients of the viscous tensors. The artificial viscosity is found to be dissipative. Besides, the cylindrical symmetry of the developed artificial viscosity is demonstrated in an equi-angular polar grid. Typical numerical cases with strong shocks show that the tensor artificial viscosity is robust and performs well in different grid types.

Introduction

Lagrangian hydrodynamics, in which the computational grid moves simultaneously with the fluid, is widely used for multi-material flow simulations. Its property of resolving sharp contact discontinuities has a broad range of applications in fields such as astrophysics and inertial confinement fusion.

Early investigations [1] led to the development of staggered grid hydrodynamics (SGH), which employs a staggered placement of variables. In SGH, the momentum equation is solved around a nodal control volume, while the energy equation is solved around a zonal control volume. A classical SGH scheme [2] uses the support operator method [3], which discretizes the governing equations in a compatible way. The advantage of this scheme is that it exactly ensures the conservation of momentum and total energy. In the presence of shock waves in the SGH, an artificial viscosity is needed to eliminate the possible numerical oscillations. For 1D calculations, Von Neumann and Richtmyer [1] introduced a viscosity with a quadratic term that is added to the pressure in the momentum and energy equations. Later, Landshoff [4] proposed an additional linear term that improved Von Neumann's method by ensuring fewer oscillations behind the shock. Another well-known artificial viscosity proposed by Kurapatenko [5] uniformly matches the quadratic term and the linear term. This artificial viscosity is a good complement of the formulation proposed by Landshoff. For 2D and 3D calculations, many artificial viscosities have been established on the basis of the 1D artificial viscosity with minor modifications. Advanced multi-dimensional artificial viscosities can be edge based [6], tensorial [7], [8], [9], or Laplacian-based [10].

A different approach designed by Godunov [11] employs a collocated placement of all variables in one control volume. This Godunov-type method is known as cell-centered hydrodynamics (CCH) in the Lagrangian framework. Classically, such schemes overcome the discontinuities at cell edges using an approximate Riemann solver. A remarkable study using CCH was conducted by Despres [12], who constructed a node solver based on a Riemann invariant relation that satisfies the time-averaged equations for momentum conservation, total energy conservation, geometric conservation law (GCL), and entropy inequality. Subsequently, Maire [13], [14] established a sophisticated node solver that exactly recovers the 1D acoustic solver by allocating two pressures on each subcell. In [15], Burton considers the ancillary equation of rotational equilibrium to establish a new node solver with increased mesh robustness. For more complex discontinuous conditions, Jia [16] improved Maire's work by recovering the numerical flux in the Harten–Lax–van Leer contact (HLLC) scheme using the finite element framework. Based on the above node solvers, important progress in CCH has been made in many areas (e.g., 2D/3D numerical schemes, planar/cylindrical coordinates, finite volume/element frameworks) [17], [18], [19], [20], [21], [22].

According to previous investigations, the artificial viscosity and approximate Riemann solution are two different shock-capturing mechanisms. However, their gaps have been bridged by the introduction of Godunov-type artificial viscosities. Related works can be traced back to the application of Godunov's artificial viscosity scheme in the early 1990s. Christensen [23] noticed that the artificial viscosity in SGH can be obtained by the HLL approximate Riemann solver under certain conditions. Luttwak et al. [24], [25] proposed to use a uniaxial tensor pseudo-viscosity based on Riemann problem in staggered meshes. Their method is known as the Staggered Mesh Godunov scheme. Later, a compatible Lagrangian discretization and associated Riemann solver-based artificial viscosity were developed in both the 2D [26], [27], [28] and 3D [29] cases. As such artificial viscosity specifies the jump condition explicitly, high-order accuracy in space can be extended through Taylor expansions of the velocity field. Detailed information can be found in the implementation of a second-order least-squares approach with a frame-invariant limiter [14]. A further potential benefit of such viscosity is that, as long as the connection between the viscosity and the Riemann solution is determined, different types of Riemann solvers (e.g., HLL, HLLC, HLLE) can capture the shock in high resolution. However, a disadvantage of the Godunov-type artificial viscosity is that little attention has been paid to the nature of its tensor. As a result, inaccuracies arise in the viscosity discretization, and these might amplify the hourglass instabilities with spurious grid distortion [8].

In this paper, the general form of the artificial viscosity is derived in the full tensor space using a discrete gradient operator method. Moreover, the relationship between the artificial viscosity and the Generalized Riemann Invariant relation is presented, allowing us to determine the viscosity coefficients. Detailed analysis shows that the developed tensor artificial viscosity is superior to the Godunov-type scalar viscosity [27] in that it is applicable for an arbitrary distribution of the velocity gradient. A series of numerical tests containing strong discontinuities demonstrates the robustness of the developed viscosity.

The remainder of this paper is organized as follows. Section 2 briefly introduces the SGH discretizations that satisfy the total energy conservation condition. In Section 3, the theory and derivation of the Godunov-type artificial viscosity are described in detail. The symmetry of the developed viscosity is proved in Section 4. In Section 5, typical numerical examples are used to verify the performance of the Godunov-type artificial viscosity. Finally, Section 6 presents the conclusions to this study.