Artificial viscosity in comoving curvilinear coordinates: towards a differential geometrically consistent implicit advection scheme

1.1 Brief introduction to conservation laws

For the sake of stringency we recapitulate some important results from the theory and numerics of conservation laws and thereby introduce a few mathematical terms needed to motivate artificial viscosity. We refer to LeVeque ([1991]) and Richtmyer and Morton ([1994]) for the complete picture.

The equations of RHD and MHD form a system of hyperbolic conservation laws that describe the interaction of a density function $\mathbf{d}(\mathbf{x},t):{\mathbb{R}}^n\times [0,\mathrm{\infty })\to {\mathbb{R}}^m$ and its flux $\mathbf{f}(\mathbf{d}):{\mathbb{R}}^m\to {\mathbb{R}}^{m\times n}$. Equation (6) shows how a concrete choice for the density and the flux field can look like in a given coordinate system.

where n is the outward oriented normal of the surface.

The system is called hyperbolic if the Jacobian matrix ${\mathrm{\nabla }}_{\mathbf{d}}\mathbf{f}$ associated with the fluxes has real eigenvalues and if there exists a complete set of eigenvectors. In case of MHD and RHD this property has a direct physical relevance (Pons et al. [2000]).

With an initial condition $\mathbf{d}(\mathbf{x},0)={\mathbf{d}}_0(\mathbf{x})$, $\mathbf{x}\in {\mathbb{R}}^n$, this is called the Cauchy problem.

The gaseous pressure tensor can be assumed to be isotropic in most applications, which means that $\mathbf{P}=g^{ij}p$ where $p(\mathbf{x},t)$ is the scalar gas pressure and $g^{ij}={{\mathbf{e}}^i}^{\mathrm{T}}{\mathbf{e}}^j$ the contravariant metric tensor. In case of adaptive grids the base vectors are time-dependent as well, i.e., ${\mathbf{e}}_i={\mathbf{e}}_i(\mathbf{x},t)$.

The function $\mathbf{d}\in L^{\mathrm{\infty }}$ is called a weak solution of the PDE (4), if it satisfies (7) and $\mathbf{d}\in U$ with ${\mathbf{d}}_0\in L^{\mathrm{\infty }}$. However, there is a small drawback. This weak solution is not necessarily unique and usually further constraints have to be imposed in order to guarantee its uniqueness. This leads us to the actual topic of this paper.

1.2 Introduction to artificial viscosity

For most physical problems it is sufficient to look for weak solutions from the function space of piecewise continuously differentiable functions. Constraining the space of solutions in this way, we call the physical variables d weak solutions of the Cauchy problem (4), if they are classical solutions wherever they are continuously differentiable, and if at discontinuities (shocks) they satisfy additional conditions in order to be physically reasonable (we elaborate on these conditions below).

and then consider the limiting case $\epsilon \to 0$. This idea is motivated by physical diffusion which broadens sincere discontinuities to differentiable steep gradients at the (microscopic) length scale of the mean free path of the particles. The physical solution of the weakly formulated problem is thus the zero diffusion limit of the diffusive problem. However, in practice this limit is difficult to calculate analytically and hence simpler conditions have to be found. A common technique to do this is motivated by continuum physics as well. Here an additional conservation law is set to hold for another quantity - the entropy of the fluid flow - as long as the solution remains smooth. Moreover, it is known that along admissible shocks this physical variable never decreases, and the conservation law for the entropy can be formulated as an inequality.

For bounded, continuous pointwise solutions ${\mathbf{d}}^{\ast }$ of (12) such that ${\mathbf{d}}^{\ast }\to \mathbf{d}$ for $\epsilon \to 0$, the vanishing viscosity solution d is a weak solution of the initial value problem (3) and fulfills entropy condition (14). Generally spoken, applying the entropy condition to systems with shock solutions unveils those propagation velocities that ensure that no characteristics rise from discontinuities which would be non-physical. For detailed motivation, stringent argumentation and proofs to mathematical techniques presented in this section we refer to Harten et al. ([1976]) and LeVeque ([1991]).

1.3 Numerical artificial viscosity

As mentioned we are looking for high-resolution methods for nonlinear PDEs derived from hyperbolic conservation laws. In the past decades major efforts have been made in developing numerical methods for these problems that are at least of second order. One patent attempt to finding such a high-resolution method is to adapt a well-known high-order method for linear problems for nonlinear problems (such as the Lax-Wendroff scheme (Lax and Wendroff [1960])).

As illustrated above we can add an artificial viscosity term to the conservation law in a way that the entropy condition is satisfied and non-physical solutions are excluded. The viscosity term has to be designed in such a manner that it affects sincere discontinuities but vanishes sufficiently elsewhere so that the order of accuracy can be maintained in those regimes where the solution is smooth. The idea of numerical artificial viscosity was inspired by physical dissipation mechanisms and dates back more than half a century to von Neumann and Richtmyer ([1950]).

where h denotes the resolution of the spatial discretization, $h=x_{j+1}-x_j$. Since the original design of the additional viscous pressure in the scalar form, $Q=c\rho {(\mathrm{\Delta }\mathbf{u})}^2$, $c\in \mathbb{R}$, as suggested in von Neumann and Richtmyer ([1950]) for one-dimensional advection ${\partial }_t\mathbf{d}+a{\partial }_x\mathbf{d}=Q{\partial }_{xx}\mathbf{d}$, it has undergone a number of modifications and generalizations. It has turned out to be numerically preferable to add a linear term (see Landshoff ([1955])) in order to control oscillations. Generalizations to multi-dimensional flows mostly retain the original analogy to physical dissipation and reformulate the velocity term accordingly, see e.g. Wilkins ([1980]).

The artificial viscosity broadens shocks to steep gradients at some characteristic length scale, but should not cause too large smearing. The concrete composition and implementation of this artificial viscosity coefficient Q depends on the application. As an example we discuss the following form of the tensor of the artificial viscosity in higher-dimensional RHD numerics. Similar forms of artificial viscosity can be found also in pure hydrodynamics and MHD calculations in 2D and 3D.

Proposition

The commonly used (see e.g. Dorfi ([1999]) in RHD, Iwakami et al. ([2008]) in MHD, Fryxell et al. ([2000]) in MHD) form of Q (16) is not compatible with these requirements since the symmetrization is only defined for lower indices, whereas the unit tensor e of a metric space is only defined for mixed indices, meaning there is no such thing as ${\delta }_{ij}$. However, the above mentioned and other authors have neglected this inconsistency since they have considered mixed indices from the start when concentrating on Cartesian or affine coordinates. Nonlinear corrections have been suggested in Benson and Schoenfeld ([1993]) albeit this approach does not explicitly concern curvilinear coordinates and is based on a TVD approach.

In several hydro- and MHD-codes that include non-Cartesian grids such as Pluto (Mignone et al. [2007]), the geometric source terms are coded explicitly for several geometries (polar, cylindrical, spherical), and not only for the artificial viscosity flux. The suggestions made e.g. in Vinokur ([1974]) lead to geometrical source terms that correct curvilinear grid effects. However the strong conservation form as elaborated in Warsi ([1981]) would need to appeal to our differential geometrically consistent approach in order to deal with the viscosity in an intrinsically consistent way. Especially when the metric tensor itself is not only a function of space but also time-dependent (as discussed in Section 1), the latter approach reaches its limits. Our correction affects curvilinear coordinates in multiple dimensions, whereas it is not necessary that the coordinates are orthogonal, i.e., the metric tensor does not need to be diagonal. Our initial motivation to study more general coordinates comes from the idea to generate problem-oriented coordinate systems for astrophysical numerical calculations. In a following paper we want to present some feasible approaches to grid generation under certain physical restrictions. Such nonlinear grids that are adaptive in multiple dimensions have time-dependent metric tensors and thus benefit directly from our consistent definition. On the contrast, when using adaptive mesh refinement, the metric tensor remains geometrically constant in time.

In order to support the theoretical results in this work, in the upcoming section we study as an example a simple velocity field with a non-vanishing divergence and visualize the according artificial viscosities for the two presented cases.

3.1 Toy model velocity field

here in Cartesian coordinates. Such self-similar solutions appear in idealized spherical models of stars, for example as shocks driven by radial stellar pulsations.

In the following section we visualize the tensor of artificial viscosity (TAV) for the velocity field (19). A uniform distribution of the leading eigenvalues over the whole domain is expected due to the symmetry of the vector field.

One easily verifies the identity $Q_i^i=0$ summing over the mixed components $Q_r^r=-2/3r$, $Q_{\theta }^{\theta }=1/3r$, $Q_{\varphi }^{\varphi }=1/3r$.

The visualization of this non-metric version of the artificial viscosity for the symmetric velocity field (19) shows obviously a field unequal in strength and direction over the whole domain. In numerical computations this would clearly lead to artificial anisotropies in the flux of the density field and destroy all efforts in constructing a higher-order conservative numerical scheme with artificial viscosity.

3.2 Visualization of scalar and tensor fields

We can see the incorrect vs. correct behavior immediately by even just displaying the major eigenvalue or the trace of the viscosity tensor. However, since the major eigenvalue represents only one degree of freedom out of the six available in the tensor field, a measure involving all six components is a more objective metric for validation.

We used the Vish Visualization Shell (Benger et al. [2007]) to numerically sample (20) and (21) on a uniform grid for analyzing scalar fields (Figures 2 and 3) and a radial sampling distribution (Figure 4) for the full tensor field.

Such glyphs shown at each sampling point provide a direct visualization of the full six degrees of freedom of the tensor properties. Reynold glyphs are also able to depict negative definite tensors, whereas a quadric surface (ellipsoids representing $P\to 1/\sqrt{Q(v(P),v(P))}$ becomes hyperbolic for negative eigenvalues and problematic for visualization purposes. The Reynold glyph directly shows the ‘directional value’ $Q(v(P),v(P))$ of the tensor field Q in the direction v around a the sampling point P - the resulting surface is intersecting the sampling point P whenever $Q(v(P),v(P))=0$, which is the case for points where the tensor is degenerating and not positive definite. In such areas the glyph will visually appear like two intersecting surfaces corresponding to the isopotential surfaces of second order spherical harmonic functions. These both surface components represent positive and negative eigenvalues of the tensor field - if both positive and negative component ‘counter-balance’ themselves they therefore indicate vanishing trace, which is the sum of the eigenvalues. If only one surface component is visible, then the tensor is either positive or negative definite on that certain point.

We used a radial sampling for the direct visualization of the tensor field in order to minimize coordinate artifacts. As depicted in Figure 4(b) and Figure 5(b) for the correct TAV all ‘modes’ are equivalently represented, indicating vanishing trace of the tensor field while being radially aligned with the underlying coordinate system. In contrast, the incorrect TAV, Figure 4(a) and Figure 5(a), exhibits a dominantly negative trace.

A.1 Adaptive grids

In fluid dynamics we distinguish two main reference systems that suit unequally for various applications. The Eulerian frame is the fixed reference system of an external observer in which the fluid moves with velocity u whereas the Lagrangian approach describes the physics in the rest frame of the fluid. Between these two systems, the transformation of an advection term for a density d (that moves with a relative velocity u) is given via the material derivative $D_t\mathbf{d}={\partial }_t\mathbf{d}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{d}$.

Hence, when we work with comoving frames, the coordinate system respectively the computational grid is time-dependent. There is a number of purposes where strict Eulerian or Lagrangian grids are suboptimal and thus we need to consider the generalized the concept of the comoving frames.

where $U^i$ denotes the contravariant velocity components relative to the moving grid, $U^i={\mathbf{e}}^i\cdot (\mathbf{u}-\dot{\mathbf{x}})$. For the full analytic derivation we refer to Thompson et al. ([1985]) again.

A.2 Set of RHD equations in strong conservation form

We exhibit the system of equations of radiation hydrodynamics in a somewhat simplified formulation. The following system has been basis for a number of implicit RHD computations (see e.g. Dorfi ([1999])). All the astrophysical assumptions, implications and simplifications can be found in Mihalas and Mihalas ([1984]). In this paper we only want to emphasize the structural form of such a set of equations in a strong conservation form for comoving curvilinear coordinates. Note that for scalar equations the only effectively remaining geometric term inside the derivatives is the volume element $\sqrt{|\mathbf{g}|}$. The vectorial equations contain however also the time-dependent base vectors.

The equations presented in the upcoming sections as well as the coordinate transformation 1 were partially generated using the computer algebra system Mathematica (Wolfram Research [2007]). The source code can be downloaded at https://bitbucket.org/.