ZalCG

The equations of compressible flow

The governing equations solved are the 3D unsteady Euler equations, governing inviscid compressible flow, written here in the flux-conservative form

where the summation convention on repeated indices has been applied, and is the density, the velocity vector, is the specific total energy, is the specific internal energy, and is the pressure. The system is closed with the ideal gas law equation of state

where is the ratio of specific heats. The exact form of the equation of state is of secondary importance, any arbitrary (i.e. analytic or tabulated) equation of state can be used to relate density and internal energy to pressure.

The numerical method

The above system is solved using an edge-based finite-element method for linear tetrahedra. Using this particular approach, the unknown solution quantities are the conserved variables and are located at the nodes of the mesh. The solution at any point within the computational domain is represented as

where is the tetrahedron containing the point , is the solution at vertex , and is a linear basis function associated with vertex and element . Applying a Galerkin, lumped-mass approximation to the Euler equations gives the following semi-discrete form of the governing equations for a vertex :

where represent all edges connected to , is the volume surrounding , is the numerical flux between and . The first term on the right-hand-side above is evaluated for all vertices within the domain. The last two terms are only evaluated for vertices and surrounding edges that lie on the domain boundary. denotes the boundary flux. The term is effectively the area-weighted normal to the dual face separating and . It is calculated as

where represents all elements attached to edge . This numerical scheme is guaranteed to be conservative since is antisymmetric. The boundary terms and are defined as

For the derivation of the above and other details, see Papers.

Numerical flux

The numerical flux between mesh points and is computed as

where denotes the edge difference vector

The above flux prescribes a Lax-Wendroff (or Taylor-Galerkin) scheme for the edge-based formulation. Monotonicity is ensured by flux-corrected transport (FCT).

Flux-corrected transport

Flux-corrected transport (FCT) is a way to circumvent the consequence of Godunov's theorem, which states that no linear scheme of order greater than 1 will yield monotonic solutions. Accordingly, FCT is a nonlinear scheme that combines a high-, and a low-order scheme using limiting.

The high-order scheme in the FCT algorithm in ZalCG is given by the Taylor-Galerkin flux above with the Euler equations discretized in time with the forward Euler method as

where is the solution at the previous time step and the notation of the numerical solution represented at mesh node is dropped for brevity. The low-order scheme is constructed from the high-order scheme in by adding mass diffusion

with the diffusion operator

where denotes a diffusion coefficient. The steps to arrive at the diffusion term for edges are analogous to those of the Laplacian operator, invoking the symmetry property of the diffusion operator for linear tetrahedra as

In the FCT procedure guarantees monotonic solutions. The above specifications lead to the following antidiffusive edge contributions (AEC)

which is then used to update the low order solution after performing the limiting procedure to arrive at the solution at the new time step

Computing the limit coefficients follows the same procedure, detailed in Lohner, Morgan, Peraire, Vahdati, Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier–Stokes equations, implemented over the edges, given below.

The limiting procedure

The procedure first aggregates to points i the signed contributions of AECs:

The limiting procedure requires the maximum and minimum nodal values of the low-order solution and the previous solution,

Another alternative is to only consider the low order solution, when computing the allowed solution bounds, which leads to the so-called 'clipping limiter' in place of the equation above. This is followed by computing the maximum and minimum nodal values of all edges connected to a node

then computing the maximum and minimum unknowns of the edges surrounding each node,

The limit coefficients will be computed (see below) based on and the maximum and minimum increments and decrements the nodal solution values are allowed to achieve,

Defining the ratios of positive and negative edge contributions for each node that ensure monotonicity as

the limit coefficient for each edge is taken as the most conservative ratio

The limited AEC is then scatter-added to nodes of the low-order solution:

The above procedure is general, works on the numerical solution (instead on fluxes or slopes), and does not require a Riemann solver. The same procedure can be used for each scalar in a system of equations. This works well for independent scalars, but for coupled system of equations additional techniques have been developed to reflect the coupled nature of the equations in the limiting procedure.