PPB Moment-Conserving Advection for Multifluid Hydrodynamics

A family of moment-conserving advection schemes in 1-D was introduced by van Leer in the mid 1970s. Of these, MUSCL advection is the best known. That advection scheme represents the advected function, which we will call a , by a linear function within each grid cell. The linear function is determined by the values of the first two moments of this density distribution within the cell, namely the mass and the center of mass of the cell. Both these moments are updated by the scheme in a process that is equivalent to projecting the detailed advected function after each time step onto the subspace of piecewise linear functions (using a weighting function that treats every point within each cell as equally important). In this sense, the MUSCL advection scheme is similar to a Galerkin finite element method (without the constraint that the resulting representation of the function should be continuous). Schemes of this type have much later been called discontinuous Galerkin methods, with Godunov's method and MUSCL the first two schemes introduced of this type. MUSCL is also essentially a spectral element method, using within each grid cell only the first 2 terms in a spectral expansion of the advected function in Legendre polynomials. Van Leer demonstrated that, for linear advection in 1-D, the resulting scheme was formally second-order accurate, but it caused error to accumulate only at the rate of a third-order accurate scheme. A next member in this family of 1-D advection schemes was introduced by van Leer that I later generalized to 2-D advection working with Rick White in the early 1980s. We will call that scheme PPB, for Piecewise-Parabolic Boltzmann method, since the original use I