Any physical variable or physical field defined in space needs to be represented or measured by one kind or a set of quantities in numerical
approximationｓ, for example the value or the gradient of a physical variable at a certain spatial point, the averaged value in respect to a
volume or a surface area. We define all of those measures as the moments.
Most conventional numerical methods for time-space variation problems can be classified into either finite difference method (FDM) or finite
volume method (FVM). Conceptually, a FDM approximates spatial differential via a consistent differencing by using the values at grid points,
while a FVM is usually cast in a formulation that predicts the integrated average value for each mesh cell by examining the net flux across all
the cell boundaries. The numerical flux on cell boundary is evaluated then by a consistent approximation based on the cell-integrated average
value. The point value in a FDM or the cell-integrated average in a FVM is actually the only quantity memorized and predicted in a numerical
model of such a kind. Thus, we call the point value the single 'moment' of the field variable used in FDM, and the cell-integrated average the
single 'moment' in FVM. In general, a high order spatial approximation in a conventional FDM or a FVM can be obtained by using extra mesh
stencils or by using a compact formulation, the latter usually results in an implicit expression.
Being an alternative to the conventional numerical methods, the CIP (Cubic-Interpolated Pseudoparticle or Constrained Interpolation Profile)
scheme makes use of the first order derivatives of the prognostic variable (the gradient) as the extra 'moments' and predicts them in the
numerical model as other dependent variables. Employing more than one moment as the dependent variables in its algorithm, the CIP method
can be distinguished from the conventional FDM and FVM and classified as the multi-moment method. Analytical and numerical studies show
that using multi moments provides more accurate numerical dispersion, more compact computational stencil, more freedoms and flexibilities in
constructing numerical models.
Multi-moment as a general concept can, in fact, lead to a new breakthrough for numerical method researches. Being a practice toward such
direction, we have developed a new numerical framework for general hydrodynamic simulations by using two integrated moments, namely
volume integrated average (VIA) and surface integrated average (SIA). The new framework, which is called VSIAM3 (Volume/Surface Integrated
Average based Multi Moment Method) show some new features that are fundamentally different from the traditional ones. For example, the
Riemann solver (either the exact one or the approximate one) is not needed in the computations for compressible flows, and a new projection
for incompressible fluid can be constructed by using both the VIA and the SIA. The VSIAM3 fluid model has been used for simulating various
hydrodynamic problems, most of the numerical outputs are very promising.
The underlying idea of the algorithm and numerical results will be shown in the presentation.