A method to solve an

advection equation of a physical quantity (for example, volume fraction of fluid in a grid cell) for an identification of the water surface is effective to reduce computational load of the interface tracking.

From this general solution, the exact pdf for the two-mode nonlinear

advection equation has been found by solving Liouville's equation [solution displayed in (ES23)].

Here, we use the unsteady linear

advection equation as a model equation and discretize this with a finite volume scheme and the implicit Euler method.

For instance, the FFD has significant numerical diffusion due to the linear interpolation used in the semi-Lagrangian solver for the

advection equation. For simplicity, the one dimensional form of the linear interpolation is as follows:

Even if the initial value of the level-set function [PHI] (x,0) is set to be the distance function, the level set function [PHI] may not remain as a distance function at t > 0 when the

advection equation, Equation (6), is solved for [PHI].

Therefore, instead we split off the

advection equation for the density (3.1), which can be handled separately.

Similarly, scheme (25) can be converted to another solution interpolation scheme for the homogeneous

advection equation:

The fractional volume is then computed and updated at each time step using the following

advection equationThe volume fraction is advected with the flow and satisfies the

advection equation as follows:

Let D = [0, 1]x[0, 1] .In the closed domain [0, T] x D consider the two-dimensional

advection equationIn this method, the energy equation is decomposed into an

advection equation and a diffusion equation as follows [8]

The Galerkin finite element methods are useful to solve the

advection equation [6].

When the 1D linear

advection equation is approximated by a numerical method, the amplification factor and relative phase error depend on only the cfl number.

In [22], a dam-break and oscillation experiments were performed to validate the 2D numerical model with the CIP method for the solution of

advection equation of Navier-Stokes equation and also for the free surface treatment.

The air/skin polymer interface and skin/core polymer interface are traced by pseudo-concentration method and governed by two

advection equations separately in the literatures [21, 22], At each time step, the

advection equations are coupled with the governing equations of the two polymer melts, respectively.