(8) In the density function method, the color function, whose value jumps from 0 to 1 across an interface, is defined, and the advection equation of the color function is solved to capture a gas-liquid interface.
Since it is caused by the nonlinearity of the advection equation, the numerical diffusion is controlled effectively by a higher-order calculation scheme of advection.
to come up with optimal coefficients for that discretization, again for the steady linear advection equation
and demonstrated that these work reasonably well for the steady Euler equations.
If the extrapolated velocity [u.sup.*] satisfies the incompressibility constraint, i.e., [nabla] * [u.sup.*] = 0, the advection equation
(3.4) and the coupled NS equations (3.5)-(3.6) are of second order accuracy in time due to the BDF2 time discretization scheme.
Modern modifications of FVM [5-8] provide well-established conservative methods for solving the governing advection equations
. Moreover, some of them were developed to treat high gradients and discontinuities of a solution [7, 8].
where [T.sub.0] is temperature at the previous time step, T* is an intermediate solution of the advection equation
, T is the temperature at the present time step, and [DELTA]t is the time step.
The originality in this work resides in the fact that expressions for the amplification factor and the relative phase error have been obtained for the WENO3 scheme discretizing the 1D linear advection equation. Very few papers have examined these properties for WENO schemes as it is cumber some to obtain the expression for the amplification factor of WENO schemes in general.
The modified equation of WENO3 when discretized by the 1D linear advection equation is given by
Takacs, "Atwo-step scheme for the advection equation
with minimized dissipation and dispersion errors," Monthly Weather Review, vol.
Austin, Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations
Austin, "Application of the Laplace decomposition method to nonlinear homogeneous and non-homogeneous advection equations
," Zeitschrift fur Naturforschung A, vol.
In the discretization of the energy and advection equations