PROC

The primitive equations (PEs) of large-scale oceanic flow formulated in
mean vorticity is proposed. In the reformulation of the PEs, the prognostic equation
for the horizontal velocity is replaced by evolutionary equations for the mean vorticity
field and the vertical derivative of the horizontal velocity. The total velocity field
(both horizontal and vertical) is statically determined by differential equations at each
fixed horizontal point. Its equivalence to the original formulation is also presented.

DCDS-B

A second order numerical method for the primitive equations
(PEs) of large-scale oceanic flow formulated in mean vorticity is proposed and
analyzed, and the full convergence in $L^2$ is established. In the reformulation of
the PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of
the horizontal velocity. The total velocity field (both horizontal and vertical)
is statically determined by differential equations at each fixed horizontal point.
The standard centered difference approximation is applied to the prognostic
equations and the determination of numerical values for the total velocity field
is implemented by FFT-based solvers. Stability of such solvers are established
and the convergence analysis for the whole scheme is provided in detail.

DCDS-B

We present a second-order-in-time finite difference scheme for the Cahn-Hilliard-Hele-Shaw equations. This numerical method is uniquely solvable and unconditionally energy stable. At each time step, this scheme leads to a system of nonlinear equations that can be efficiently solved by a nonlinear multigrid solver. Owing to the energy stability, we derive an $\ell^2 (0, T; H_h^3)$ stability of the numerical scheme. To overcome the difficulty associated with the convection term $\nabla · (\phi \mathit{\boldsymbol{u}})$, we perform an $\ell^∞ (0, T; H_h^1)$ error estimate instead of the classical $\ell^∞ (0, T; \ell^2)$ one to obtain the optimal rate convergence analysis. In addition, various numerical simulations are carried out, which demonstrate the accuracy and efficiency of the proposed numerical scheme.

DCDS-B

A class of upwind flux splitting methods in the
Euler equations of compressible flow
is considered in this paper. Using the property that Euler flux $F(U)$
is a homogeneous function of degree one in $U$,
we reformulate the splitting fluxes with $F^{+}=A^{+} U$,
$F^{-}=A^{-} U$, and the corresponding matrices
are either symmetric or symmetrizable and keep only
non-negative and non-positive eigenvalues.
That leads to the conclusion that the first order schemes
are positive in the sense of Lax-Liu [18],
which implies that it is $L^2$-stable
in some suitable sense. Moreover, the second order scheme
is a stable perturbation of the first order scheme,
so that the positivity of the second order schemes
is also established, under
a CFL-like condition. In addition, these splitting methods preserve
the positivity of density and energy.

DCDS

We present unconditionally stable and convergent numerical sche- mes for gradient flows with energy of the form $ \int_\Omega( F(\nabla\phi(\x)) + \frac{\epsilon^2}{2}|\Delta\phi(\x)|^2 )$dx. The construction of the schemes involves an appropriate extension of Eyre's idea of convex-concave decomposition of the energy functional. As an application, we derive unconditionally stable and convergent schemes for epitaxial film growth models with slope selection (F(y)= 1/4(|y|^{2}-1)^{2}) and without slope selection (F(y)= -1/2ln(1+|y|^{2})). We conclude the paper with some preliminary computations that employ the proposed schemes.

DCDS-B

In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.