My research work in this direction is focused on the subject of
CFD (computational fluid dynamics), which is the design
and analysis of numerical methods to approximate incompressible
flow as described by the NavierStokes Equations (NSE).
There has been a tremendous amount of work on this subject in
the past. Some wellknown difficulties,
such as enforcement of the incompressibility constraint,
lack of a dynamic equation and a boundary condition for
the pressure, and implementation of the vorticity boundary condition,
have existed for both the primitive variable
formulation and the vorticity formulation.
In addition, one faces serious challenges in dealing with a flow
with a high Reynolds number, due to the sharp gradient of the flow,
lack of resolution and inefficiency
of the numerical scheme. Furthermore, the application
of many classical high order methods (such as spectral methods)
to a numerical example with a nonperiodic boundary condition may lead to
a high computational cost, and spurious effects are often observed.
Stability presents another big concern for these classical approaches.
In a sequence of my articles, new approaches were proposed
and implemented to overcome
the abovementioned difficulties. In particular, the proposed
numerical schemes avoid a Stokes solver, using
only a Poisson solver at each time step (stage).
As a result, these schemes are highly efficient and are capable of
producing highly resolved solutions for
complicated structural transitions of the flow at a reasonable
computational cost.
Moreover, spurious effects or numerical
artifacts around the boundary, due to the stability and accuracy
of the scheme, are not observed.
Similar ideas can be applied to deal with geophysical flows,
such as the Primitive Equations (PEs) and
the Planetary Geostrophic Equations (PGEs),
and the 3D QuasiGeostrophic equations (QGEs).
These models represent the fundamental governing equations for
atmospheric and oceanic flow under different physical scales.
A special property for the horizontal velocity field, which is a
nonlocal constraint, reflects the fact that the flow with large scale or
mesoscale is essentially twodimensional.
This makes possible the efficient solution of these equations
avoiding the numerical difficulties which typically arise in
threedimensional flows.
The PEs can be formulated in either surface pressure Poisson
equation or mean vorticity. Both 3D staggered MAC
(marker and cell) grids and regular numerical
grids are extensively discussed, with both second
and fourth order accuracy.
The PGEs with inviscid geostrophic balance are formulated in
an alternate form.
Moreover, collocation spectral methods can be applied to deal with
GFD models, with a careful treatment of hydrostatic and geostrophic
balance equations. The purpose for utilizing these high order methods is
to improve the accuracy within the limited resolution
due to the enormous scale of the threedimensional setting.
The calculation of thermocline evolution in large
scale oceanic motion was also undertaken.
Direct impacts of the numerical methods include more
detailed simulation of weather systems (both global and local),
and more efficient use of supercomputer resources.
If successful, a better understanding of the detailed process
of weather dynamics is expected due to the efficiency of the
reformulations and the high accuracy and stability of the
proposed methods.
As another application, these numerical schemes can be used to
study the structural stability and bifurcation of
incompressible fluid. Solutions of the incompressible fluid equations
in different physical
formulations, either NSE, Boussinesq equations, or other physical
models, can be viewed as oneparameter families
of divergencefree vector fields with time as the parameter.
The structural analyses of divergencefree velocity vector fields,
including topological equivalence classes, stability conditions,
and structural bifurcation criterion,
have attracted much attention. Such analyses can be carried out
both theoretically and numerically. Some theoretical results were
recently obtained in related references.
My work in this direction is focused
to a detailed numerical study of both structural
stability and structural bifurcation, i.e., change of topological
equivalence class, for 2D incompressible flow
under different physical setups.

1. Convergence of gauge method for incompressible flow.
C. Wang, J.G. Liu, Mathematics of Computation,
69, (2000), 13851407.
PDF file

2. Analysis of finite difference schemes for unsteady
NavierStokes equations in vorticity formulation.
C. Wang, J.G. Liu, Numerische Mathematik, 91, (2002), 543576.
PDF file

3. A fourth order scheme for incompressible
Boussinesq equations.
J.G. Liu, C. Wang, H. Johnston,
Journal of Scientific Computing, 18, (2003), 253285.
PDF file

4. Positivity property of second order fluxsplitting schemes
of compressible Euler equations.
C. Wang, J.G. Liu, Discrete and Continuous
Dynamical SystemsSeries B, 3, (2003), 201228.
PDF file

5. A fast finite difference method for solving NavierStokes
equations on irregular domains.
Z. Li, C. Wang, Communications in Mathematical Sciences,
1, (2003), 181197.
PDF file

6. Fourth order convergence of compact difference solver
for incompressible flow.
C. Wang, J.G. Liu, Communications in Applied Analysis,
7, (2003), 171191.
PDF file

7. Surface pressure Poisson equation formulation
of the primitive equations: Numerical schemes.
R. Samelson, R. Temam, C. Wang, S. Wang,
SIAM Journal of Numerical Analysis, 41, (2003), 11631194.
PDF file

8. The primitive equations formulated in mean vorticity.
C. Wang, Discrete and Continuous Dynamical Systems,
Proceeding of ``International Conference on
Dynamical Systems and Differential Equations'', 2003, 880887.
PDF file

9. High order finite difference methods for unsteady
incompressible flows in multiconnected domains.
J.G. Liu, C. Wang, Computers and Fluids, 33, (2004), 223255.
PDF file

10. Analysis of a fourth order finite difference
method for incompressible Boussinesq equations.
C. Wang, J.G. Liu, H. Johnston,
Numerische Mathematik, 97, (2004), 555594.
PDF file

11. Convergence analysis of the numerical method for the
primitive equations formulated in mean vorticity on
a Cartesian grid.
C. Wang, Discrete and Continuous Dynamical SystemsSeries B,
4, (2004), 11431172.
PDF file

12. Boundarylayer separation and adverse pressure gradient for
2D viscous incompressible flow.
M. Ghil, J.G. Liu, C. Wang, S. Wang,
Physica D, 197, (2004), 149173.
PDF file

13. Global weak solution of the planetary
geostrophic equations with inviscid geostrophic balance
J. Liu, R. Samelson, C. Wang,
Applicable Analysis, 85, (2006), 593606.
PDF file

14. A fourth order numerical method for the planetary
geostrophic equations with inviscid geostrophic balance.
R. Samelson, R. Temam, C. Wang, S. Wang,
Numerische Mathematik, 107, (2007), 669705.
PDF file

15. A fourth order numerical method for the
primitive equations formulated in mean vorticity.
with J.G. Liu, C. Wang,
Communications in Computational Physics, 4, (2008), 2655.
PDF file

16. A fourth order difference scheme for the
Maxwell equations on Yee grid.
A. Fathy, C. Wang, J. Wilson, S. Yang,
Journal of Hyperbolic Differential Equations,
5, (2008), 613642
PDF file

17. A general stability condition for multistage
vorticity boundary conditions in incompressible fluids.
C. Wang, Methods and Applications of Analysis,
15, (2008), 469476.
PDF file

18. Structural stability and bifurcation for 2D
divergencefree vector with symmetry.
C. Hsia, J.G. Liu, C. Wang,
Methods and Applications of Analysis,
15, (2008), 495512.
PDF file

19. Long time stability of a classical efficient
scheme for two dimensional NavierStokes equations.
S. Gottlieb, F. Tone, C. Wang, X. Wang, D. Wirosoetisno,
SIAM Journal on Numerical Analysis,
50, (2012), 126150.
PDF file

20. Stability and convergence analysis of
fully discrete Fourier collocation spectral method
for 3D viscous Burgers' equation.
S. Gottlieb, C. Wang,
Journal of Scientific Computing,
53, (2012), 102128.
PDF file

21. A local pressure boundary condition spectral
collocation scheme for the threedimensional
NavierStokes equations.
H. Johnston, C. Wang, J.G. Liu,
Journal of Scientific Computing,
60, (2014), 612626.
PDF file

22. A Fourier pseudospectral method
for the ``Good" Boussinesq equation with
secondorder temporal accuracy.
K. Cheng, W. Feng, S. Gottlieb, C. Wang,
Numerical Methods for Partial Differential Equations,
31, (2015), 202224.
PDF file

23. Simple finite element numerical simulation of incompressible flow over nonrectangular domains and the superconvergence analysis.
Y. Xue, C. Wang, J.G. Liu, Journal of Scientific Computing, 65 (2015), 11891216.
PDF file

24. Long time stability of high order multistep numerical schemes for twodimensional incompressible NavierStokes equations.
K. Cheng, C. Wang, SIAM Journal on Numerical Analysis, 54 (2016), 31233144.
PDF file

25. A second order operator splitting numerical scheme for the ``Good" Boussinesq equation.
C. Zhang, H. Wang, J. Huang, C. Wang, X. Yue, Applied Numerical Mathematics, (2017), submitted and in review.

26. On the operator splitting and integral equation preconditioned deferred correction methods for the ``Good" Boussinesq equation.
C. Zhang, J. Huang, C. Wang, X. Yue, Journal of Scientific Computing, (2017), submitted and in review.

27. The StrongStabilityPreserving (SSP) scheme applied to the
Integrating Factor (IF) form of Exponential Time Differencing (ETD) problems
A second order convex splitting scheme for the CahnHilliardHeleShaw equation.
S. Gottlieb, Z. Grant, C. Wang, (2017), in preparation.

28. A spectral collocation method for twodimensional
incompressible fluid flows in vorticity formulation.
H. Johnston, C. Wang, J.G. Liu,
(2017), in preparation.

Flow in a cooling system
Please send any comments or suggestions to:
cwang1@umassd.edu,
11/05/16