Qingshan Chen, Ming-Cheng Shiue, Roger Temam et al.
A new set of nonlocal boundary conditions are proposed for the higher modes of the 3D inviscid primitive equations. Numerical schemes using the splitting-up method are proposed for these modes. Numerical simulations of the full nonlinear primitive equations are performed on a nested set of domains, and the results are discussed.
Qingshan Chen, Zhen Qin, Roger Temam
A new method is proposed to improve the numeri- cal simulation of time dependent problems when the initial and boundary data are not compatible. Unlike earlier methods limited to space dimension one, this method can be used for any space dimension. When both methods are applicable (in space dimen- sion one), the improvements in precision are comparable, but the method proposed here is not restricted by dimension.
Qingshan Chen, Zhen Qin, Roger Temam
The incompatibilities between the initial and boundary data will cause singularities at the time-space corners, which in turn adversely affect the accuracy of the numerical schemes used to compute the solutions. We study the corner singularity issue for nonlinear evolution equations in 1D, and propose two remedy procedures that effectively recover much of the accuracy of the numerical scheme in use. Applications of the remedy procedures to the 1D viscous Burgers equation, and to the 1D nonlinear reaction-diffusion equation are presented. The remedy procedures are applicable to other nonlinear diffusion equations as well.
Qingshan Chen, Lili Ju, Roger Temam
An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived, by approximating the Hamiltonian formulation, based on the Poisson brackets and the vorticity-divergence variables, of the inviscid shallow water flows. The conservation of the energy and/or enstrophy stems from skew-symmetry of the Poisson brackets, which is retained in the discrete approximations. These schemes operate on unstructured orthogonal dual meshes, over bounded or unbounded domains, and they are also shown to possess the same optimal dispersive wave relations as those of the Z-grid scheme.
Arthur Bousquet, Mickaël D. Chekroun, Youngjoon Hong et al.
We aim to study a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions to the system of equations. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. The resulting scheme allows for a significant reduction of the errors near the topography when compared to more standard finite volume schemes. In the numerical simulations, we first present the associated good convergence results that are satisfied by the solutions simulated by our scheme when compared to particular analytic solutions. We then report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated. The numerical results show that such a forcing is responsible for recurrent large-scale patterns to emerge in the temperature and velocity fields.
Daozhi Han, Yifeng Hou, Roger Temam
In this article,we study theoretically and numerically the interaction of a vortex induced by a rotating cylinder with a perpendicular plane. We show the existence of weak solutions to the swirling vortex models by using the Hopf extension method, and by an elegant contradiction argument, respectively. We demonstrate numerically that the model could produce phenomena of swirling vortex including boundary layer pumping and two-celled vortex that are observed in potential line vortex interacting with a plane and in a tornado.