arXiv subjects

Qingshan Chen

Publications and source records attributed to Qingshan Chen.

15 recordsLinked to original sources

arXiv2026

A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain

The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existence and uniqueness of a generalized solution is proven, provided that the initial potential vorticity (PV) field is essentially bounded. If the initial PV field is differentiable, then the solution is shown to satisfy the system in the classical sense.

math.AP↗

arXiv2020

Conservative numerical schemes with optimal dispersive wave relations -- Part II. Numerical evaluations

A new energy and enstrophy conserving scheme is evaluated using a suite of test cases over the global spherical domain or bounded domains. The evaluation is organized around a set of pre-defined properties: accuracy of individual opeartors, accuracy of the whole scheme, conservation, control of the divergence variable, representation of the energy and enstrophy spectra, and simulation of nonlinear dynamics. The results confirm that the scheme is between the first and second order accurate, and conserves the total energy and potential enstrophy up to the time truncation errors. The scheme is capable of producing more physically realistic energy and enstrophy spectra, indicating that the new scheme can help prevent the unphysical energy cascade towards the finest resolvable scales. With an optimal representation of the dispersive wave relations, the scheme is able to keep the flow close to being non-divergent, maintain the geostrophically balanced structures with large-scale geophysical flows over long-term simulations.

math.NA↗

arXiv2020

A Non-Stationary VVLC MIMO Channel Model for Street Corner Scenarios

In recent years, the application potential of visible light communication (VLC) technology as an alternative and supplement to radio frequency (RF) technology has attracted people's attention. The study of the underlying VLC channel is the basis for designing the VLC communication system. In this paper, a new non-stationary geometric street corner model is proposed for vehicular VLC (VVLC) multiple-input multiple-output (MIMO) channel. The proposed model takes into account changes in vehicle speed and direction. The category of scatterers includes fixed scatterers and mobile scatterers (MS). Based on the proposed model, we derive the channel impulse response (CIR) and explore the statistical characteristics of the VVLC channel. The channel gain and root mean square (RMS) delay spread of the VVLC channel are studied. In addition, the influence of velocity change on the statistical characteristics of the model is also investigated. The proposed channel model can guide future vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) optical communication system design.

eess.SP↗

arXiv2019

Conservative numerical schemes with optimal dispersive wave relations -- Part I. Derivations and analyses

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.

math.NA↗

arXiv2018

On the well-posedness of the inviscid multi-layer quasi-geostrophic equations

The inviscid multi-layer quasi-geostrophic equations are considered over an arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions are imposed to accommodate the free fluctuations of the top and layer interfaces. Using the barotropic and baroclinic modes in the vertical direction, the elliptic system governing the streamfunctions and the potential vorticity is decomposed into a sequence of scalar elliptic boundary value problems, where the regularity theories from the two-dimensional case can be applied. With the initial potential vorticity being essentially bounded, the multi-layer quasi-equations are then shown to be globally well-posed, and the initial and boundary conditions are satisfied in the classical sense.

math.AP↗

arXiv2017

Conservative finite volume schemes for the quasi-geostrophic equation on coastal-conforming unstructured primal-dual meshes

In this paper we propose finite volume schemes for solving the inviscid and viscous quasi-geostrophic equations on coastal-conforming unstructured primal-dual meshes. Several approaches for enforcing the boundary conditions are also explored along with these schemes. The pure transport part in these schemes are shown to conserve the potential vorticity along fluid paths in an averaged sense, and conserve the potential enstrophy up to the time truncation errors. Numerical tests based on the centroidal Voronoi-Delaunay meshes are performed to confirm these properties, and to distinguish the dynamical behaviors of these schemes. Finally some potential applications of these schemes in different situations are discussed.

math.NA↗

arXiv2017

The barotropic quasi-geostrophic equation under a free surface

The inviscid barotropic quasi-geostrophic equation with a free surface is considered. The free surface mandates a non-standard boundary condition. The global existence existence and uniqueness of a weak solution is established, thanks to the uniform in time bounds on the potential vorticity. The solution is also shown to satisfy the initial and boundary conditions in the classical sense.

math.AP↗

arXiv2016

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower-order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured non-uniform meshes. This combined approach is first applied to the one-dimensional elliptic boundary value problem on non-uniform meshes, and a first-order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered MAC scheme for the two-dimensional incompressible Stokes problem on unstructured meshes. A first-order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements.

math.NA↗

arXiv2016

Stable and convergent approximation of two-dimensional vector fields on unstructured meshes

A new framework is proposed for analyzing staggered-grid finite difference finite volume methods on unstructured meshes. The new framework employs the concept of external approximation of function spaces, and gauge convergence of numerical schemes through the quantities of vorticity and divergence, instead of individual derivatives of the velocity components. The construction of a stable and convergent external approximation of a simple but relevant vector-valued function space is demonstrated, and the new framework is applied to establish the convergence of the MAC scheme for the incompressible Stokes problem on unstructured meshes.

math.NA↗

arXiv2011

A scale-invariant formulation of the anticipated potential vorticity method

The long-term success of climate models that operate on multi-resolution grids will depend on access to subgrid parameterizations that act appropriately across a wide range of spatial and temporal scales. As the first step in a series of efforts to obtain such scale-aware subgrid parametrizations, in this article we focus on the anticipated potential vorticity method (APVM) on quasi-uniform grids with varying resolutions. By a scale analysis technique and the phenomenological theories for two-dimensional turbulent flows, we derive a new formulation of the APVM, which depends on a single parameter that is, in principle, invariant with respect to the time step size, the grid resolution and the flow itself. Results of numerical experiments with this new formulation demonstrate that the optimal parameter of the APVM is invariant with respect to the time step size, insensitive to the flows, and is only weakly dependent on the grid resolution.

physics.ao-ph↗

arXiv2010

Treatment of Incompatible Initial and Boundary Data for Parabolic Equations in Higher Dimension

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.

math.NA↗

arXiv2010

Numerical Approximation of the Inviscid 3D Primitive Equations in a Limited Domain

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.

math.NA↗

arXiv2010

Numerical Resolution near t = 0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

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.

math.NA↗

arXiv2010

Goal-oriented A Posteriori Error Estimation for Finite Volume Methods

A general framework for goal-oriented a posteriori error estimation for finite volume methods is presented. The framework does not rely on recasting finite volume methods as special cases of finite element methods, but instead directly determines error estimators from the discretized finite volume equations. Thus, the framework can be ap- plied to arbitrary finite volume methods. It also provides the proper functional settings to address well-posedness issues for the primal and adjoint problems. Numerical results are presented to illustrate the validity and effectiveness of the a posteriori error estimates and their applicability to adaptive mesh refinement.

math.NA↗

arXiv2010

Partial and spectral-viscosity models for geophysical flows

Two models based on the hydrostatic primitive equa- tions are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions is estab- lished. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.

math.AP↗