Moving-Water Equilibria Preserving HLL-Type Schemes for the Shallow Water Equations

We construct new HLL-type moving-water equilibria preserving upwind schemes for the one-dimensional Saint-Venant system of shallow water equations with nonflat bottom topography.The designed first-and secondorder schemes are tested on a number of numerical examples,in which we verify the well-balanced property as well as the ability of the proposed schemes to accurately capture small perturbations of moving-water steady states.

Spectrophotometric determination of the formation constants of Calcium(Ⅱ) complexes with 1,2-ethylenediamine,1,3-propanediamine and 1,4-butanediamine in acetonitrile

In this work, with the purpose to explore the coordination chemistry of calcium complexes which could work as a partial model of manganese-calcium cluster, a spectrophotometric study to evaluate the stability of the complexes: Calcium(Ⅱ)-1,2-ethylendiamine, Calcium(Ⅱ)-1,3-propanediamine and Calcium(Ⅱ)-1,4-butanediamine in acetonitrile, were carried on. By processing the spectrophotometric data with the HypSpec program allows the determination of the formation constants. The logarithmic values of the formation constants obtained for Calcium(Ⅱ)-1,2-ethylendiamine, Calcium(Ⅱ)-1,3-propanediamine and Calcium(Ⅱ)-1,4-butanediamine were log β_(110) = 4.69, log β_(110)= 5.25 and log β_(110) = 4.072, respectively.

Wind-Driven,Double-Gyre,Ocean Circulation in a Reduced-Gravity,2.5-Layer,Lattice Boltzmann Model

A coupled lattice Boltzmann （LB） model with second-order accuracy is applied to the reduced-gravity, shallow water, 2.5-layer model for wind-driven double-gyre ocean circulation. By introducing the secondorder integral approximation for the collision operator, the model becomes fully explicit. The Coriolis force and other external forces are included in the model with second-order accuracy, which is consistent with the discretization accuracy of the LB equation. The feature of the multiple equilibria solutions is found in the numerical experiments under different Reynolds numbers based on this LB scheme. With the Reynolds number increasing from 3000 to 4000, the solution of this model is destabilized from the anti-symmetric double-gyre solution to the subtropic gyre solution and then to the subpolar gyre solution. The transitions between these equilibria states are also found in some parameter ranges. The time-dependent variability of the circulation based on this LB simulation is also discussed for varying viscosity regimes. The flow of this model exhibits oscillations with different timescales varying from subannual to interannual. The corresponding statistical oscillation modes are obtained by spectral analysis. By analyzing the spatiotemporal structures of these modes, it is found that the subannual oscillation with a 9-month period originates from the barotropic Rossby basin mode, and the interarmual oscillations with periods ranging from 1.5 years to 4.6 years originate from the recirculation gyre modes, which include the barotropic and the baroclinic recirculation gyre modes.

A Well-Balanced Partial Relaxation Scheme for the Two-Dimensional Saint-Venant System

We develop a new moving-water equilibria preserving partial relaxation(PR)scheme for the two-dimensional(2-D)Saint-Venant systemof shallowwater equations.The new scheme is a 2-D generalization of the one-dimensional(1-D)PR scheme recently proposed in[X.Liu,X.Chen,S.Jin,A.Kurganov,andH.Yu,SIAMJ.Sci.Comput.,42(2020),pp.A2206–A2229].Our scheme is based on the PR approximation,which is designed in two steps.First,the geometric source terms are incorporated into the discharge fluxes,which results in a hyperbolic system with global fluxes.Second,the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary equation.The obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finitevolume method combined with an efficient semi-implicit ODE solver.The new 2-D PR scheme inherits the main advantages of the 1-D PR scheme:(i)no special treatment of the geometric source terms is required,(ii)no nonlinear(cubic)equations should be solved to obtain the point values of the water depth out of the reconstructed equilibriumvariables.The performance of the proposed PR scheme is illustrated on a number of numerical examples,in which we demonstrate that the PR scheme not only capable of exactly preserving quasi 1-D moving-water steady states and accurately capturing their small perturbations,but can also handle genuinely 2-D steady states and their small perturbations in a non-oscillatory manner.